kered two-dimensional
crystal, see \fig{1} (a, top). Its tight-binding model is given by
\begin{equation}
  H = \sum_{\braket{i,j}} t_{ij}\ket{i}\bra{j} + \textrm{H.c.},
  \label{Hc}
\end{equation}
where $\ket{i}$ are the atomic states localized on the phosphorous atoms at
position $\vec{r}_i$. The sum takes into account first and second nearest
neighbors, which are coupled by the energies $t_1=-1.22\un{eV}$ and
$t_2=3.665 \un{eV}$, respectively \cite{Betancur-Ocampo2019, Rudenko, Rudenko2,
  Cakir, Sisakht, Ezawa, Lewenkopf, Sousa2017}. The tight-binding model of
phosphorene can be understood more easily by projecting the atoms to the $xy$
plane, keeping their couplings constant, see \fig{1} (a, bottom). This
projection shows that the phosphorene lattice can be understood as a strongly
deformed hexagonal lattice with lattice constants $a = 4.42$, $b = 3.27 $ and
$\delta=0.8 \un{\AA}$ which connects the two atoms of the unit cell
\cite{Brown}.

Using a plane-wave ansatz, the tight-binding Hamiltonian can be written as
\begin{equation}
  H(\vec{k}) = \left(
    \begin{array}{@{}*{2}{c}@{}}
      0 & g^*(\vec{k})\\
      g(\vec{k}) & 0
    \end{array}
  \right),
  \label{H}
\end{equation}
where
\begin{equation}
  \label{gofk}
  g(\vec{k}) = \textrm{e}^{-\mathrm{i} k_{\delta}}[t_2 + 2t_1\textrm{e}^{\mathrm{i} k_a/2}\cos(k_b/2)]
\end{equation}
with $k_{\delta} = \vec{k}\cdot\vec{\delta}$, $k_{a} = \vec{k}\cdot\vec{a}$
$k_{b} = \vec{k}\cdot\vec{b}$. The energy bands are given by
\begin{equation}
  \label{eofk}
  \begin{array}{c}
    E(\vec{k})= s \abs{g(\vec{k})} \\[2mm]
    = s \sqrt{t_2^2 + 4 t_1 t_2 \cos(k_a/2)\cos(k_b/2) + 4 t_1^2 \cos^2(k_b/2)},
  \end{array}
\end{equation}
where $s=\text{sgn}(E)= \pm 1$. They show a band gap $2\Delta = 4t_1 + 2t_2$ and
strong anisotropy around the $\Gamma$ point. In the armchair direction ($x$-axis
in \fig{1}~(a), $k_b=0$), the electrons behave approximately as massive Dirac
Fermions. Their dispersion relation is quadratic at low energies, followed by a
long linear regime. In the zigzag direction ($y$-axis, $k_a=0$), the electrons
have the parabolic dispersion of Schrödinger Fermions. Moreover, it has been
shown that this anisotropy leads to negative reflection and anti-super-Klein
tunneling in phosphorene pn junctions \cite{Betancur-Ocampo2019}. The
eigenfunctions are given by
\begin{equation}
  \Psi_{\vec{k}}(\vec{r}) = \frac{1}{\sqrt{2}}\left(
    \begin{array}{@{}*{1}{c}@{}}
      1 \\
      s\textrm{e}^{\mathrm{i} \phi(\vec{k})}
    \end{array}\right)
  \textrm{e}^{\mathrm{i} \vec{k}\cdot \vec{r}},
  \label{wf}
\end{equation}
where the phase (or pseudo-spin) reads
\begin{eqnarray}
  \phi(\vec{k}) & = &
    \arctan\left(\frac{\textrm{Im}[g(\vec{k})]}{\textrm{Re}[g(\vec{k})]}\right) \nonumber\\[2mm]
                     & = &
    \arctan\left(\frac{\lambda_1(\vec{k})\cos(k_{\delta}) -\lambda_2(\vec{k})\sin(k_{\delta})}
                      {\lambda_2(\vec{k})\cos(k_{\delta}) +\lambda_1(\vec{k})\sin(k_{\delta})}\right),
   \label{phis}
\end{eqnarray}
with $\lambda_1(\vec{k}) = 2t_1\sin(k_a/2)\cos(k_b/2)$ and
$\lambda_2(\vec{k}) = t_2 + 2t_1\cos (k_a/2)\cos (k_b/2)$.


\section{Phosphorene pn and pnp junctions}


A phosphorene pn junction, which will be the main building block to construct a
phosphorene waveguide, is constituted by two regions of different electrostatic
potential (or doping), which shifts the energy bands, see \fig{1}~(c). In order
to understand the electron scattering at the interface of such pn junction, we
make the following ansatz for the wavefunction in region I
\begin{equation}
  \Psi_{\text{I}}(\vec{r}) = \frac{1}{\sqrt{2}}\left(
    \begin{array}{c}
      1\\
      s\textrm{e}^{\mathrm{i} \phi_i}
    \end{array}\right)
  \textrm{e}^{\mathrm{i} \vec{k}_i\cdot\vec{r}} +
  \frac{r}{\sqrt{2}}\left(
    \begin{array}{c}
      1\\
      s\textrm{e}^{\mathrm{i} \phi_r}
    \end{array}\right)
  \textrm{e}^{\mathrm{i} \vec{k}_r\cdot\vec{r}}
\label{pwr1}
\end{equation} 
and in region II
\begin{equation}
  \Psi_{\text{II}}(\vec{r}) = \frac{t}{\sqrt{2}}\left(
    \begin{array}{c}
      1\\
      s'\textrm{e}^{\mathrm{i} \phi_t}
    \end{array}\right)\textrm{e}^{\mathrm{i} \vec{k}_t\cdot\vec{r}}.
\label{pwr2}
\end{equation}
The continuity of the wavefunction at the interface allows to calculate the
transmission coefficient
\begin{equation}
  T = \frac{2\sin[\phi_t -(\phi_r+\phi_i)/2]\sin[(\phi_i-\phi_r)/2]}{ss'-\cos(\phi_t-\phi_r)}
\label{T}
\end{equation}
as a function of the pseudo-spin angles $\phi_i$, $\phi_r$, and $\phi_t$ of the
incident, reflected and transmitted electrons \cite{Betancur-Ocampo2019}.

\begin{figure}[t]
  \centering
  \includegraphics[scale= 0.6]{fig2}
  \caption{Reflection coefficient $R=1-T$ at the interface of a phosphorene pn
    junction ($V=2E$) as a function of the angle of incidence $\theta$ (see
    \fig{3}~(a)). Total reflection, $R=1$, is obtained for all $\theta$ and all
    electron energies $E$, if the junction is oriented parallel to the armchair
    edge. If the junction is parallel to the zigzag edge, the reflection is
    minimal for normal incidence but increases strongly for grazing $\theta$ and
    lower $E$.}
  \label{fig:2}
\end{figure}

\begin{figure}[t]
  \centering
 \includegraphics[scale= 0.4]{fig3}
 \caption{(a) Schematic representation of a perfect electron waveguide based on
   a phosphorene pnp junction. Electrons are injected in the n region
   (orange). At the interface to the p region (blue), they are totally reflected
   due to anti-super-Klein tunneling. (b) Electronic band structure across the
   potential well that generates the pnp junction.}
  \label{fig:3}
\end{figure}

Total reflection $R=1-T=1$