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One-way excess channel system





In this subsection, we consider the conductance of zigzag nanoribbons in the clean limit, which is simply given by the number of the conducting channel. As can be seen in figure 6(a), there is always one excess left-going channel in the right valley (K +) within the energy window of | E | 6 1. Analogously, there is one excess right-going channel in the left valley (K −) within the same energy window. Although the number of right-going and left-going channels are balanced as a whole system, if we focus on one of two valleys, there is always one excess channel in one direction, i.e. a chiral mode.

Now let us consider the injection of electrons from left to right 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, (21)

where n = 0,1,2,.... The quantization rule in the K −-valley is

g K + = n +1. (22)

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. (23)

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 +, and that of the left valley (K −) as −, the scattering matrix can be rewritten as

Here we should note that the dimension of each column vector is not identical. 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 = 1 at E = E 0 in figure 6(a). Figure 6(b) shows the schematic figure of scattering geometry for K + and K − points. Thus the dimension of the column vectors is given as follows:

dim n c +1,

(25)

, dim n c,

and

n c +1, dim n c,

(26)

n c, dim .

Subsequently, the reflection matrices have the following matrix structures:

The reflection matrices become non-square when the inter-valley 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 inter-valley scattering is suppressed, a system with an excess channel is realized in the K −-valley. Thus, for single valley transport, the r −− and are n c ×(n c +1) and (n c +1)× n c matrices, respectively, and t −− and t 0are (n c +1)×(† n c +1) and n c × n † c matrices, respectively. Noting the dimensions of r −− and, we find that r −− r −− and r 0−− r 0−−† have a single zero eigenvalue.† Combining this property with the flux conservation relation (S S = S S = 1), we arrive at the conclusion that t −− t −−† has an eigenvalue equal to unity, which indicates the presence of a† PCC only in the right-moving channels.

When the electrons are injected from the left lead of the sample and the inter-valley scattering is suppressed, a system with an excess channel is realized in the K −-valley. Thus, for single valley transport, the r −− and are n c ×(n c +1) and (n c +1)× n c matrices, respectively, and t −− and t 0are (n c +1)×(† n c +1) and n c × n † c matrices, respectively. Noting the dimensions of r −− and, we find that r −− r −− and r 0−− r 0−−† have a single zero eigenvalue.† Combining this property with the flux conservation relation (S S = S S = 1), we arrive at the conclusion that t −− t −−† has an eigenvalue equal to unity, which indicates the presence of a

We see that g K +1. Since the overall TRS of the system guarantees the following relation:

g 0 K + = g K −,

(31)

g ,

the conductance g = g K + + g K (right-moving) and g 0(left-moving) are equivalent.

If the probability distribution of { Ti } is obtained as a function L, we can describe the statistical properties of g as well as g 0. The evolution of the distribution function with increasing L is described by the DMPK (Dorokhov–Mello–Pereyra–Kumar) equation for transmission eigenvalues [65].

In the following, the presence of a PCC in disordered graphene nanoribbons will be demonstrated with the help of numerical calculation. Recently Hirose et al pointed out that the Chalker–Coddington model, which possesses non-square reflection matrices with unitary symmetry, gives rise to a PCC [64]. However, systems with an excess channel in one direction were believed difficult to realize. Therefore disordered graphene zigzag nanoribbons with LRI might constitute the first realistic example. It is possible to extend the discussion to a generic multiple-excess channel model, where the m -PCCs (m = 2,3,...) appear [65]. Such systems can be realized by stacking zigzag nanographene ribbons [78]. The electronic transport due to PCC resembles the electronic transport due to a chiral mode in the quantum Hall system. However, it should be noted that PCC due to edge states in zigzag ribbons occurs even without the magnetic field [79, 80].

Date: 2015-05-09; view: 553; Нарушение авторских прав; Помощь в написании работы --> СЮДА...



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