Transmission of waves across atomic step discontinuities in discrete nanoribbon structures

Abstract

Scalar wave propagation across a semi-infinite step or step-like discontinuity on any one boundary of the square lattice waveguides is considered within nearest-neighbour interaction approximation. An application of the Wiener–Hopf method does yield an exact solution of the discrete scattering problem, using which, as the main result of the paper, the transmission coefficients for energy flux are obtained. It is assumed that a wave mode is incident from either side of the step and the question addressed is what fraction of incident energy is transmitted across the atomic step discontinuity. A total of ten configurations are presented that arise due to various placements of discrete Dirichlet and Neumann boundary conditions on the waveguide. Numerical illustrations of a measure of ‘conductance’ are provided.

Appendices

A Auxiliary expressions

1.1 A.1

It is easily conceivable that there are four combinatorial variants of the square lattice waveguides (also the same are studied in detail by [38]) which are denoted by self-explanatory notation where the superscript on ${{\mathfrak {S}}}$ represents the upper boundary and the subscript represents the lower boundary. Indeed, in terms of the Chebyshev polynomials, using the definition (2.7), the dispersion relations for square lattice waveguides of width ${\mathtt {N}}$ [36, 38] are:

(A.1a)

(A.1b)

(A.1c)

Also (A.1c), using the Chebyshev polynomials ${\mathtt {V}}_n$ of the third kind, can be expressed as ${\mathtt {V}}_{{\mathtt {N}}}({\vartheta })=0$. Moreover, the expressions of the numerator and denominator of the kernel can be further expanded as stated in Table 1; in particular, (2.10) transforms to (2.13) to (2.17) to (2.18) to (2.27) to (2.30) to (2.35) to (2.36) to (2.45) to and (2.48) to .

1.2 A.2

By application of the discrete Fourier transform (1.5), the discrete Helmholtz equation (1.4), for all ${\mathtt {y}}\in {\mathbb {Z}}$ with ${\mathtt {y}}$ away from the boundary of the given lattice waveguide, is expressed as

(A.2a)

(A.2b)

(A.2c)

(A.2d)

The complex functions , , and ${\lambda }$ are defined on ${\mathbb {C}}\setminus {{\mathcal {B}}}$ where ${{\mathcal {B}}}$ denotes the union of branch cuts for ${{\lambda }}$, borne out of the chosen branch for and such that $|{{\lambda }}({{z}})|\le 1, {{z}}\in {\mathbb {C}}\setminus {{\mathcal {B}}},$ as ${\upomega }_2$ in (A.2c) is positive. Note that . The general solution of (A.2a) is given by the expression

$$\begin{aligned} \begin{aligned} {\mathtt {u}}_{{\mathtt {y}}}^F={ c }_1 {\lambda }^{{\mathtt {y}}}+{ c }_2 {\lambda }^{-{\mathtt {y}}}, \end{aligned} \end{aligned}$$

(A.3)

where ${ c }_{1, 2}$ are arbitrary analytic functions of z in ${{{\mathcal {A}}}}$ (to be specified in the Wiener–Hopf formulation for each case).

B Reflectance and transmittance using the numerical solution

The reflectance ${{{\mathscr {R}}}}$ (resp. transmittance ${{\mathscr {T}}}$) is the ratio of the energy flux in the outgoing wave ahead (resp. behind) of the cracks to the energy flux carried by the incident wave [43]. The energy flux is calculated across two vertical segments ${\mathscr {S}}_{{A},{B}}$ between $\mathtt {x}=\pm X$ and $\mathtt {x}=\pm X\pm 1$ (X is taken much larger than ${\mathtt {N}}$ so that ${\mathscr {S}}_{A,B}$ are far away from the location of step). The segment ${\mathscr {S}}_{A}$ intersects ${\mathtt {N}}$ horizontal bonds as it is located on the portion ahead of the step, while the segment ${\mathscr {S}}_{B}$ intersects ${\mathtt {N}}-{\mathtt {N}}_{step}$ horizontal bonds as it is located on the portion behind the step. Assume that the wave mode is incident from the portion ahead of the step (i.e., ${s}={A}$). Then, following [43], the energy flux carried by the incident wave across ${\mathscr {S}}_{A}$ is

$$\begin{aligned} \begin{aligned} \mathtt {W}^i({{z}}_{{\text {in}}})&=\mathfrak {R}\sum \limits _{n=1}^{{{\mathtt {N}}}} ((\mathtt {u}_{X+1,n}^i-\mathtt {u}_{X,n}^i )\overline{(-i \omega \mathtt {u}_{X,n}^i)}), \end{aligned} \end{aligned}$$

(B.1)

and the energy flux in the outgoing wave reflected back across ${\mathscr {S}}_{A}$ can be written as

$$\begin{aligned} \begin{aligned} \mathtt {W}^r({{z}}_{{\text {in}}})&=\mathfrak {R}\sum \limits _{n=1}^{{{\mathtt {N}}}} ((\mathtt {u}_{X,n}-\mathtt {u}_{X+1,n})\overline{(-i\omega \mathtt {u}_{X+1,n})}). \end{aligned} \end{aligned}$$

(B.2)

Similarly, the energy flux in the outgoing wave behind the step, that is, the energy flux carried by the transmitted waves across ${\mathscr {S}}_{B}$ is determined as

$$\begin{aligned} \begin{aligned} \mathtt {W}^t({{z}}_{{\text {in}}})&=\mathfrak {R}\sum \limits _{n={{{\mathtt {N}}}_{step}} +1}^{\mathtt {N}}((\mathtt {u}_{-X,n}^t-\mathtt {u}_{-X-1,n}^t)\overline{(-i\omega \mathtt {u}_{-X-1,n}^t)}. \end{aligned} \end{aligned}$$

(B.3)

Using (B.1), (B.2), and (B.3), the reflectance and transmittance can be written as

$$\begin{aligned} {{{\mathscr {R}}}}({{z}}_{{\text {in}}})=\frac{\mathtt {W}^r({{z}}_{{\text {in}}})}{\mathtt {W}^i({{z}}_{{\text {in}}})}, {{\mathscr {T}}}({{z}}_{{\text {in}}})=\frac{\mathtt {W}^t({{z}}_{{\text {in}}})}{\mathtt {W}^i({{z}}_{{\text {in}}})}, \end{aligned}$$

(B.4)

respectively. Analogous expressions holds for the incident from the portion behind the step (i.e., ${s}={B}$).