3D scattering by 2D periodic zero-range potentials: total reflection/transmission and threshold effects

Abstract

Phenomena of total reflection and total transmission are studied in the case of quantum particle 3D scattering on 2D periodic zero-range potentials. The general case is considered where the target is a “molecular monolayer” obtained by 2D periodic mapping of N different zero-range potential scattering centers. This is an infinite multicenter and multichannel scattering problem. It turns out that many of the global properties of reflection and transmission coefficients are determined by phenomena which occur at thresholds for opening of new scattering channels. We have performed detailed numerical studies for the cases when $N=1$ (atomic monolayers) and $N=2$ (planar and non-planar diatomic monolayers, that is atomic bilayers) on square and honeycomb lattices. In these cases, many interesting predictions can be derived analytically. We have also addressed the problem of the quasi-2D localized states embedded in continuum, that is, the band structures above the vacuum level.

Graphic abstract

Appendices

Appendix A: Lattice sums

For numerical calculations of lattice sums we used formulas (32) and (33) from [13] obtained by Ewald’s method (taking into account that our definition of lattice sums has an extra factor $\exp (i\mathbf{q}\cdot {\mathbf {\rho }})$):

$$\begin{aligned}&S_{k,\mathbf{q}}(\mathbf {\rho }, z)=\sum _{\mathbf{n}} \frac{\exp [-i\mathbf{q}\cdot \mathbf{a_n}]}{2(|\mathbf {\rho }+\mathbf{a_n}|^2+|z|^2)^{1/2}} \nonumber \\&\quad \times \left\{ \exp [ ik(|\mathbf {\rho }+\mathbf{a_n}|^2+|z|^2)^{1/2}] \right. \nonumber \\&\quad \times \mathrm{erfc}\left[ \frac{ik}{2\eta } +(|\mathbf {\rho }+\mathbf{a_n}|^2+|z|^2)^{1/2}\eta \right] \nonumber \\&\quad +\exp [-ik(|\mathbf {\rho }+\mathbf{a_n}|^2+|z|^2)^{1/2}] \nonumber \\&\quad \times \left. \mathrm{erfc}\left[ -\frac{ik}{2\eta } +(|\mathbf {\rho }+\mathbf{a_n}|^2+|z|^2)^{1/2}\eta \right] \right\} \nonumber \\&\quad +\frac{\pi i}{A}\sum _{\mathbf{m}} \frac{\exp (i\mathbf{q_m}\cdot \mathbf {\rho })}{k_{z\mathbf{m}}} \nonumber \\&\quad \times \left[ \exp (ik_{z\mathbf{m}}|z|)\mathrm{erfc}\left( -\frac{ik_{z\mathbf{m}}}{2\eta }-|z|\eta \right) \right. \nonumber \\&\quad \left. +\exp (-ik_{z\mathbf{m}}|z|)\mathrm{erfc}\left( -\frac{ik_{z\mathbf{m}}}{2\eta }+|z|\eta \right) \right] , \end{aligned}$$

(A.1)

$$\begin{aligned}&S'_{k,\mathbf{q}}(\mathbf{0}, 0)=\sum _{\mathbf{n} \ne \mathbf{0}}\frac{\exp (-i\mathbf{q}\cdot \mathbf{a_n})}{2|\mathbf{a_n}|} \nonumber \\&\quad \times \left[ \exp ( ik|\mathbf{a_n}|)\mathrm{erfc}\left( \frac{ik}{2\eta } +|\mathbf{a_n}|\eta \right) +\exp (- ik|\mathbf{a_n}|)\right. \nonumber \\&\quad \times \left. \mathrm{erfc}\left( -\frac{ik}{2\eta } +|\mathbf{a_n}|\eta \right) \right] +\frac{2\pi i}{A}\sum _{\mathbf{m}} \frac{\mathrm{erfc}\left( -\frac{ik_{z\mathbf{m}}}{2\eta }\right) }{k_{z\mathbf{m}}}\nonumber \\&\quad -ik \mathrm{erf}\left( \frac{ik}{2\eta }\right) -\frac{2\eta }{\sqrt{\pi }}\exp \left( \frac{k^2}{4\eta ^2}\right) -ik. \end{aligned}$$

(A.2)

The presence of complementary error functions makes the above sums rapidly convergent. The optimal choice of Ewald’s separation parameter, $\eta _{opt}=\sqrt{\pi /A}$ guarantees that approximately equal number of terms in sums over $\mathbf{n}$ and $\mathbf{m}$ are necessary to achieve the convergence. However, as discussed in [18], at high energies the loss of numerical accuracy occurs due to subtractions of large terms of the order of $\exp (k^2/(4\eta ^2))$. To limit this exponent to maximum value of $\exp (H^2)$ one has to choose

$$\begin{aligned} \eta _\mathrm{opt}=\mathrm{max}\left( \sqrt{\frac{\pi }{A}},\frac{k}{2H}\right) . \end{aligned}$$

(A.3)

In all our calculations, we obtained stable results with $H=3$.

Another useful form of $S_{k,\mathbf{q}}'(\mathbf{0}, 0)$ was derived in [11] (equation 1.6.26, p.214),

$$\begin{aligned} S_{k,\mathbf{q}}'(\mathbf{0}, 0)=\lim _{\omega \rightarrow \infty }\left( \frac{2\pi i }{A}\sum _{\mathbf{m},|\mathbf{q_{\mathbf{m}}}|\le \omega } \frac{1}{k_{z\mathbf{m}}}-\omega \right) -ik, \end{aligned}$$

(A.4)

where $k_{z\mathbf{m}}$ is defined in (14). For the scattering problems $q^2< k^2<+\infty$, we have

$$\begin{aligned}&\Re S_{k,\mathbf{q}}'(\mathbf{0}, 0)\nonumber \\&\quad =\lim _{\omega \rightarrow \infty }\left( \frac{2\pi }{A}\sum _{\mathbf{m}, k<|\mathbf{q_{\mathbf{m}}}|\le \omega }\frac{1}{\sqrt{|\mathbf{q_m}|^2-k^2}}-\omega \right) , \end{aligned}$$

(A.5)

$$\begin{aligned}&\Im S_{k,\mathbf{q}}'(\mathbf{0}, 0)= \frac{2\pi }{A}\sum _{\mathbf{m},|\mathbf{q_{\mathbf{m}}}|\le k} \frac{1}{\sqrt{k^2-| \mathbf{q_m}|^2}}-k. \end{aligned}$$

(A.6)

For band structures, in the interval $0<k^2<q^2$, we have

$$\begin{aligned}&\Re S_{k,\mathbf{q}}'(\mathbf{0}, 0)\nonumber \\&\quad =\lim _{\omega \rightarrow \infty }\left( \frac{2\pi }{A}\sum _{\mathbf{m}, |\mathbf{q_{\mathbf{m}}}|\le \omega }\frac{1}{\sqrt{|\mathbf{q_m}|^2-k^2}}-\omega \right) , \end{aligned}$$

(A.7)

$$\begin{aligned}&\Im S_{k,\mathbf{q}}'(\mathbf{0}, 0)=-k, \end{aligned}$$

(A.8)

while in the interval $-\infty<k^2<0$ ($k=i|k|$) we have

$$\begin{aligned}&\Re S_{k,\mathbf{q}}'(\mathbf{0}, 0)\nonumber \\&\quad =\lim _{\omega \rightarrow \infty }\left( \frac{2\pi }{A}\sum _{\mathbf{m}, |\mathbf{q_{\mathbf{m}}}|\le \omega }\frac{1}{\sqrt{|\mathbf{q_m}|^2-k^2}}-\omega \right) +|k|, \end{aligned}$$

(A.9)

$$\begin{aligned}&\Im S_{k,\mathbf{q}}'(\mathbf{0}, 0)=0. \end{aligned}$$

(A.10)

Appendix B: Zeroth threshold

For the zeroth threshold ($t=0$) we have $k_0=0, k_{z 0}=k_z, N_0=1, \mathbf{m}_1^0=(0,0),\tilde{\mathbf{q}}_{\mathbf{m}^0_1}=0, \tilde{S}^0_{jj'}=1$. Consequently, $\det \tilde{\mathbf{S}}^0=1,0$ for $N=1,N>1$ and $\det \tilde{\mathbf{S}}^{jj'0}=1,1,0$ for $N=1,2,N>2$. Direct calculations using (45) for $N=1$ and $N=2$ show that ${\tilde{A}}_{\mathbf{0}}^{\pm }= -1$ (see also Sects. 6 and 7). In the case $N>2$, we find

$$\begin{aligned}&F_N(\tilde{\mathbf{M}}^0, \tilde{\mathbf{S}}^0)= \left( \frac{Ak_{z}}{2\pi i}\right) ^{N-1}\sum _{\mu =1}^N\det [\tilde{\mathbf{M}}^0_{N-1}|\tilde{\mathbf{S}}^0_1]_\mu \nonumber \\&\qquad +\left( \frac{Ak_{z}}{2\pi i}\right) ^N\det \tilde{\mathbf{M}}^0, \end{aligned}$$

(B.11)

$$\begin{aligned}&F_{N-1}(\tilde{\mathbf{M}}^{jj'0}, \tilde{\mathbf{S}}^{jj'0})\nonumber \\&\quad = \left( \frac{Ak_{z}}{2\pi i}\right) ^{N-2}\sum _{\mu =1}^{N-1}\det [\tilde{\mathbf{M}}^{jj'0}_{N-2}|\tilde{\mathbf{S}}^{jj'0}_1]_\mu \nonumber \\&\qquad +\left( \frac{Ak_{z}}{2\pi i}\right) ^{N-1}\det \tilde{\mathbf{M}}^{jj'0}. \end{aligned}$$

(B.12)

It can be shown that

$$\begin{aligned}&\sum _{jj'}(-1)^{j+j'}\sum _{\mu =1}^{N-1}\det [\tilde{\mathbf{M}}^{jj'0}_{N-2}|\tilde{\mathbf{S}}^{jj'0}_1]_\mu =0, \end{aligned}$$

(B.13)

$$\begin{aligned}&\sum _{jj'}(-1)^{j+j'}\det \mathbf{M}^{j'j0}=\sum _{j=1}^N\det [\tilde{\mathbf{M}}^0_{N-1}|\tilde{\mathbf{S}}^0_1]_j, \end{aligned}$$

(B.14)

and consequently

$$\begin{aligned}&\sum _{jj'}(-1)^{j+j'}F_{N-1}(\tilde{\mathbf{M}}^{j'j0}, \tilde{\mathbf{S}}^{j'j0})\nonumber \\&\quad =\left( \frac{Ak_{z}}{2\pi i}\right) ^{N-1}\sum _{\mu =1}^N\det [\tilde{\mathbf{M}}^0_{N-1}|\tilde{\mathbf{S}}^0_1]_\mu . \end{aligned}$$

(B.15)

Taking into account that the exponent in (45) is equal to 1 at $k=0$, and substituting (B.11) and (B.15) in (45), we find that ${\tilde{A}}_{\mathbf{0}}^{\pm }=-1$.