Edge states transport with impurity/defect in the quantum limit: applied to the relativistic quasiparticles systems

Abstract

Using the extended scattering matrix method, the authors explore the interference of edge states in the nanoribbon resonant structure with the relativistic quasiparticles in the quantum limit. By varying the chemical potential in the gated region, the structure of the junction, and the number of the scattering impurities/defects, the scattering coefficients of the edge states can be tuned efficiently. In particular, the Fabry–Pérot like interference in the present structure has been found and the phase accumulated highly depends on both the modes propagating vertically to the interface and the modes propagating along the interface. The feature is fundamentally different from the case of the conventional material where the longitudinal resonant states of the narrow constriction give rise to a clear conductance dips. These results are crucial and useful for engineering nanoelectronic devices based on the relativistic quasiparticles systems.

Graphical Abstract

Appendix

The overall scattering matrix $S\left( {L,R} \right)$ through the system can be given as:

$\begin{aligned}\left( {\begin{array}{*{20}c}{r_{{n,n^{\prime}}} } \\ {t_{{n,n^{\prime}}} } \\ \end{array} }\right) = S\left( {L,R} \right)\left( {\begin{array}{*{20}c} I \\{0} \\ \end{array} } \right){ = }\left( {\begin{array}{*{20}c}{S\left( {L,R} \right)_{{{11}}} } & {S\left( {L,R} \right)_{{{12}}}} \\ {S\left( {L,R} \right)_{{{21}}} } & {S\left( {L,R}\right)_{{{22}}} } \\ \end{array} } \right)\left({\begin{array}{*{20}c} I \\ {0} \\ \end{array} }\right)\end{aligned}$ with $I$ is the dependent 2 × 2 unit matrix. The scattering matrix $S\left( {L,R}\right)$ can be obtained by an iteration process from the leftmost interface to the leftmost interface. In this manner, the scattering matrix $S\left( {L,j}\right)$ can be defined for the subsystem up to the jth region as following

$$\begin{aligned} S\left( {L,j}\right)_{11} &= \left[ {1 - M\left( {j - 1,i} \right)_{11}^{ - 1}S(L,j - 1)_{12} M\left( {j - 1,j} \right)_{21} } \right]^{ - 1}\\ &\quad M\left( {j - 1,j} \right)_{11}^{ - 1} S(L,j - 1)_{11} // S\left({L,j} \right)_{21}\\ &= \left[ {1 - M\left( {j - 1,j} \right)_{11}^{ -1} S(L,j - 1)_{12} M\left( {j - 1,j} \right)_{21} } \right]^{ - 1}\end{aligned}$$

$$\begin{aligned}& \left[ M\left( {j - 1,j} \right)_{11}^{ -1} S(L,j - 1)_{{1{2}}} M\left( {j - 1,j} \right)_{{{22}}} { -}\right.\\ &\left.M\left( {j - 1,j} \right)_{11}^{ - 1} M\left( {j - 1,j}\right)_{{{12}}} \right]\end{aligned} $$

(A1)

$$ \begin{aligned}S\left({L,j} \right)_{{{2}1}} &= S(L,j - 1)_{{{2}2}} M\left( {j - 1,j}\right)_{21} \\ &\quad S(L,j)_{11} { + }S(L,j - 1)_{{{2}1}}\end{aligned}$$

$$ \begin{aligned}S\left({L,j} \right)_{{{22}}} &= S(L,j - 1)_{{{2}2}} M\left( {j - 1,j}\right)_{21} S(L,j)_{{1{2}}}\\&\quad { + }S(L,j - 1)_{{{22}}} M\left( {j -1,j} \right)_{{2{2}}} \end{aligned}$$

where $M(j -1,j)$ is the transmission matrix can be defined by the odd and even interfaces, respectively.

For odd interface, the wave functions in the adjacent regions are expressed as

$$ \left\{ {\begin{array}{*{20}c} {\Psi _{j} = \sum\limits_{{n = 1}}^{\infty } {\left( {p_{n}^{{\prime j}} e^{{ik_{n} y}} d_{{n,n^{\prime}}}^{j} + f_{n}^{{\prime j}} e^{{ - ik_{n} y}} d_{{n,n^{\prime}}}^{{\prime j}} } \right)\sqrt {2/w_{1} } } \sin \left[ {n\pi \left( {y/w_{1} + 1/2} \right)} \right],} \\ \begin{gathered} \Psi _{{j + 1}} = \sum\limits_{{n = 1}}^{\infty } {\left( {p_{n}^{{j + 1,B}} e^{{ik_{n}^{B} y}} d_{{n,n^{\prime}}}^{{j + 1,B}} + f_{n}^{{j + 1,B}} e^{{ - ik_{n}^{B} y}} d_{{n,n^{\prime}}}^{{\prime j + 1,B}} } \right)\sqrt {2/w_{2} } \sin \left[ {n\pi \left( {\left( {y - w_{1} /2 + w_{2} } \right)/w_{2} } \right)} \right]} \hfill \\ + \sum\limits_{{n = 1}}^{\infty } {\left( {p_{n}^{{j + 1,C}} e^{{ik_{n}^{C} y}} d_{{n,n^{\prime}}}^{{j + 1,C}} + f_{n}^{{j + 1,C}} e^{{ - ik_{n}^{C} y}} d_{{n,n^{\prime}}}^{{\prime j + 1,C}} } \right)\sqrt {2/w_{2} } \sin \left[ {n\pi \left( {\left( {y + w_{1} /2 - w_{2} } \right)/w_{2} } \right)} \right]} , \hfill \\ \end{gathered} \\ \end{array} } \right. $$

(A2)

where $k_{n}$and $k_{n}^{B/C}$are the nth mode wave vectors of the quasiparticles in the $y$direction in the $N$ and $P$regions, respectively. Besides, $k_{n}^{B}$and $k_{n}^{C}$indicate the wave vectors in the up and down red $P$regions, respectively.

To solve the scattering processes, we require the wave functions and its derivative to be continuous at the interface. Moreover, multiplying both sides by $\sqrt {2/w_{1} } \sin \left[ {n\pi \left( {y/w_{1} + 1/2} \right)} \right]$ or $\sqrt {2/w_{2} } \sin \left[ {n\pi \left( {\left( {y \pm (w_{1} /2 - w_{2} )} \right)/w_{2} } \right)} \right]$ and integrating over $y$, we obtain.

$$ \sum\limits_{n = 1}^{\infty } {\rho_{nm} \left( {p_{n}^{\prime j} e^{{ik_{n} l_{1} }} d_{{n,n^{\prime}}}^{j} + f_{n}^{\prime j} e^{{ - ik_{n} l_{1} }} d_{{n,n^{\prime}}}^{\prime j} } \right)} = p_{m}^{j + 1,B} d_{{n,n^{\prime}}}^{j + 1,B} + f_{m}^{j + 1,B} d_{{n,n^{\prime}}}^{\prime j + 1,B} , $$

$$ \sum\limits_{n = 1}^{\infty } {\rho_{nm} \left( {p_{n}^{\prime j} e^{{ik_{n} l_{1} }} d_{{n,n^{\prime}}}^{j} + f_{n}^{\prime j} e^{{ - ik_{n} l_{1} }} d_{{n,n^{\prime}}}^{\prime j} } \right)} = p_{m}^{j + 1,C} d_{{n,n^{\prime}}}^{j + 1,C} + f_{m}^{j + 1,C} d_{{n,n^{\prime}}}^{\prime j + 1,C} , $$

(A3)

$$ k_{m} \left( {p_{m}^{\prime j} e^{{ik_{n} l_{1} }} d_{{n,n^{\prime}}}^{j} - f_{m}^{\prime j} e^{{ - ik_{n} l_{1} }} d_{{n,n^{\prime}}}^{\prime j} } \right) = \sum\limits_{n = 1}^{\infty } {\rho_{mn} k_{n}^{B} \left( {p_{n}^{j + 1,B} d_{{n,n^{\prime}}}^{j + 1,B} - f_{n}^{j + 1,B} d_{{n,n^{\prime}}}^{\prime j + 1,B} } \right)} + \sum\limits_{n = 1}^{\infty } {\rho_{mn} k_{n}^{C} \left( {p_{n}^{j + 1,C} d_{{n,n^{\prime}}}^{j + 1,C} - f_{n}^{j + 1,C} d_{{n,n^{\prime}}}^{\prime j + 1,C} } \right)} , $$

where $\rho _{{mn}} = {2 / {\sqrt {w_{1} w_{2} } }}\int\limits_{{w_{1} /2 - w_{2} }}^{{w_{1} /2}} \sin \left[ {n\pi \left( {y/w_{1} + 1/2} \right)} \right]\sin \left[ {n\pi \left( {\left( {y - w_{1} /2 + w_{2} } \right)/w_{2} } \right)} \right]dy$ and $l_{1}$denote the length of region without the gate voltage. The above formula can be written in matrix form as

$$ \rho^{T} \left({p^{\prime j} d^{j} e^{{ikl_{1} }} + f^{\prime j} d^{\prime j} e^{{- ikl_{1} }} } \right) = p^{j + 1,B} d^{j + 1,B} + f^{j + 1,B}d^{\prime j + 1,B} $$

$$ \rho^{T} \left( {p^{\prime j} d^{j} e^{{ikl_{1} }} + f^{\prime j} d^{\prime j} e^{{ - ikl_{1} }} } \right) = p^{j + 1,C} d^{j + 1,C} + f^{j + 1,C} d^{\prime j + 1,C} $$

(A4)

$$ k\left( {p^{\prime j} d^{j} e^{{ikl_{1} }} - f^{\prime j} d^{\prime j} e^{{ - ikl_{1} }} } \right) = \rho k^{B} \left( {d^{j + 1,B} p^{j + 1,B} - d^{\prime j + 1,B} f^{j + 1,B} } \right) + \rho k^{C} \left( {d^{j + 1,C} p^{j + 1,B} - d^{\prime j + 1,C} f^{j + 1,B} } \right) $$

By introducing the auxiliary coefficient $p^{j + 1} = p^{j + 1,B} + p^{j + 1,C}$ and $f^{j + 1} = f^{j + 1,B} + f^{j + 1,C}$, we can get the transmission matrix with the help $k^{B} = k^{C}$, $d^{j + 1,B} = d^{j + 1,C}$, and $d^{\prime j + 1,B} = d^{\prime j + 1,C}$ as

$$ \left( {\begin{array}{*{20}c} {p^{\prime j}} \\ {f^{\prime j} } \\ \end{array} } \right) = M(j,j + 1)\left({\begin{array}{*{20}c} {p^{j + 1} } \\ {f^{j + 1} } \\ \end{array} }\right) $$

(A5)

where $M(j,j + 1) = \left( {\begin{array}{*{20}l}{e^{{ikl_{1} }} } & 0 \\ 0 & {e^{{ - ikl_{1} }} } \\ \end{array} }\right)^{ - 1} \left( {\begin{array}{*{20}l} {2\rho^{T} d^{j} } &{2\rho^{T} d^{\prime j} } \\ {kd^{j} } & { - kd^{\prime j} } \\\end{array} } \right)^{ - 1}$

$\left( {\begin{array}{*{20}l} {d^{j +1,B} } & {d^{\prime j + 1,B} } \\ {\rho d^{j + 1,B} k^{B} } & { -\rho d^{j + 1,B} k^{B} } \\ \end{array} }\right)$.

For even interface, the expansion coefficients $p^{j}$ and $f^{j}$ in the region $j$ and the expansion coefficients $p^{\prime j + 1}$ and $f^{\prime j + 1}$ in the region $j + 1$ are related by a transmission matrix $M(j,j + 1)$ and it is can be given in a similar manner as

$$ \left( {\begin{array}{*{20}c} {p^{j} } \\ {f^{j} } \\ \end{array} } \right) = M(j,j + 1)\left( {\begin{array}{*{20}c} {p^{\prime j + 1} } \\ {f^{\prime j + 1} } \\ \end{array} } \right) $$

(A6)

where $M(j,j + 1) = \left( {\begin{array}{*{20}c}{e^{{ik^{B} l_{2} }} } & 0 \\ 0 & {e^{{ - ik^{B} l_{2} }} } \\\end{array} } \right)^{ - 1} \left( {\begin{array}{*{20}c} {d^{j,B}} & {d^{\prime j,B} } \\ {\rho d^{j,B} k^{B} } & { - \rho d^{j,B}k^{B} } \\ \end{array} } \right)^{ - 1}$

$\left({\begin{array}{*{20}c} {2\rho^{T} d^{j + 1} } & {2\rho^{T} d^{\prime j + 1} } \\ {kd^{j + 1} } & { - kd^{\prime j + 1} } \\ \end{array} }\right)$ and $l_{2}$ denote the length of region with the gate voltage.