#  Coherent-scatterer enhancement and Klein-tunneling suppression by potential
barriers in gapped graphene with chirality-time-reversal symmetry

Farhana Anwar1111Email<EMAIL_ADDRESS><EMAIL_ADDRESS>Andrii
Iurov2222Email<EMAIL_ADDRESS><EMAIL_ADDRESS>Danhong
Huang3, Godfrey Gumbs4,5, and Ashwani Sharma3,6,7 1Lawrence Berkeley National
Laboratory, 1 Cyclotron Rd, Berkeley, CA 94720
2Department of Physics and Computer Science, Medgar Evers College of the City
University of New York, Brooklyn, NY 11225, USA
3Air Force Research Laboratory, Space Vehicles Directorate, Kirtland Air Force
Base, NM 87117, USA
4Department of Physics and Astronomy, Hunter College of the City University of
New York, 695 Park Avenue, New York, NY 10065, USA
5Donostia International Physics Center (DIPC), P de Manuel Lardizabal, 4,
20018 San Sebastian, Basque Country, Spain
6Center for High Technology Materials, University of New Mexico, 1313 Goddard
SE, Albuquerque, NM 87106, USA
7Department of Electrical and Computer Engineering, University of New Mexico,
Albuquerque, NM 87106, USA

###### Abstract

We have utilized the finite-difference method to investigate electron
tunneling in gapped graphene across various electrostatic-potential barriers.
Specifically, we analyze Gaussian and triangular envelope functions to compare
with a square potential barrier. The transmission coefficient was calculated
numerically for each case and applied to corresponding tunneling conductance.
It is well known that Klein tunneling in graphene will be greatly reduced in a
gapped graphene. Our results further demonstrated that such a decrease of
transmission can be significantly enhanced for spatially-modulated potential
barriers. Additionally, we investigated the effect from a bias field applied
to those barrier profiles, from which we showed that it enables the control of
electron flow under normal incidence. The suppression of Klein tunneling was
found to be more severe for a non-square barrier and exhibits a strong
dependence on bias-field polarity for all kinds of barriers. The roles played
by a dilute distribution of point impurities on electron transmission and
conductance were analyzed with a sharp peak appearing in the electron
conductance as an impurity atom is placed in the middle of a square barrier.
However, for narrow triangular and Gaussian barriers, the conductance peaks
become substantially broadened, associated with an enhancement of the
tunneling conductance.

##  I Introduction

The unusual properties of Dirac quasiparticles have become one of the most
popular topics in fundamental research as well as a promising source for new
technological applications Geim and Novoselov (2010); Neto et al. (2009); Geim
(2009). Being a zero-bandgap semiconductor and possessing a specific chiral
wave function simultaneously will result in full transparency for any
potential barrier for normally-incident electrons. Beenakker (2008);
Katsnelson et al. (2006) A massless Dirac fermion which is scattered by an
electrostatic potential is able to tunnel with certainty for normal incidence,
as known Klein paradox, regardless of the potential-barrier height or width.
Klein (1929); Calogeracos and Dombey (1999) Graphene is also known for its
specific magnetic-field properties, Goerbig (2011); Gumbs et al. (2014a);
Pyatkovskiy and Gusynin (2011); Checkelsky and Ong (2009) as well as its
strong decrease of magneto-resistance Huang et al. (2019); Kim and Kim (2008)
through $p$-$n$ junctions. Cheianov and Fal’ko (2006) While the metallic band
structure and high mobility are obvious advantages of graphene, most
applications in electronics require a small and tunable bandgap Pereira et al.
(2008) in order to keep its charged carriers confined within a finite area of
an electronic device. Berger et al. (2006) This feature could be achieved by
opening a finite bandgap Ni et al. (2008) which directly leads to a
suppression of the Klein tunneling in graphene. Iurov et al. (2011);
Kindermann et al. (2012); Zhou et al. (2007); Iurov et al. (2013)

There are various efficient ways for creating a sizable ($\sim 100\,$meV and
above) and tunable bandgap in graphene. Most of them are generated byadding a
dielectric Si-based substrate Ni et al. (2008) or a substrate with broken
inversion symmetry between sublattices. Zhou et al. (2007) It was just
recently discovered that a tunable gap can be achieved after a Dirac-cone
hexagonal two-dimensional (2D) lattice has been irradiated with a circularly-
polarized Iurov et al. (2019); Oka and Aoki (2009); Liu et al. (2011) and off-
resonance field Iurov et al. (2020a); Pastrana-Martinez et al. (2012); Usaj et
al. (2014); Iurov et al. (2017a); Kibis (2010); Pervishko et al. (2015);
Portnoi et al. (2008); Kibis et al. (2017); Iurov et al. (2017b); Dey and
Ghosh (2018). In addition, strain engineering, finite-width graphene
nanoribbon (GNR) systems, Brey and Fertig (2006), structural and topological
defects or insertion of impurity atoms can also produce a bandgap in graphene.
Pereira et al. (2008) Physically, the presence and size of an induced bandgap
in graphene and other hexagon lattices directly affect tunneling and transport
properties Kristinsson et al. (2016); Iurov et al. (2020b), and play a crucial
role in graphene-based electronics, such as field-effect transistors since
their on/off current ratios Low and Appenzeller (2009) can be tuned through
tunneling control. Also, the optical response of gapped graphene acquire
attractive features in opto-electronics and optical spectroscopy. Pedersen et
al. (2009) Particularly, these responses have been proved to be sensitive to
localized and trapped states within the bandgap of systems considered. Pereira
et al. (2006); Gumbs et al. (2014b); Castro et al. (2008) The effect due to
mesoscopic fluctuations appearing in the conductance of a gapped-graphene
strip arising from a random electrostatic potential landscape is believed to
be important but has not been investigated thoroughly as for gapless graphene.
Rycerz et al. (2007) The same goes for the effect of an electrostatic field
applied across a potential barrier. Jang et al. (2013)

Although a considerable amount of research has been published on graphene
band-transport characteristics, investigations on tunneling across a smoothly-
varying potential barrier in a gapped Dirac system have been much less dealt
with. In fact, tunneling transport through a steep-slope potential profile,
involving either single or double square barrier, has been investigated and
reported in Refs. [Dahal and Gumbs, 2017; Azarova and Maksimova, 2014;
Navarro-Giraldo and Quimbay, 2018]. However, analytical solutions for a
finite-slope barrier are still inaccessible. On the other hand, electron
transmission in graphene and other newly discovered Dirac materials was also
computed based on Wentzel-Kramers-Brillouin (WKB) semiclassical theory. Sonin
(2009); Weekes et al. (2021) Furthermore, it has been shown within WKB theoru
that if electron-hole transition is considered inside a non-square potential
barrier, tunneling transmission will increase with the slope of a potential
profile. Technically, a smooth finite-slope potential barrier becomes more
realistic since it matches better with the experimentally accessible situation
Stander et al. (2009). Importantly, such a case also presents several
intriguing phenomena and properties, e.g., Klein collimator Libisch et al.
(2017) as well as a possibility for building up a collimated interferometer or
reflector Young and Kim (2009); Wang et al. (2019). Besides, these smooth-
barrier profiles also display unusual tunneling features upon applying an
electric or a magnetic field. Mouhafid and Jellal (2013); Shytov et al.
(2008); Young and Kim (2009); Anwar et al. (2017); Anwar (2020)

Compared with tunneling transport of incident electrons through a potential
barrier in graphene, impurity scattering Titov (2007); Ando and Koshino
(1984); Rycerz et al. (2007); Anwar et al. (2020) in gapped graphene turns out
to be another important research topic that has not been adequately studied.
Disorder embedded within an electrostatic-potential barrier in a gapped
graphene system Huang et al. (1999), which exhibits either disorder-assisted
or disorder-impeded tunneling, is even less known. Specifically, the spatial
position, strength and polarity of an embedded scatterer and its effect on
tunneling conductance of electrons for various electrostatic potential
barriers Rycerz et al. (2007); Titov (2007) have become the most important
aspect because they decide whether the conductance of the system is enhanced
or reduced by different impurity configurations. As a result, it is of
paramount importance to study the effect of impurity scattering in gapped
graphene and reveal the condition for suppressed back-scattering of incident
electrons by appropriately distributing scatterers within a barrier region,
and this makes a crucial part of our current work.

The key issue addressed in this paper is quantifying electron transmission and
conductance across non-square potential barriers with a finite slope in gapped
graphene subject to a DC electric field by using a discretized Dirac equation
Hernández and Lewenkopf (2012); Tworzydło et al. (2008) based on a finite-
difference method Anwar et al. (2020); Huang et al. (2020). Our numerical
method presented here has proven to be crucial for studying a gapped-graphene
system with a smooth disorder potential. Based on our established numerical
procedure, we hsvr accurately calculate the effect of a disorder potential on
electron tunneling and conductance. The remainder of the paper is organized as
follows. We first validate our discretized model by calculating the
transmission coefficient of gapped graphene across a potential barrier, and
then compare the obtained numerical results with previous analytical solutions
Setare and Jahani (2010) in some limiting cases. By employing this numerical
method, we further explore quantitatively how the barrier-potential profile,
applied bias with different polarities and barrier-embedded disorders affect
the conductance as a function of incident-electron kinetic energy for various
graphene $p$-$n$ junction (GPNJ) within a gapped monolayer graphene.

##  II General Formalism

The low-energy states of gapped graphene can be described fully by a Dirac
Hamiltonian with an additional gap term, Setare and Jahani (2010); Iurov et
al. (2011); Pyatkovskiy (2008) given by

$\hat{\mathcal{H}}(\mbox{\boldmath$r$})=-iv_{F}\,\left[\hat{\mbox{\boldmath$\Sigma$}}\cdot\mbox{\boldmath$\nabla$}\right]_{(x,y)}+V_{B}(x)\,\hat{\Sigma}_{0}+\Delta_{G}\,\hat{\Sigma}_{z}\
,$ (1)

where $\hat{\Sigma}_{x,y,z}$ are three two-dimensional Pauli matrices,
$\hat{\Sigma}_{0}$ is a $2\times 2$ unit matrix, $v_{F}$ is the Fermi
velocity, $\Delta_{G}$ is the gap parameter, and $V_{B}(x)$ represents a
spatially non-uniform barrier potential. For constant potential
$V_{B}(x)=V_{0}$, the Dirac Hamiltonian in Eq. (1) gives rise to a finite
energy bandgap $E_{G}=2\Delta_{G}$ between the valence and conduction bands
which are symmetric with respect to the Dirac point. In addition, the energy
dispersions are calculated as $\varepsilon_{\gamma}(k)=\gamma\sqrt{(\hbar
v_{F}k)^{2}+\Delta_{G}^{2}}$ and $\mbox{\boldmath$k$}=\\{k_{x},k_{y}\\}$ is a
two-dimensional wave vector of electrons. For varied $V_{B}(x)$, however, we
have $\gamma=\text{sign}[\varepsilon_{\gamma}(k)-V_{B}(x)]=\pm 1$,
corresponding to electron and hole states, respectively, where
$\text{sign}(x)$ is a sign function. This implies that the carrier could go
through an electron-hole (or inverse) transition inside the barrier region.

In this paper, we look forward to finding scattering-state solutions
$\Phi_{\gamma}(\mbox{\boldmath$r$})$ for the Hamiltonian in Eq. (1) in the
form two-component (spinor) type of wave function, i.e.

$\Phi_{\gamma}(\mbox{\boldmath$r$})=\texttt{exp}(ik_{y}y)\,\Psi_{\gamma}(x)=\,\texttt{exp}(ik_{y}y)\,\left[\begin{array}[]{c}\phi_{A}(\gamma,x)\\\
\phi_{B}(\gamma,x)\end{array}\right]\ ,$ (2)

where unlike gapless graphene both components rely on the electron-hole index
$\gamma$.

As a special case, if $V_{B}(x)=V_{0}$ is a constant, the translational
symmetry of the system is preserved in both $x$ and $y$ directions, and then
the Hamiltonian in Eq. (1) can be greatly simplified as

$\hat{\mathcal{H}}^{(0)}_{g}(k\,|\,\theta_{\bf
k})=\left[\begin{array}[]{cc}V_{0}+\Delta_{G}&\hbar v_{F}k_{-}\\\ \\\ \hbar
v_{F}k_{+}&V_{0}-\Delta_{G}\end{array}\right]\ ,$ (3)

where $\mbox{\boldmath$k$}=(k_{x}^{(0)},k_{y})$, $k_{\pm}=k_{x}^{(0)}\pm
ik_{y}$, $\gamma=\text{sign}(\varepsilon_{0}(k)-V_{0})$ within the barrier
region, and $\varepsilon_{0}(k)=\hbar v_{F}\sqrt{(k_{x}^{(0)})^{2}+k_{y}^{2}}$
is the given energy of an incident electron. In this case, the scattering-
state wave function associated with the Hamiltonian in Eq. (3) takes the
explicit form Pyatkovskiy (2008); Iurov et al. (2011, 2017c)

$\Psi_{\gamma}^{(0)}(\mbox{\boldmath$r$})=\frac{1}{\sqrt{2\gamma\,\delta\varepsilon_{0}(k)}}\left[\begin{array}[]{c}\sqrt{|\delta\varepsilon_{0}(k)+\Delta_{G}|}\\\
\\\
\gamma\sqrt{|\delta\varepsilon_{0}(k)-\Delta_{G}|}\,\texttt{e}^{i\theta_{\bf
k}}\end{array}\right]\texttt{exp}(ik^{(0)}_{x}x+ik_{y}y)\ ,$ (4)

where $\theta_{\bf k}=\tan^{-1}(k_{y}/k_{x}^{(0)})$,
$\delta\varepsilon_{0}(k)\equiv\varepsilon_{0}(k)-V_{0}\geq\Delta_{G}$ for an
electron with $\gamma=+1$ and $\delta\varepsilon_{0}(k)\leq-\Delta_{G}$ for a
hole with $\gamma=-1$. Here, two components of the wave function spinor in Eq.
(4) are not the same and their ratio further depends on the electron/hole
index $\gamma=\pm 1$.

Here, we concentrate on studying the electron transmission and conductance
through a biased barrier of different geometries in gapped graphene.
Therefore, we limit our investigation only to experimentally-available
symmetrical barrier profiles, e.g., triangular and Gaussian shapes. For this
reason, we would like to compare our results first with the well-studied case
having a square barrier

$\displaystyle\frac{V_{s}(x)}{V_{0}}=\Theta(x)\,\Theta(W_{B}-x)\,,$ (5)

where $\Theta(x)$ is the Heaviside step function, meaning $V_{s}(x)=V_{0}$ if
$0<x<W_{B}$ and is equal to zero otherwise. As adopted in Figs. 1 and 2, the
selection of using a square potential barrier makes it simple to compare our
numerical results for a general varying-slope barrier system with some known
analytical cases and discern quantitatively the effect of a finite slope of a
potential barrier.

Next, for comparison we would introduce a triangular barrier $V_{t}(x)$ with
both positive and negative slopes present on its two sides, yielding

$\frac{V_{t}(x)}{V_{0}}=\left\\{\begin{array}[]{ccc}0&\hskip
7.22743pt\text{for}&\hskip 7.22743ptx<a\ ,\\\ (x-a)/(b-a)&\hskip
7.22743pt\text{for}&\hskip 7.22743pta<x<b\ ,\\\ (c-x)/(c-b)&\hskip
7.22743pt\text{for}&\hskip 7.22743ptb<x<c\ ,\\\ 0&\hskip
7.22743pt\text{for}&\hskip 7.22743ptx>c\ .\\\ \end{array}\right.$ (6)

For our numerical computations, we adopt a symmetric triangular barrier
configuration which corresponds to $c-b=b-a$, as used in Fig. 3.

For comparison with experimentally-available setup, we also consider a
(symmetric) Gaussian profile with the potential voltage profile varying
according to

$\displaystyle\frac{V_{g}(x)}{V_{0}}=\pm\texttt{exp}\left[-\left(\frac{x-x_{0}}{l}\right)^{2}\right]\,,$
(7)

where $l$ is employed to define the effective width $\sqrt{\pi}l$ of our
barrier, $x_{0}$ is its symmetric center where the highest possible potential
$V_{0}$ is reached, as employed in Fig. 3. Depending on the $\pm$ sign in Eq.
(7), this profile could describe either a barrier ($+V_{0}$) or a trap
($-V_{0}$).

For numerical computations, using the Hamiltonian in Eq. (1) we obtain a pair
of coupled scattering-state equations within the barrier region, leading to

$\displaystyle\frac{\partial}{\partial
x}\,\phi_{B}(x)+k_{y}\,\phi_{B}(x)=\frac{i}{\hbar
v_{F}}\,\left[\varepsilon_{0}(k)-V_{B}(x)+V_{d}\,\delta(x-x_{s})-\Delta_{G}\right]\,\phi_{A}(x)\
,$ (8) $\displaystyle\frac{\partial}{\partial
x}\,\phi_{A}(x)-k_{y}\,\phi_{A}(x)=\frac{i}{\hbar
v_{F}}\,\left[\varepsilon_{0}(k)-V_{B}(x)+V_{d}\,\delta(x-x_{s})+\Delta_{G}\right]\,\phi_{B}(x)\
.$ (9)

In Eqs. (8) and (9), when we consider a tilted barrier under an applied
electric field ${\cal E}_{0}$, we should replace $V_{B}(x)$ by
$V_{B}(x)-e{\cal E}_{0}x$ within the barrier region, where ${\cal E}_{0}$ can
be either positive or negative. Additionally, $k_{y}$ remains conserved during
a tunneling process for all regions considered, i.e., on the left of the
barrier $(1)$, $x<0$, inside $(2)$, $0<x<W_{B}$, and to right of the barrier
$(3)$, $x>W_{B}$. We also introduce in Eqs. (8) and (9) a single scatterer of
strength $\pm V_{d}$ at the position $x=x_{s}$ within the barrier region for
$0<x_{s}<W_{B}$. The scattering problem is such that there could be both
transmitted and reflected waves with the amplitudes $t_{i}(\varepsilon_{0})$
and $r_{i}(\varepsilon_{0})$, $i=1,2$, in regions $(1)$ and $(2)$ but only the
transmitted wave $t_{3}(\varepsilon_{0})$ in region $(3)$.

From a physical point of view, it is very instructive to note that under the
replacements $k_{y}\to-k_{y}$ and $\Delta_{G}\to-\Delta_{G}$ for
$\phi_{A}(x)\to\phi_{B}(x)$ as well, Eq. (9) transforms to Eq. (8) and vice
versa. This implies a hidden chirality-time-reversal (CTR) symmetry for
electron tunneling in this gapped-graphene system if its scatterer-embedded
potential $V_{B}(x)-V_{d}\,\delta(x-x_{s})$ remains unchanged under the
barrier transformation with respect to $x\to W_{B}-x$ for fixed barrier width
$W_{B}$, i.e., $V_{B}(x)-V_{d}\,\delta(x-x_{s})$ should acquire a mirror
symmetry with respect to its midpoint $x=W_{B}/2$ under zero bias ${\cal
E}_{0}=0$ condition.

##  III Results and Discussions

In order to ensure that our numerically calculated results based on the
finite-difference approach (FDA) are accurate and valid, we first compare them
with some previously known cases having analytical solutions Setare and Jahani
(2010) for a square barrier with $V_{B}(x)=V_{0}\,\Theta(x)\,\Theta(W_{B}-x)$.
As a validation, we have presented in Fig. 1 the transmission coefficient
$T(\varepsilon,\phi_{\bf k}\,|\,{\cal E}_{0})$ for a square barrier using both
our FDA results and the known analytical solutions Setare and Jahani (2010).
This direct comparison clearly indicates that the FDA will be valid for an
arbitrary biased barrier potential profile $V_{B}(x)$ on the order of
$10^{2}$-$10^{3}\,$meV, including both Gaussian and triangular potential
barriers embedded with a single scatterer.

The analytically-calculated transmission coefficients $T(\varepsilon,\phi_{\bf
k}=0\,|\,W_{B})$ for a 1D square-barrier potential $V_{s}(x)$ are plotted in
Fig. 2 as functions of bandgap parameter $\Delta_{G}$ for various barrier
widths $W_{B}$ and incoming particle energies $\varepsilon$. We see clearly
from Fig. 2 that the transmission coefficient $T(\varepsilon,\phi_{\bf
k}=0\,|\,W_{B})$ for head-on collision will be completely suppressed once
$\Delta_{G}$ exceeds an $\varepsilon$-dependent threshold value, which becomes
largely independent of the barrier width $W_{B}$. However, this threshold
value for $\Delta_{G}$ reduces with increasing incident-electron energy
$\varepsilon$.

We now turn our attention to numerical computations of the transmission
probability $T(\varepsilon,\phi_{\bf k}\,|\,{\cal E}_{0})$ by introducing
general FDA method, as described in Eqs. (8) and (9), for arbitrary shape of
barrier potential profiles under an applied bias field and with embedded point
scatterers.

We first present results for the effect of a barrier potential profile on the
transmission probability $T(\varepsilon,\phi_{\bf k})$ in the absence of a
bias field ${\cal E}_{0}=0$. We have specifically selected square, triangular
and Gaussian as three distinctive barrier profiles in Fig. 3 in order to
acquire a full comparison among them. From the left panel of Fig. 3, we find a
great suppression of Klein tunneling in the presence of a finite gap by a
triangular-barrier potential $V_{t}(x)$ (red curve). Meanwhile, the
transmission for this case is only limited to a very narrow angular region
around $\phi_{\bf k}=0$. For a square barrier potential $V_{s}(x)$ (black
curve), on the contrary, the transmission is distributed widely within a broad
angular region bounded by $|\phi_{\bf k}|\leq\pi/3$, and meanwhile the
transmission for head-on collision at $\phi_{\bf k}=0$ remains strong. The
transmission for a Gaussian potential barrier $V_{g}(x)$ (brown curve)
somewhat stands between the previous two cases with a limited angular
distribution as well as a greatly enhanced strength at $\phi_{\bf k}=0$
compared to a square-barrier and triangular barrier potentials, respectively.

In order to obtain a better and convincing understanding of the effect due to
chosen potential barrier-profiles on the transmission probability, we present
in Fig. 4 the density plot for $T(\varepsilon,\phi_{\bf k}\,|\,V_{B})$ as
functions of incident-electron energy $\varepsilon$ and incident angle
$\phi_{\bf k}$ for various barrier profiles. Comparing Figs. 4$(a)$ and
4$(b)$, we identify a dominant feature in this figure for the tunneling of
electrons in gapped graphene, i.e., the transmission probability can be
substantially reduced and nearly goes to zero as the incident energy
approaches the barrier height at normal incidence, where the incident particle
acquires a small or even imaginary momentum within the barrier region.
Additionally, the electron transmission is modified significantly for two
different slowly-varying barrier profiles considered in Figs. 4$(c)$ and
4$(d)$. Here, many layered sharp resonant features of the transmission
probability observed in Fig. 4$(b)$ for an under square-barrier incidence
disappear in both Figs. 4$(c)$ and 4$(d)$, leaving only a single energy range
around $\phi_{\bf k}=0$ in Fig. 4$(d)$.

From a technology perspective, we know that the control of an electrical
current flow in graphene devices becomes crucial for their applications, such
as current modulation, amplification and signal processing. For this reason,
we compare in Fig. 5 the changes of scaled tunneling conductance
$\sigma(\varepsilon\,|\,V_{B})/\sigma_{0}$ as functions of incoming-particle
energy $\varepsilon$ in gapped graphene for three distinct barrier profiles.
As shown in Fig. 5, for $\varepsilon$ within the range of $50$-$230\,$meV, a
square barrier gives rise to a square-like highest conductivity (black curve)
which, however, is decreased as $\varepsilon$ approaches
$V_{0}-\Delta_{G}=235\,$meV. For a triangular barrier (red curve). On the
other hand, we find a strongly-oscillating conductance with multiple peaks and
valleys in the same range. Quite differently, Gaussian barrier (brown curve)
leads to the lowest weakly-oscillating conductivity for all barrier profiles
considered within this energy range, but it produces the highest step-rising
conductivity above this energy range. Meanwhile, unlike the square barrier,
the conductivity associated with either triangular or Gaussian potential
barrier rises quickly for $\varepsilon>235\,$meV although its increase is not
as rapid as that for the square potential barrier around
$\varepsilon=350\,$meV.

In Fig. 6, for three chosen barrier-potential profiles, we compare the
obtained numerical results for the transmission probability
$T(\varepsilon,\phi_{\bf k}\,|\,V_{B})$ as a function of the angle of
incidence $\phi_{\bf k}$ with a series of gap parameters $\Delta_{G}$. For the
Gaussian barrier profile in Fig. 6$(b)$, the tunneling is confined well within
a small angle region. With increasing $\Delta_{G}$, the tunneling amplitude at
$\phi_{\bf k}=0$ is enhanced quickly to unity which is accompanied by the
expanded angle region around $\phi_{\bf k}=0$. Interestingly, very strong
focusing of tunneling with respect to $\phi_{\bf k}=0$ is developed for a
triangular barrier profile in Fig. 6$(c)$ which is supplemented by the
appearance of two symmetrical sharp side features. Additionally, in this case
the tunneling amplitude at $\phi_{\bf k}=0$ is rapidly reduced with increasing
$\Delta_{G}$, which further goes together with the suppression of the two side
features. For the square barrier profile in Fig. 6$(a)$, lots of side peaks
occur and their angle distributions are shrunken slowly as $\Delta_{G}$ is
increased.

From a device point of view, tuning the tunneling conductance in gapped
graphene is an effective technique and very important for its functionality
and application. Here, we demonstrate tunable tunneling conductance for gapped
graphene by applying a bias field ${\cal E}_{0}$ to a general barrier
potential $V_{B}(x)$, which is in interplay with the opened bandgap of
graphene. We start with a square barrier and present calculated polar plots of
$T(\varepsilon,\phi_{\bf k}\,|\,{\cal E}_{0})$ in Fig. 7$(a)$-$(b)$ for
forward and both backward biases from which we find the suppression of Klein
tunneling at $\phi_{\bf k}=0$ due to electron-hole transition resulting from a
finite $\Delta_{G}$ chosen but it is still robust against the applied bias
field ${\cal E}_{0}$. It is interesting to note that the suppression of
transmission under normal incidence appears only for a positive biase but not
for a negative bias, which exhibits a strong asymmetry with respect to the
polarity of ${\cal E}_{0}$ or broken CTR symmetry in the system. In
particular, enhancements of $T(\varepsilon,\phi_{\bf k}\,|\,{\cal E}_{0})$
near $\phi_{\bf k}\approx\pm 90^{\rm o}$ are seen only for ${\cal
E}_{0}=-1\,$kV/cm. On the other hand, there exists an insulating zero-
conductance gap for incident electron energy $\varepsilon$ which follows from
Fig. 7$(c)$-$(d)$, but its two edges shift in opposite directions with the
polarity of ${\cal E}_{0}$. In this way, we can easily switch the electron
tunneling conductance between the conducting and insulating phases in gapped
graphene by properly selecting the polarity and magnitude of an applied bias
field for any fixed kinetic energy of incident electrons.

We have established that for a finite bandgap $\Delta_{G}$ of graphene, the
tuning of tunneling conductance by an applied bias field also depends on the
selected shape of a barrier potential profile, such as a triangular or
Gaussian barrier. For the triangular barrier under a reverse bias in Fig.
8$(a)$, we find the conductance as a function of $\varepsilon$ remains at zero
for $\varepsilon<50\,$meV. However, both the dominant higher and the secondary
lower conductance peaks move upward as a function of incoming electron energy
$\varepsilon$ with increasing $|{\cal E}_{0}|$. On the contrary, these two
conductance peaks are shifted to smaller values of $\varepsilon$ with
increased ${\cal E}_{0}$ under forward biases, as found from Fig. 8$(b)$. In
order to gain an overall picture regarding the tuning of tunneling conductance
$\sigma(\varepsilon,{\cal E}_{0}\,|\,V_{B})$ by a bias field ${\cal E}_{0}$
for different barrier profiles $V_{B}(x)$, we compare corresponding density
plots of scaled $\sigma(\varepsilon,{\cal E}_{0})/\sigma_{0}$ as functions of
both incoming-particle energy $\varepsilon$ and ${\cal E}_{0}$ in Figs.
9$(a)$-9$(d)$. Generally speaking, the case for biased Gaussian barriers in
Figs. 9$(a)$-9$(b)$ only gives rise to two relatively-weak peaks in tunneling
conductance $\sigma(\varepsilon,{\cal E}_{0})$ within two separate ranges for
kinetic energy $\varepsilon$. The bias field ${\cal E}_{0}$, on the other
hand, only shifts those conductance peaks in $\sigma(\varepsilon,{\cal
E}_{0})$ downward in $\varepsilon$ under the forward-bias condition but shifts
it upward as a function of $\varepsilon$ in the reverse-bias condition. For a
triangular barrier, we find enhanced features for peak shifting both upward
and downward with increasing $|{\cal E}_{0}|$, as displayed in Figs.
9$(c)$-9$(d)$.

In the remaining part of this Section, we will address the effect of a single
scatterer embedded at various positions within an unbiased barrier of
different shapes on tunneling conductance of electrons in gapped graphene. In
Fig. 10, we plot $\sigma(\varepsilon,V_{d}\,|\,V_{B})/\sigma_{0}$ for an
unbiased square barrier with different locations for a single scatterer. As a
comparison, we also display the result with no scatterer in Fig. 10$(a)$ under
${\cal E}_{0}=0$ for a square barrier potential. From Fig. 10$(b)$ with
$x_{s}/W_{B}=0.1$, we observe that $\sigma(\varepsilon,V_{d})$ acquires
several consecutive peaks and valleys within the low-energy range of
$0.3<\varepsilon/V_{0}<0.8$ in the presence of a single scatterer. In
addition, a zero-conductance gap exists for the intermediate energy range
$0.8<\varepsilon/V_{0}<1.3$, and meanwhile, a number of peaks and valleys
develop for enhanced $\sigma(\varepsilon,V_{d})$ in the high-energy region
$1.3<\varepsilon/V_{0}<1.8$. As $x_{s}/W_{B}=0.5$ in Fig. 10$(c)$ for the
scatterer located right in the middle of thr square barrier, a sharp peak in
$\sigma(\varepsilon,V_{d})$ occurs at $\varepsilon/V_{0}=1$ within this zero-
conductance gap, despite the sign and magnitude of $V_{d}$. For
$x_{s}/W_{B}=0.9$ in Fig. 10$(d)$, we verify CTR symmetry with respect to the
center $x_{s}=W_{B}/2$ of a one-dimensional (1D) barrier, i.e.,
$\sigma(\varepsilon,V_{d})$ for a scatterer at $x_{s}$ is the same as
$\sigma(\varepsilon,V_{d})$ for a scatterer at $W_{B}-x_{s}$.

In order for us to understand quantitatively the interplay between effects due
to a scatterer and the shape of a barrier potential, we compare 2D plots in
Figs. 11 and 12 for the scaled tunneling conductance
$\sigma(\varepsilon,V_{d}\,|\,V_{B})/\sigma_{0}$ of gapped graphene in the
presence of a single scatterer at different positions within a triangular and
Gaussian barrier regions. First, for a triangular barrier in Fig. 11, we
observe a conductance peak at $\varepsilon/V_{0}=0.5$ for $x_{s}/W_{B}=0.1$
instead of $\varepsilon/V_{0}=1$ as in Fig. 10$(b)$. Moreover,
$\sigma(\varepsilon,V_{d}\,|\,V_{B})$ remains zero within the energy ranges of
$0<\varepsilon/V_{0}<0.4$ as well as $0.6<\varepsilon/V_{0}<1$. As the
scatterer is shifted to $x_{s}/W_{B}=0.3$, we find a weak conductance peak
appearing near $\varepsilon/V_{0}=0.5$ in Fig. 11$(c)$. Furthermore, when
$x_{s}/W_{B}=0.5$, we reveal a new strong conductance peak for
$\varepsilon/V_{0}=0.85$ in Fig. 11$(d)$ due to constructive superposition of
two individual peaks. In particular, the CTR symmetry with respect to the
center $x_{s}=W_{B}/2$ of a 1D barrier in Fig. 10 is still maintained in Figs.
11$(e)$ and 11$(f)$ in comparison with Figs. 11$(c)$ and 11$(b)$,
respectively, with a switched sign for $V_{d}$.

Finally, for a Gaussian potential barrier, we learn from Fig. 12 that there
exist two weak conductance peaks at $\varepsilon/V_{0}=0.3$ and
$\varepsilon/V_{0}=0.65$ for $x_{s}/W_{B}=0.1$ in Fig. 12$(b)$. Additionally,
zero-conductance gap is still present within the energy ranges of
$0<\varepsilon/V_{0}<0.25$ and $0.25<\varepsilon/V_{0}<0.65$. Interestingly,
there is a suppression of $\sigma(\varepsilon,V_{d}\,|\,V_{B})$ for incident
energy $\varepsilon$ below the barrier height $V_{0}$ as $x_{s}/W_{B}=0.3$, as
shown in Fig. 12$(c)$. After $x_{s}/W_{B}$ increases to $0.5$ at the peak
position of a Gaussian barrier in Fig. 12$(d)$, the shape of
$\sigma(\varepsilon,V_{d}\,|\,V_{B})$ as a function of $\varepsilon$ changes
drastically by displaying a high and wide constructive conductance peak at
$\varepsilon/V_{0}=0.85$ accompanied by an overall enhancement of conductance
in the energy range of $\varepsilon/V_{0}>0.4$. Furthermore, the CTR symmetry
associated with $x_{s}=W_{B}/2$ for a 1D barrier is retained in Figs. 12$(e)$
and 12$(f)$ compared to Figs. 12$(c)$ and 12$(b)$ with a switched sign for
$V_{d}$.

##  IV Summary and Concluding Remarks

In a summary, we have thoroughly investigated tunneling and calculated the
crucial transmission properties of electrons across square, triangular and
Gaussian potential barriers embedded with a single scatterer for gapped
graphene. For this, we employed a finite difference approach since their
computations are not accessible by standard analytical solution techniques. We
have also addressed the effect of a bias and point scatterer located within
the barrier. It is known that the transmission and conduction in gapped
graphene are largely suppressed as the particle energy lies inside the bandgap
inside a barrier region. Due to the existence of a finite bandgap between
valence and conduction bands, the Klein tunneling for head-on collision (i.e.,
with incident angle $\phi_{\bf k}=0$) is suppressed for the case with a square
potential barrier. Simultaneously, the side resonances for electron tunneling,
which are associated with finite incident angles, are also reduced
significantly in gapped graphene. These suppression effects become even more
pronounced for smooth barriers. However, we have demonstrated multiple ways to
modify or even break this low-conduction condition and substantially improve
the collimation of transmitted electron beam by employing the approach
described above. Using the obtained transmission probability, we have further
calculated the tunneling conductance which also displays a suppression for a
finite gap in graphene. In fact, we have found that both the transmission and
conductance display a strong dependence on the barrier profile, its slope and
curvature. Meanwhile, we have also shown that the application of a bias field
and its polarity greatly affect the Klein tunneling suppression resulting from
broken CTR symmetry of the system, and at the same time shift conductance
peaks in energy for all barrier types.

Under a bias field, a zero-conductance gap occurs for a square barrier in a
range of selected incident-electron energy below the top of the barrier. For
positive/negative bias, two edges of this zero-conductance gap are
respectively dragged to lower/higher energies for incident electrons with
increasing absolute value $|{\cal E}_{0}|$ of the bias. For a triangular
barrier, on the other hand, we have found only one dominant and another
secondary peaks in higher and lower energy ranges for incident electrons.
Similar behaviors of peak shifting have been seen with increasing $|{\cal
E}_{0}|$. By introducing a single scatterer to an energy barrier of gapped
graphene, we have revealed that its strength, polarity and position can affect
the conductance profile of gapped graphene. As a scatterer is moved to the
midpoint (or the symmetry point) of an energy barrier, the resulting
conductance always acquires either a peak within this zero-conductance gap or
a significant enhancement from a constructive scattering contribution due to
CTR symmetry of the system. Specifically, for a square barrier, we have
observed appearance of a sharp conductance peak for $\varepsilon/V_{0}\sim 1$
within the zero-conductance gap. For Gaussian and triangular potential
barriers, however, such a conductance peak around $\varepsilon/V_{0}\sim 0.85$
becomes much broadened, especially for smooth Gaussian barriers with a smaller
curvature.

Finally, we note that studying the conduction properties of gapped graphene
and their possible alteration is very important and timely from the beginning.
Practical use of graphene in device applications has already been witnessed
recently. Our results are directly associated with creating spatial
confinement for graphene electrons within designated areas of an electronic
device modulated by a bias voltage. Apart from that, gapped graphene itself is
also related to some newly discovered materials with an intrinsic spin-orbit
gap, such as silicene, germanene and molybdenum disulfide. We believe that our
current works could be applied to these materials as well.

###### Acknowledgements.

D.H. would like to acknowledge the financial supports from Air Force Office of
Scientific Research (AFOSR). G.G. would like to acknowledge the support from
Air Force Research Laboratory (AFRL) through Contract #FA9453-18-1-0100.

## References

  * Geim and Novoselov (2010) A. K. Geim and K. S. Novoselov, in _Nanoscience and technology: a collection of reviews from nature journals_ (World Scientific, 2010), pp. 11–19.
  * Neto et al. (2009) A. C. Neto, F. Guinea, N. M. Peres, K. S. Novoselov, and A. K. Geim, Reviews of modern physics 81, 109 (2009).
  * Geim (2009) A. K. Geim, science 324, 1530 (2009).
  * Beenakker (2008) C. Beenakker, Reviews of Modern Physics 80, 1337 (2008).
  * Katsnelson et al. (2006) M. Katsnelson, K. Novoselov, and A. Geim, Nature physics 2, 620 (2006).
  * Klein (1929) O. Klein, Zeitschrift für Physik 53, 157 (1929).
  * Calogeracos and Dombey (1999) A. Calogeracos and N. Dombey, Contemporary physics 40, 313 (1999).
  * Goerbig (2011) M. Goerbig, Reviews of Modern Physics 83, 1193 (2011).
  * Gumbs et al. (2014a) G. Gumbs, A. Iurov, D. Huang, and L. Zhemchuzhna, Physical Review B 89, 241407 (2014a).
  * Pyatkovskiy and Gusynin (2011) P. Pyatkovskiy and V. Gusynin, Physical Review B 83, 075422 (2011).
  * Checkelsky and Ong (2009) J. G. Checkelsky and N. Ong, Physical Review B 80, 081413 (2009).
  * Huang et al. (2019) D. Huang, A. Iurov, H.-Y. Xu, Y.-C. Lai, and G. Gumbs, Physical Review B 99, 245412 (2019).
  * Kim and Kim (2008) W. Y. Kim and K. S. Kim, Nature nanotechnology 3, 408 (2008).
  * Cheianov and Fal’ko (2006) V. V. Cheianov and V. I. Fal’ko, Physical review b 74, 041403 (2006).
  * Pereira et al. (2008) V. M. Pereira, V. N. Kotov, and A. C. Neto, Physical Review B 78, 085101 (2008).
  * Berger et al. (2006) C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, T. Li, J. Hass, A. N. Marchenkov, et al., Science 312, 1191 (2006).
  * Ni et al. (2008) Z. H. Ni, T. Yu, Y. H. Lu, Y. Y. Wang, Y. P. Feng, and Z. X. Shen, ACS nano 2, 2301 (2008).
  * Iurov et al. (2011) A. Iurov, G. Gumbs, O. Roslyak, and D. Huang, Journal of Physics: Condensed Matter 24, 015303 (2011).
  * Kindermann et al. (2012) M. Kindermann, B. Uchoa, and D. L. Miller, Physical Review B 86, 115415 (2012).
  * Zhou et al. (2007) S. Y. Zhou, G.-H. Gweon, A. Fedorov, d. First, PN, W. De Heer, D.-H. Lee, F. Guinea, A. C. Neto, and A. Lanzara, Nature materials 6, 770 (2007).
  * Iurov et al. (2013) A. Iurov, G. Gumbs, O. Roslyak, and D. Huang, Journal of Physics: Condensed Matter 25, 135502 (2013).
  * Iurov et al. (2019) A. Iurov, G. Gumbs, and D. Huang, Physical Review B 99, 205135 (2019).
  * Oka and Aoki (2009) T. Oka and H. Aoki, Physical Review B 79, 081406 (2009).
  * Liu et al. (2011) Y. Liu, G. Bian, T. Miller, and T.-C. Chiang, Physical review letters 107, 166803 (2011).
  * Iurov et al. (2020a) A. Iurov, G. Gumbs, and D. Huang, Journal of Physics: Condensed Matter 32, 415303 (2020a).
  * Pastrana-Martinez et al. (2012) L. M. Pastrana-Martinez, S. Morales-Torres, V. Likodimos, J. L. Figueiredo, J. L. Faria, P. Falaras, and A. M. Silva, Applied Catalysis B: Environmental 123, 241 (2012).
  * Usaj et al. (2014) G. Usaj, P. M. Perez-Piskunow, L. F. Torres, and C. A. Balseiro, Physical Review B 90, 115423 (2014).
  * Iurov et al. (2017a) A. Iurov, L. Zhemchuzhna, G. Gumbs, and D. Huang, Journal of Applied Physics 122, 124301 (2017a).
  * Kibis (2010) O. Kibis, Physical Review B 81, 165433 (2010).
  * Pervishko et al. (2015) A. Pervishko, O. V. Kibis, S. Morina, and I. Shelykh, Physical Review B 92, 205403 (2015).
  * Portnoi et al. (2008) M. Portnoi, O. Kibis, and M. R. Da Costa, Superlattices and Microstructures 43, 399 (2008).
  * Kibis et al. (2017) O. Kibis, K. Dini, I. Iorsh, and I. Shelykh, Physical Review B 95, 125401 (2017).
  * Iurov et al. (2017b) A. Iurov, G. Gumbs, and D. Huang, Journal of Modern Optics 64, 913 (2017b).
  * Dey and Ghosh (2018) B. Dey and T. K. Ghosh, Physical Review B 98, 075422 (2018).
  * Brey and Fertig (2006) L. Brey and H. Fertig, Physical Review B 73, 235411 (2006).
  * Kristinsson et al. (2016) K. Kristinsson, O. V. Kibis, S. Morina, and I. A. Shelykh, Scientific reports 6, 1 (2016).
  * Iurov et al. (2020b) A. Iurov, L. Zhemchuzhna, D. Dahal, G. Gumbs, and D. Huang, Physical Review B 101, 035129 (2020b).
  * Low and Appenzeller (2009) T. Low and J. Appenzeller, Physical Review B 80, 155406 (2009).
  * Pedersen et al. (2009) T. G. Pedersen, A.-P. Jauho, and K. Pedersen, Physical Review B 79, 113406 (2009).
  * Pereira et al. (2006) V. M. Pereira, F. Guinea, J. L. Dos Santos, N. Peres, and A. C. Neto, Physical review letters 96, 036801 (2006).
  * Gumbs et al. (2014b) G. Gumbs, A. Balassis, A. Iurov, and P. Fekete, The Scientific World Journal 2014 (2014b).
  * Castro et al. (2008) E. V. Castro, N. Peres, J. L. dos Santos, A. C. Neto, and F. Guinea, Physical review letters 100, 026802 (2008).
  * Rycerz et al. (2007) A. Rycerz, J. Tworzydło, and C. Beenakker, EPL (Europhysics Letters) 79, 57003 (2007).
  * Jang et al. (2013) M. S. Jang, H. Kim, Y.-W. Son, H. A. Atwater, and W. A. Goddard, Proceedings of the National Academy of Sciences 110, 8786 (2013).
  * Dahal and Gumbs (2017) D. Dahal and G. Gumbs, Journal of Physics and Chemistry of Solids 100, 83 (2017).
  * Azarova and Maksimova (2014) E. Azarova and G. Maksimova, Physica E: Low-dimensional Systems and Nanostructures 61, 118 (2014).
  * Navarro-Giraldo and Quimbay (2018) J. Navarro-Giraldo and C. Quimbay, Journal of Physics: Condensed Matter 30, 265304 (2018).
  * Sonin (2009) E. Sonin, Physical Review B 79, 195438 (2009).
  * Weekes et al. (2021) N. Weekes, A. Iurov, L. Zhemchuzhna, G. Gumbs, and D. Huang, Physical Review B 103, 165429 (2021).
  * Stander et al. (2009) N. Stander, B. Huard, and D. Goldhaber-Gordon, Physical review letters 102, 026807 (2009).
  * Libisch et al. (2017) F. Libisch, T. Hisch, R. Glattauer, L. Chizhova, and J. Burgdörfer, Journal of Physics: Condensed Matter 29, 114002 (2017).
  * Young and Kim (2009) A. F. Young and P. Kim, Nature Physics 5, 222 (2009).
  * Wang et al. (2019) K. Wang, M. M. Elahi, L. Wang, K. M. Habib, T. Taniguchi, K. Watanabe, J. Hone, A. W. Ghosh, G.-H. Lee, and P. Kim, Proceedings of the National Academy of Sciences 116, 6575 (2019).
  * Mouhafid and Jellal (2013) A. E. Mouhafid and A. Jellal, arXiv preprint arXiv:1303.0559 (2013).
  * Shytov et al. (2008) A. V. Shytov, M. S. Rudner, and L. S. Levitov, Physical review letters 101, 156804 (2008).
  * Anwar et al. (2017) F. Anwar, C. Carlos, V. Saraswat, V. Mangu, M. Arnold, and F. Cavallo, Aip Advances 7, 115015 (2017).
  * Anwar (2020) F. Anwar (2020).
  * Titov (2007) M. Titov, EPL (Europhysics Letters) 79, 17004 (2007).
  * Ando and Koshino (1984) T. Ando and M. Koshino, Phys. Rev. Lett 53, 2449 (1984).
  * Anwar et al. (2020) F. Anwar, A. Iurov, D. Huang, G. Gumbs, and A. Sharma, Physical Review B 101, 115424 (2020).
  * Huang et al. (1999) D. Huang, A. Singh, and D. Cardimona, Physics Letters A 259, 488 (1999).
  * Hernández and Lewenkopf (2012) A. R. Hernández and C. H. Lewenkopf, Physical Review B 86, 155439 (2012).
  * Tworzydło et al. (2008) J. Tworzydło, C. Groth, and C. Beenakker, Physical Review B 78, 235438 (2008).
  * Huang et al. (2020) D. Huang, F. Anwar, A. Iurov, G. Gumbs, and A. Sharma, Bulletin of the American Physical Society 65 (2020).
  * Setare and Jahani (2010) M. Setare and D. Jahani, Physica B: Condensed Matter 405, 1433 (2010).
  * Pyatkovskiy (2008) P. Pyatkovskiy, Journal of Physics: Condensed Matter 21, 025506 (2008).
  * Iurov et al. (2017c) A. Iurov, G. Gumbs, D. Huang, and L. Zhemchuzhna, Journal of Applied Physics 121, 084306 (2017c).

Figure 1: (color online) Polar plots for the transmission coefficient
$T(\varepsilon,\phi_{\bf k}\,|\,W_{B})$ as a function of incidence angle
$\phi_{\bf k}$ in graphene with a bandgap $\Delta_{G}=50\,$meV for a square
potential barrier $V_{s}(x)=V_{0}\,\Theta(x)\,\Theta(W_{B}-x)$ with various
barrier widths $W_{B}$. Here, both the analytical expressions (solid curves)
and numerical results (dashed curves) obtained by using the FDA method are
plotted together for comparisons. Panel $(a)$ displays calculated
$T(\varepsilon,\phi_{\bf k}\,|\,W_{B})$ for $W_{B}=110\,$nm, while panel $(b)$
shows $T(\varepsilon,\phi_{\bf k}\,|\,W_{B})$ for $W_{B}=50\,$nm, where the
barrier height is $V_{0}=285\,$meV, the bias field ${\cal E}_{0}=0$, and the
incident-electron energy $\varepsilon=80\,$meV are assumed.  Figure 2: (color
online) Calculated transmission coefficients $T(\varepsilon,0\,|\,W_{B})$ for
the head-on collision $\phi_{\bf k}=0$ through a square-barrier potential
$V_{s}(x)$ as functions of bandgap parameter $\Delta_{G}$ with various barrier
widths $W_{B}$ in panel $(a)$ and with different incident-electron energies
$\varepsilon$ in panel $(b)$. Here, $V_{0}=0.2\,$eV and the unit-energy
$E_{0}=0.1\,$eV are chosen for both panels. Figure 3: (color online) (Left)
comparison of the FDA calculated transmission coefficients
$T(\varepsilon,\phi_{\bf k})$ as a function of the angle of incidence
$\phi_{\bf k}$ for three different unbiased barrier-potential profiles with
$V_{0}=285\,$meV, ${\cal E}_{0}=0$, $\Delta_{G}=50\,$meV, and
$\varepsilon=80\,$meV, including a square (black), a triangular (red) and
Gaussian (brown) potential barrier. (Right) corresponding potential-barrier
profiles $V_{B}(x)$ chosen for our computation results presented in the left
panel. Figure 4: (color online) Density plots for the transmission coefficient
$T(\varepsilon,\phi_{\bf k}\,|\,V_{B})$ as a function of the incoming-particle
energy $\varepsilon$ and incidence angle $\phi_{\bf k}$ for various barrier
profiles $V_{B}(x)$ in four different cases: (a) a square barrier without gap;
(b) a square barrier with a gap $\Delta_{G}=50\,$meV; (c) a triangular barrier
with a gap $\Delta_{G}=50\,$meV; (d) Gaussian potential barrier with a gap
$\Delta_{G}=50\,$meV. For all plots, the bias field ${\cal E}_{0}=0$, the
barrier width is $W_{B}=110\,$nm, and its height is $V_{0}=285\,$meV. Figure
5: (color online) Relative conductance $\sigma/\sigma_{0}$ as a function of
the incident-electron energy $\varepsilon$ for three different barrier
profiles, i.e., square (black), triangular (red) and Gaussian (brown) as
depicted, in gapped graphene. Here, we set the barrier height
$V_{0}=285\,$meV, the barrier width $W_{B}=100\,$nm, the gap parameter
$\Delta_{G}=50\,$meV, and the bias field ${\cal E}_{0}=0$, similar to those
used in Fig. 3. The unit for the conductance is chosen to be
$\sigma_{0}=2\,e^{2}/\pi\hbar$.  Figure 6: (color online) Transmission
coefficient $T(\varepsilon,\phi_{\bf k}\,|\,V_{B})$, calculated by FDA method,
as a function of the angle of incidence $\phi_{\bf k}$ for various bandgaps in
graphene and different barrier profiles $V_{B}(x)$: $(a)$ square barrier;
$(b)$ Gaussian barrier; $(c)$ triangular barrier. Here, we set the barrier
width $W_{B}=110\,$nm, the barrier height $V_{0}=285\,$meV, the bias field
${\cal E}_{0}=0$, and the incident-electron energy $\varepsilon=80\,$meV.
Figure 7: (color online) Polar plots of $T(\varepsilon,\phi_{\bf k}\,|\,{\cal
E}_{0})$ as a function of $\phi_{\bf k}$ for gapped graphene with
$\Delta_{G}=50\,$meV over a square barrier with its height $V_{0}=285\,$meV,
width $W_{B}=110\,$nm, as well as various applied bias fields ${\cal E}_{0}$.
Here, Panels $(a)$-$(b)$ show the results for fixed $\varepsilon=80\,$meV
corresponding to a reverse and a forward bias, respectively. Panels
$(c)$-$(d)$ display the change of relative conductance $\sigma/\sigma_{0}$ as
functions of $\varepsilon$ for a reverse and forward bias, separately. The
unit for the conductance in panels $(c)$ and $(d)$ is
$\sigma_{0}=2e^{2}/\pi\hbar$.  Figure 8: (color online) Numerically-calculated
relative conductance $\sigma/\sigma_{0}$ as a function of $\phi_{\bf k}$ for
gapped graphene with $\Delta_{G}=50\,$meV over a triangular barrier with its
height $V_{0}=285\,$meV, width $W_{B}=110\,$nm, as well as various applied
bias fields ${\cal E}_{0}$. Panels $(a)$ and $(b)$ display the change of
relative conductance $\sigma/\sigma_{0}$ as functions of $\varepsilon$ for a
reverse and a forward bias, separately. The unit for the conductance in these
two panels is $\sigma_{0}=2e^{2}/\pi\hbar$.  Figure 9: (color online) 2D plots
of tunneling conductance as functions of applied bias ${\cal E}_{0}$ and
incident-electron energy $\varepsilon$. Here, four panels correspond to $(a)$
Gaussian barrier with forward bias; $(b)$ Gaussian barrier with reverse bias;
$(c)$ triangular barrier with forward bias; $(d)$ triangular barrier with
reverse bias. For all cases, we choose the barrier width $W_{B}=110\,$nm, the
barrier height $V_{0}=285\,$meV, and the energy gap $\Delta_{G}=50\,$meV. The
unit for the conductance in panels $(a)$-$(d)$ is set as
$\sigma_{0}=2e^{2}/\pi\hbar$.  Figure 10: (color online) 2D plots for
$\sigma/\sigma_{0}$ as functions of incident energy $\varepsilon$ and
scattering strength $V_{d}$ (both positive and negative) for the case of a
square barrier with width $W_{B}=100\,$nm, height $V_{0}=285\,$meV and zero
bias ${\cal E}_{0}=0$. Here, panel $(a)$ presents the result for the situation
with no scatterer, while panels $(b)$-$(d)$ correspond to $\sigma/\sigma_{0}$
with a single scatterer sitting at various positions
$x_{s}/W_{B}=0.1,\,0.5,\,0.9$. The unit for the conductance in panels
$(a)$-$(d)$ is set as $\sigma_{0}=2e^{2}/\pi\hbar$. Figure 11: (color online)
2D plots for $\sigma/\sigma_{0}$ as functions of incident energy $\varepsilon$
and scattering strength $V_{d}$ (both positive and negative) for the case of a
triangular barrier with width $W_{B}=100\,$nm, height $V_{0}=285\,$meV and
zero bias ${\cal E}_{0}=0$. Here, panel $(a)$ presents the result for the
situation with no scatterer, while panels $(b)$-$(f)$ correspond to
$\sigma/\sigma_{0}$ with a single scatterer sitting at various positions
$x_{s}/W_{B}=0.1,\,0.3,\,0.5,\,0.7,\,0.9$, respectively. The unit for the
conductance in panels $(a)$-$(f)$ is set as $\sigma_{0}=2e^{2}/\pi\hbar$.
Figure 12: (color online) 2D plots for $\sigma/\sigma_{0}$ as functions of
incident energy $\varepsilon$ and scattering strength $V_{d}$ (both positive
and negative) for the case of Gaussian barrier with width $W_{B}=100\,$nm,
height $V_{0}=285\,$meV and zero bias ${\cal E}_{0}=0$. Here, panel $(a)$
presents the result for the situation with no scatterer, while panels
$(b)$-$(f)$ correspond to $\sigma/\sigma_{0}$ with a single scatterer sitting
at various positions $x_{s}/W_{B}=0.1,\,0.3,\,0.5,\,0.7,\,0.9$, respectively.
The unit for the conductance in panels $(a)$-$(f)$ is set as
$\sigma_{0}=2e^{2}/\pi\hbar$.


# On real-time multi-stage speech enhancement systems

###### Abstract

Recently, multi-stage systems have stood out among deep learning-based speech
enhancement methods. However, these systems are always high in complexity,
requiring millions of parameters and powerful computational resources, which
limits their application for real-time processing in low-power devices.
Besides, the contribution of various influencing factors to the success of
multi-stage systems remains unclear, which presents challenges to reduce the
size of these systems. In this paper, we extensively investigate a lightweight
two-stage network with only 560k total parameters. It consists of a Mel-scale
magnitude masking model in the first stage and a complex spectrum mapping
model in the second stage. We first provide a consolidated view of the roles
of gain power factor, post-filter, and training labels for the Mel-scale
masking model. Then, we explore several training schemes for the two-stage
network and provide some insights into the superiority of the two-stage
network. We show that the proposed two-stage network trained by an optimal
scheme achieves a performance similar to a four times larger open source model
DeepFilterNet2 [1].

Index Terms—  Speech enhancement, real-time, multi-stage network, deep
learning

##  1 Introduction

Speech enhancement, intended to improve the quality of speech signals
corrupted by additive noise, is a fundamental task in audio signal processing
with various applications, including mobile telecommunication, robust
automatic speech and speaker recognition [2], and hearing aids. Over the past
decade, speech enhancement approaches based on deep learning have seen large
attention and success. The reported deep speech enhancement models either
predict the magnitude-only target such as magnitude spectrum and ideal ratio
mask [3, 4, 5], or phase-related targets like complex ratio mask [6, 7],
phase-sensitive mask [8], and complex spectrum [9]. Recently, a series of
multi-stage systems dominates the mainstream benchmarks [10, 11, 12, 13]. They
usually predict magnitude-only targets in the first stage and then predict
complex spectrum in the second stage. However, these reported multi-stage
networks normally have an overall size larger than 2M, which hinders their
deployment on battery-operated devices like earbuds and headsets. Besides, the
contributions of individual components to the overall performance of the
multi-stage systems remain unclear, which poses challenges for resizing the
multi-stage networks. Focusing on real-time and on-device applications, we
provide a consolidated view of the roles of each individual component of
multi-stage systems based on a lightweight two-stage network with 560k total
parameters.

The Mel-scale is a perceptual scale imitating the frequency resolution of the
human hearing system, which intendeds to tackle the perceptual nonlinearity in
the linear frequency scale [14]. It compresses the redundant high-frequency
information and amplifies the critical low-frequency information. By using the
Mel-scale spectrum, we can use fewer frequency bands without losing much
important frequency information, and thus significantly reduce the model size
while not degrading the performance a lot compared with the linear scale
model. Based on this idea, we first extensively investigated a masking model
that predicts the oracle gains [4, 5] in Mel scale. After magnitude masking,
gain post-filters are frequently used [1, 4, 5]. However, sometimes the gain
post-filters instead cause performance degradation. By understanding the roles
of post-filters and the gain power factor in the training loss, we found that
the post-filter only contributes positively when it performs a complementary
role to the gain power factor. Besides, inspired by [15], we include the post-
processing steps of Mel-scale masking into the training graph and supervise
the model training with time-domain loss functions, which significantly
reduces the speech distortion and further enhances the perceptual speech
quality, especially in low signal-to-noise ratio (SNR) scenarios.

Nonetheless, there exist two sources of distortion inherent in Mel-scale
masking. The first is a frequency information loss caused by Mel compression,
while the second is phase-related loss due to using a noisy phase when
reconstructing the enhanced speech. The former will be shown to be
insignificant compared to (uncompressed) frequency-scale masking. However, the
phase-related distortion will be demonstrated as significant, especially in
low SNR scenarios. Thus, a second-stage model, which predicts the clean
complex spectrum from the first-stage enhanced spectrum and the original noisy
spectrum, is introduced to reduce the phase-related distortions and further
suppress the residual noise. After a consolidated exploration of two-stage
training schemes, we obtained a two-stage model that significantly improved
the enhanced speech quality and reduced speech distortions beyond the single-
stage models. Moreover, the proposed two-stage model performs similarly to a
four times larger open-source two-stage model DeepFilterNet2 [1].

Fig. 1: Two-stage speech enhancement pipeline. Top: masking-based magnitude
stage, bottom: mapping-based complex stage.

##  2 Light-weight Two stage speech enhancement system

###  2.1 Speech enhancement pipeline

The two-stage speech enhancement pipeline is shown in Figure 1. After the
Short-Time Fourier Transform (STFT) of the noisy signal, the derived magnitude
spectrum in linear frequency scale is converted to Mel scale by applying Mel
filter bank. The Mel spectrum is compressed through a natural logarithm before
being fed into the neural network. The first stage, the Mel-scale masking
model, is trained to predict the Mel gains. The oracle Mel gains are defined
by Eq. (1), where $|S|$ denotes the target speech magnitude, $|X|$ denotes the
noisy mixture magnitude, and the subscript $Mel$ denotes corresponding
quantities in the Mel scale.

$g_{Mel}=\frac{|S|_{Mel}}{|X|_{Mel}}$ (1)

The predicted Mel gains must be interpolated back to the linear frequency
scale before being applied to the noisy spectrum. The interpolation is done
through the transpose of the Mel filter bank. The mask can be applied directly
or after post-filtering. After masking the noisy magnitude, the noisy phase is
combined with enhanced magnitude to obtain the enhanced spectrum and then to
obtain the first stage enhanced speech audio.

After the first stage network, the enhanced complex spectrum is concatenated
with the original, noisy complex spectrum and then fed into the second stage
network to predict the clean complex spectrum. We separate the complex
spectrum into the real and imaginary parts and then concatenate the real and
imaginary parts for input and output of the network. Thus, the second stage
network is still real-valued.

###  2.2 Model architecture

As audio signals are time-dependent objects, recurrent networks would be
natural options for deep speech enhancement models [2, 16]. However, Sach et
al. [17] demonstrated that the convolutional recurrent networks show a
significant advantage compared with recurrent networks. Tan et al. [18]
reported an efficient architecture that combines Unet convolution layers with
long short-term memory networks (LSTM). Inspired by this work, we designed an
Unet-GRU architecture, replacing the heavy LSTM with more efficient Gated
Recurrent Units (GRU). Focusing on real-time applications, causal convolution
is used in both networks. The kernel size is 1 (the time axis) and 3 (the
frequency axis). Skip connections of Unet are designed to avoid the gradient
vanishing issues.

As shown in Figure 1, the Unet-GRU architecture is used in both stages. The
first stage network has 300k trainable parameters. There are five downsample
blocks and five corresponding upsample blocks. Each downsample block contains
two convolution layers and 1 batch normalization layer. Each upsample block
contains two transposed convolution layers and 1 batch normalization layer.
The input is single-channel with 64 Mel bands, and the initial channel after
the first downsample block is 8. There are two recurrent layers between the
encoder and decoder, with 64 GRUs in each layer. The second stage model takes
the input of 512 linear bands (full band) with four channels and has 260k
parameters. It has 16 initial channels after the first downsample block, three
downsample blocks, and three upsample blocks. There are 16 GRUs in the first
recurrent layer and 64 GRUs in the second recurrent layer.

###  2.3 Loss functions and gain post-filter

For the magnitude masking models, the gain loss function is commonly used [4,
5]. It is defined by (2), which is the mean square error of oracle gains $g$
and estimated gains $\hat{g}$ to the power of $p$. As gain values range in
$[0,1]$, the power factor $p$ controls which bins should be emphasized during
learning. For example, if $p=0.5$, the contribution of the relatively noisy
bins where $g$ is close to $0$ is amplified, while the contribution of
relatively clean bins where $g$ is close to $1$ is reduced. In this case, the
model is encouraged to attenuate more noise. On the other hand, if $p=2$, the
model is trained to mildly attenuate noise while paying more attention to
remaining speech.

$L_{g}=\frac{1}{TF}\sum_{t}^{T}\sum_{f}^{F}(g^{p}_{t,f}-\hat{g}^{p}_{t,f})^{2}$
(2)

After predicting the Mel gains and interpolating back to the frequency scale,
an element-wise post-filter [4, 1], defined in Eq. (3), is sometimes used
before applying gains on the noisy magnitude. The post-filter is designed to
attenuate noise in the noisy frequency bins further while maintaining the
predicted gain values in the other bins. Thus, the post-filter has a similar
role of the gain loss function with $p=0.5$ and contributes complementarily to
the gain loss with $p=2$.

$g^{\prime}=\hat{g}\sin{\left(\frac{\pi}{2}\hat{g}\right)}$ (3)

(a) Spectral

(b) Qualitative

Fig. 2: Illustration of phase importance: a) Spectral ($N$: additive noise;
$S$: target speech; $X$: noisy mixture; $\hat{S}_{1}$: ideal spectral vector
estimated by using magnitude-only loss; $\hat{S}_{2}$: spectral vector using
noisy phase and closest to target speech), b) Qualitative (phase-related loss
obtained with oracle inference).

In addition to the post-filter Eq. (3), other DSP-based post-filter approaches
can be applied in the first stage network. As part of this work, we have also
applied a spectral weighting method described in [19]. By exploiting the
masking properties of the human auditory system, the spectral weighting rule
is designed to mask the distortions of the residual noise. In our case,
however, this post-filter has not given better performance compared to the
“sin” post-filter.

Supervising the Mel-scale masking model by gain loss will ignore all the
phase-related information, which could lead the model to a suboptimal point
with respect to the distance to the target speech, as shown in Figure 2a. The
bias could be evident when the angle $\alpha$ is large, i.e., in low SNR
scenarios. In fact, the differentiability of the interpolation and inverse
STFT makes it possible to supervise the Mel-scale masking model by time-domain
losses like SI-SNR defined in Eq. (7), where $s$ and $\hat{s}$ are the target
and estimated audio signals, respectively. Beyond $L_{SI-SNR}$, magnitude loss
$L_{mag}$ and asymmetric loss $L_{asym}$ are also added following [10]. Note
that the $\beta$ in $L_{mag}$ and $L_{asym}$ is the power compression factor
set to 0.5 in this work. It is intended to compress the range of magnitude so
that the model is easier to resolve the low power noise, which is different
from the gain power factor $p$.

$L_{mag}=\frac{1}{TF}\sum_{t}^{T}\sum_{f}^{F}(|S|^{\beta}_{t,f}-\hat{|S|}_{t,f}^{\beta})^{2}$
(4)

$h(x)=\begin{cases}0&\quad\text{if}\ x\leq 0\\\ x&\quad\text{if}\ x>0\\\
\end{cases}$ (5)
$L_{asym}=\frac{1}{TF}\sum_{t}^{T}\sum_{f}^{F}\left|h(|S|_{t,f}^{\beta}-\hat{|S|}_{t,f}^{\beta})\right|^{2}$
(6)

$L_{\text{SI-SNR}}=-10\log_{10}\left(\frac{\|\kappa s\|^{2}}{\|\kappa
s-\hat{s}\|^{2}}\right)$ (7) $\kappa=\arg\min_{\kappa}\|\kappa
s-\hat{s}\|^{2}=\frac{\hat{s}^{T}s}{\|s\|^{2}}$ (8)
$L_{1}=(L_{mag}+L_{asym})\cdot F+2L_{\text{SI-SNR}}$ (9)

There are also many loss options for complex spectrum learning. Sebastian et
al. [20] made a consolidated view for the commonly used loss items and
concluded that combining magnitude-only with phase-aware objectives always
leads to improvements. Thus, we also combine magnitude loss and phase-aware
loss as our second-stage loss $L_{2}$, which is also the final loss of our
two-stage network. The phase-aware loss is defined in Eq. (10), where
$\theta_{t,f}$ and $\hat{\theta}_{t,f}$ are the phase of target and estimated
spectrum in each bin.

$L_{phase}=\frac{1}{TF}\sum_{t}^{T}\sum_{f}^{F}(|S|^{p}_{t,f}\exp(i\theta_{t,f})-\hat{|S|}_{t,f}^{p}\exp(i\hat{\theta}_{t,f}))^{2}$
(10) $L_{2}=L_{mag}+L_{phase}$ (11)

##  3 Experiment results and analysis

###  3.1 Experiment setup

The training samples and test samples are generated from the DNS Challenge
dataset by randomly mixing the speech samples and noise samples. We have
$72000$ samples for training and $500$ samples for validation during training.
Each audio is in 10 seconds. Thus, the total length of training samples is 200
hours. The sample rate is 32 kHz, and the hop length is 320. Thus, there are
1000 time frames in the spectrum. We used the standard Hann window with a
length of 640. For test samples, as we would like to observe the performance
under different SNR scenarios, we generate a noisy mixture with specific SNR.
We have 100 independent mixtures of speech and noise, and each mixture has 8
SNR bands ranging from -5 to 30 with a step of 5. Thus, we have 800 test
samples in total. The enhanced speech performance is evaluated by PESQ [21]
and SI-SDR [22].

###  3.2 Mel-scale masking model investigation

In order to understand the systematic distortions of the Mel-scale masking
model, we first conducted oracle inference, i.e., the performance enhanced by
oracle gains. The oracle inference results are shown in Figure 2b. The gap
between oracle gains and oracle Mel gains is the loss caused by compressing
512 linear bands to 64 Mel bands, which is much smaller than the loss caused
by ignoring phase information. The phase-related loss is especially
significant in low SNR scenarios. The oracle inference results partially
demonstrated the rationality of Mel compression, which remarkably reduces the
model size while causing relatively small systematic distortions. The oracle
inference results confirm the importance of phase information, particularly in
low SNR scenarios.

The inference results for the Mel-scale masking model are shown in Figure 3.
As discussed before, when the gain power factor $p$ is set to 0.5, the model
pays more attention to noisy bins and tends to attenuate the noise
aggressively. In this case, applying the post-filter could degrade the
enhanced speech quality and increase speech distortion due to their
overlapping roles. On the other hand, as the model trained by gain power
factor $p=2$ pays more attention to cleaner bins, the noise is not
sufficiently canceled. Thus, the enhanced speech quality is significantly
lower than the model trained by $p=0.5$ in terms of PESQ. However, after
applying the post-filter to attenuate more noise in noisy bins based on the
prior knowledge from the model, the PESQ is significantly enhanced. More
interestingly, due to the complementarity, the postfiltered gains perform even
better than the aggressive denoising model trained by $p=0.5$. As the model
trained by $L_{1}$ can see the phase information, the speech distortions are
evidently reduced according to SI-SDR results, and PESQ is also improved in
low SNR compared with the one learned by gain loss. The models could degrade
the SI-SDR in very high SNR as SI-SDR is not sensitive to residual noise and
mainly measures speech distortions.

(a) Speech quality

(b) Speech distortion ratio

Fig. 3: The impact of the gain power factor $p$, the postfilter $pf$ and loss
functions on: a) PESQ, b) SI-SDR

###  3.3 Two-stage training schemes comparison

The inference results for different two-stage training schemes are listed in
Table 1. Direct joint training schemes perform moderately because the first
stage model could not be optimized sufficiently due to the gradient vanishing
issues. The separate training scheme with the first stage model trained by
gain loss performs even worse than direct joint training and significantly
worse than other schemes. In contrast, the separate training scheme with the
first stage model trained by $L_{1}$ performs similar to the optimal one.
These indicate that the first stage model trained by $L_{g}$ still has a
distance to the two stage global optimum while the one trained by $L_{1}$
almost reaches the global optimum. Other training schemes performs similarly,
which indicates that these two-stage models almost reach the optimum. By
comparing both PESQ and SI-SDR, the best training scheme is considered to be
training the first-stage model with $L_{1}$ and then jointly training the two-
stage model and the trained two-stage model is used for the following
comparison.

Table 1: Two stage training schemes comparison (”$\pm$” represents 95% confidence intervals) Training schemes | PESQ | SI-SDR
---|---|---
joint |  2.53 $\pm$ 0.05 |  17.15 $\pm$ 0.55
stage1($L_{g}$)-stage2 |  2.52 $\pm$ 0.07 |  16.89 $\pm$ 0.59
stage1($L_{g}$)-joint |  2.57 $\pm$ 0.06 |  17.51 $\pm$ 0.56
stage1($L_{g}$)-stage2-joint |  2.59 $\pm$ 0.07 |  17.44 $\pm$ 0.58
stage1($L_{1}$)-stage2 |  2.57 $\pm$ 0.05 |  17.57 $\pm$ 0.56
stage1($L_{1}$)-joint |  2.58 $\pm$ 0.06 |  17.58 $\pm$ 0.55
stage1($L_{1}$)-stage2-joint |  2.59 $\pm$ 0.07 |  17.51 $\pm$ 0.58

To demonstrate the superiority of our proposed two-stage network, we trained a
single-stage Unet-GRU complex spectrum mapping model, and then we tested the
model as well as a baseline model DeepFilterNet2 [1] on our test dataset. The
DeepFilterNet2 is an open source model also based on two-stage learning. It
was originally proposed for real-time speech enhancement but also applied in
hearing aids [23]. The state-of-the-art single stage complex model was
proposed by Ristea et al.[24], but the size exceeds 7.5M. Hence we still use
our Unet-GRU based single stage complex model for comparison. The overall
results with statistical significance test are listed in Table 2. We found
that the two-stage model significantly outperforms both single-stage models in
terms of PESQ and SI-SDR. Besides, our two-stage model performs similarly to
DeepFilterNet2, which has 2M parameters. The complex spectrum without Mel
transformation contains much redundant information, which hinders sufficient
learning under limited model capacity. These results potentially demonstrate
the superiority of a two-stage framework with Mel-scale masking in the first
stage and complex spectrum learning in the second stage.

Table 2: Comparison of two-stage model, single-stage models, and baseline model (”$\pm$” represents 95% confidence intervals) | PESQ | SI-SDR | #Params
---|---|---|---
noisy |  1.81 $\pm$ 0.05 |  12.50 $\pm$ 0.82 | -
DeepFilter2 | 2.60 $\pm$ 0.05 | 16.99 $\pm$ 0.50 | 2.1M
Mel($L_{1}$) |  2.43 $\pm$ 0.06 |  16.99 $\pm$ 0.53 | 300k
Complex |  2.45 $\pm$ 0.07 |  17.01 $\pm$ 0.54 | 600k
Two-stage | 2.58 $\pm$ 0.06 | 17.58 $\pm$ 0.55 | 560k

##  4 Conclusions

This paper provides a consolidated view of the roles of individual components
in a multi-stage speech enhancement system. Focusing on real-time and on-
device applications, we propose a lightweight two-stage network. Through the
study on the Mel-scale masking model, we demonstrate the positive contribution
that could be made by complementarily combining over-attenuation post-filter
and gain power factor. We also show that supervising the Mel-scale masking
model with a phase-visible objective could improve speech quality and
significantly reduce speech distortions compared with a magnitude-only
objective due to the significant systematic phase loss in low SNR scenarios.
Besides, exploring two-stage training schemes also confirms the global
optimality of the phase-visible objective for first-stage model training. We
also show that training the first stage model and then conducting joint
training could be a potential best two-stage training scheme. Finally, we
demonstrate that our two-stage model outperforms single-stage models and
performs similarly to a baseline open source model DeepFilterNet2, with a four
times larger size, which shows the superiority of the proposed two-stage
speech enhancement framework.

## References

  * [1] Hendrik Schröter, A Maier, AN Escalante-B, and T Rosenkranz,  “Deepfilternet2: Towards real-time speech enhancement on embedded devices for full-band audio,”  in 2022 International Workshop on Acoustic Signal Enhancement (IWAENC). IEEE, 2022, pp. 1–5.
  * [2] DeLiang Wang and Jitong Chen,  “Supervised speech separation based on deep learning: An overview,”  IEEE/ACM Transactions on Audio, Speech, and Language Processing, vol. 26, no. 10, pp. 1702–1726, 2018.
  * [3] Yuxuan Wang, Arun Narayanan, and DeLiang Wang,  “On training targets for supervised speech separation,”  IEEE/ACM transactions on audio, speech, and language processing, vol. 22, no. 12, pp. 1849–1858, 2014.
  * [4] Jean-Marc Valin, Umut Isik, Neerad Phansalkar, Ritwik Giri, Karim Helwani, and Arvindh Krishnaswamy,  “A perceptually-motivated approach for low-complexity, real-time enhancement of fullband speech,”  arXiv preprint arXiv:2008.04259, 2020.
  * [5] Jean-Marc Valin,  “A hybrid dsp/deep learning approach to real-time full-band speech enhancement,”  in 2018 IEEE 20th international workshop on multimedia signal processing (MMSP). IEEE, 2018, pp. 1–5.
  * [6] Yanxin Hu, Yun Liu, Shubo Lv, Mengtao Xing, Shimin Zhang, Yihui Fu, Jian Wu, Bihong Zhang, and Lei Xie,  “Dccrn: Deep complex convolution recurrent network for phase-aware speech enhancement,”  arXiv preprint arXiv:2008.00264, 2020.
  * [7] Donald S Williamson, Yuxuan Wang, and DeLiang Wang,  “Complex ratio masking for monaural speech separation,”  IEEE/ACM transactions on audio, speech, and language processing, vol. 24, no. 3, pp. 483–492, 2015.
  * [8] Hakan Erdogan, John R Hershey, Shinji Watanabe, and Jonathan Le Roux,  “Phase-sensitive and recognition-boosted speech separation using deep recurrent neural networks,”  in 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2015, pp. 708–712.
  * [9] Ke Tan and DeLiang Wang,  “Complex spectral mapping with a convolutional recurrent network for monaural speech enhancement,”  in ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2019, pp. 6865–6869.
  * [10] Yukai Ju, Wei Rao, Xiaopeng Yan, Yihui Fu, Shubo Lv, Luyao Cheng, Yannan Wang, Lei Xie, and Shidong Shang,  “Tea-pse: Tencent-ethereal-audio-lab personalized speech enhancement system for icassp 2022 dns challenge,”  in ICASSP 2022-2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2022, pp. 9291–9295.
  * [11] Andong Li, Wenzhe Liu, Xiaoxue Luo, Chengshi Zheng, and Xiaodong Li,  “Icassp 2021 deep noise suppression challenge: Decoupling magnitude and phase optimization with a two-stage deep network,”  in ICASSP 2021-2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2021, pp. 6628–6632.
  * [12] Yukai Ju, Shimin Zhang, Wei Rao, Yannan Wang, Tao Yu, Lei Xie, and Shidong Shang,  “Tea-pse 2.0: Sub-band network for real-time personalized speech enhancement,”  in 2022 IEEE Spoken Language Technology Workshop (SLT). IEEE, 2023, pp. 472–479.
  * [13] Xiaopeng Yan, Yindi Yang, Zhihao Guo, Liangliang Peng, and Lei Xie,  “The npu-elevoc personalized speech enhancement system for icassp2023 dns challenge,”  in ICASSP 2023-2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2023, pp. 1–2.
  * [14] Tae Gyoon Kang, Jong Won Shin, and Nam Soo Kim,  “Dnn-based monaural speech enhancement with temporal and spectral variations equalization,”  Digital Signal Processing, vol. 74, pp. 102–110, 2018.
  * [15] Ashutosh Pandey and DeLiang Wang,  “A new framework for supervised speech enhancement in the time domain.,”  in Interspeech, 2018, pp. 1136–1140.
  * [16] Ashutosh Pandey, Ke Tan, and Buye Xu,  “A Simple RNN Model for Lightweight, Low-compute and Low-latency Multichannel Speech Enhancement in the Time Domain,”  in Proc. INTERSPEECH 2023, 2023, pp. 2478–2482.
  * [17] Marvin Sach, Jan Franzen, Bruno Defraene, Kristoff Fluyt, Maximilian Strake, Wouter Tirry, and Tim Fingscheidt,  “EffCRN: An Efficient Convolutional Recurrent Network for High-Performance Speech Enhancement,”  in Proc. INTERSPEECH 2023, 2023, pp. 2508–2512.
  * [18] Ke Tan and DeLiang Wang,  “A convolutional recurrent neural network for real-time speech enhancement.,”  in Interspeech, 2018, vol. 2018, pp. 3229–3233.
  * [19] Stefan Gustafsson, Peter Jax, and Peter Vary,  “A novel psychoacoustically motivated audio enhancement algorithm preserving background noise characteristics,”  Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP ’98 (Cat. No.98CH36181), vol. 1, pp. 397–400 vol.1, 1998.
  * [20] Sebastian Braun and Ivan Tashev,  “A consolidated view of loss functions for supervised deep learning-based speech enhancement,”  in 2021 44th International Conference on Telecommunications and Signal Processing (TSP). IEEE, 2021, pp. 72–76.
  * [21] Antony W Rix, John G Beerends, Michael P Hollier, and Andries P Hekstra,  “Perceptual evaluation of speech quality (pesq)-a new method for speech quality assessment of telephone networks and codecs,”  in 2001 IEEE international conference on acoustics, speech, and signal processing. Proceedings (Cat. No. 01CH37221). IEEE, 2001, vol. 2, pp. 749–752.
  * [22] Jonathan Le Roux, Scott Wisdom, Hakan Erdogan, and John R Hershey,  “Sdr–half-baked or well done?,”  in ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2019, pp. 626–630.
  * [23] Hendrik Schröter, Tobias Rosenkranz, Alberto N. Escalante-B., and Andreas Maier,  “Deep Multi-Frame Filtering for Hearing Aids,”  in Proc. INTERSPEECH 2023, 2023, pp. 3869–3873.
  * [24] Nicolae Catalin Ristea, Evgenii Indenbom, Ando Saabas, Tanel Pärnamaa, Jegor Guzhvin, and Ross Cutler,  “DeepVQE: Real Time Deep Voice Quality Enhancement for Joint Acoustic Echo Cancellation, Noise Suppression and Dereverberation,”  in Proc. INTERSPEECH 2023, 2023, pp. 3819–3823.