# Scattering of topological kink-antikink states in bilayer graphene
structures

Nassima Benchtaber  Institute of Interdisciplinary Physics and Complex Systems
IFISC (CSIC-UIB), E-07122 Palma, Spain  David Sánchez  Institute of
Interdisciplinary Physics and Complex Systems IFISC (CSIC-UIB), E-07122 Palma,
Spain  Department of Physics, University of the Balearic Islands, E-07122
Palma, Spain  Llorenç Serra  Institute of Interdisciplinary Physics and
Complex Systems IFISC (CSIC-UIB), E-07122 Palma, Spain  Department of Physics,
University of the Balearic Islands, E-07122 Palma, Spain

###### Abstract

Gapped bilayer graphene can support the presence of intragap states due to
kink gate potentials applied to the graphene layers. Electrons in these states
display valley-momentum locking, which makes them attractive for topological
valleytronics. Here, we show that kink-antikink local potentials enable
modulated scattering of topological currents. We find that the kink-antikink
coupling leads to anomalous steps in the junction conductance. Further, when
the constriction detaches from the propagating modes, forming a loop, the
conductance reveals the system energy spectrum. Remarkably, these kink-
antikink devices can also work as valley filters with tiny magnetic fields by
tuning a central gate.

##  I Introduction

For many years, there has been considerable interest in providing with
reliable platforms that can create, manipulate and detect qubits. Eventually,
these quantum information processing tasks are to be supplemented with
protected communication channels to transmit quantum states between distant
sites. Graphene has emerged as an excellent candidate in scalable solid-state
architectures due to its ultra long decoherence times for spin qubits and its
ability to host additional isospin (valley) degrees of freedom Trauzettel  _et
al._ (2007); Pereira  _et al._ (2007); Rycerz  _et al._ (2007); Recher and
Trauzettel (2010); Gunlycke and White (2011). These emerge as the $K$ and
$K^{\prime}$ points from the Dirac cones in the reciprocal space of the
graphene hexagonal lattice.

However, pristine graphene lacks a bandgap, which handicaps potential
applications of this material for nanoelectronics. This circumstance can be
surpassed with the employment of two graphene sheets (hereafter, bilayer
graphene or BLG) McCann and Koshino (2013); Rozhkov  _et al._ (2016).
Interlayer coupling in a Bernal stacking structure (common also to graphite)
generates a huge band splitting of the order of 380 meV, although two bands
still remain degenerate at the neutrality point. Further application of a
perpendicular electric field creating a potential difference between the two
layers finally lifts the electronic degeneracy Zhang  _et al._ (2009); Overweg
_et al._ (2018a). The gap thus opened can now be used to design tunnel
barriers and quantum point contacts Overweg  _et al._ (2018b); Kraft  _et al._
(2018). Unlike monolayer point contacts that are fabricated by etching Terrés
_et al._ (2016); Clericò  _et al._ (2019) and show trapped states due to edge
roughness, BLG quantum wires display clear conductance quantization steps.
Further, if two of these barriers are connected in a series the device works
as a quantum dot Eich  _et al._ (2018); Kurzmann  _et al._ (2019); Banszerus
_et al._ (2020, 2021). Therefore, robust spin or valley qubits can form in BLG
dots showing a discrete spectrum.

An even more exciting possibility arises in BLG systems. When the
perpendicular electric field becomes inhomogeneous by changing its sign in
different regions of the BLG, the domain wall separating the two opposite
fields holds topological states propagating next to the wall (edge states)
Martin  _et al._ (2008); Zarenia  _et al._ (2011). These types of domain wall
and propagating states are known as kink and kink states, respectively. The
topological character originates from the field induced band inversion and
confinement. Interestingly, the valley index remains a good quantum number
Zhang  _et al._ (2013) and as a consequence the kink states become chiral with
different valleys traveling in opposite directions along the kink Ju  _et al._
(2015). This is a consequence of the chiral symmetry that relates states with
opposite energies, valleys and propagation directions. Crucially, such
propagating states exist at zero magnetic field, in which case time-reversal
symmetry is fulfilled. Hence, if electrons are injected from the side into a
straight kink using a small dc bias, the output flux becomes valley polarized.
If the structure is built zero dimensional like a dot, the bound states are
valley degenerate but chiral. These topological states can even show Luttinger
behavior Killi  _et al._ (2010) or become massless Dirac modes Killi  _et al._
(2011) in the presence of interactions or periodic potentials, respectively.

The challenge then is how to probe and manipulate these unique kink states.
This can be achieved with pairs of gates whose voltage is tuned independently
in both the top and bottom layers Li  _et al._ (2016); Chen  _et al._ (2020).
Experimentally, a conductance of $4e^{2}/h$ is observed, Ju  _et al._ (2015);
Li  _et al._ (2016); Chen  _et al._ (2020) which demonstrates the presence of
two current-carrying spin-degenerate valley-polarized modes. Alternate
combinations of dual gates can be implemented for guiding these modes in
valley valves and beam splitters Xiao  _et al._ (2007); Qiao  _et al._ (2011);
Li  _et al._ (2018). However, signatures of disorder are detected, inducing
backscattering and intervalley mixing. What is needed is a controlled source
of backscattering that would allow to shape ballistic beam splitters for,
e.g., topological valleytronic interferometry Cheng  _et al._ (2018). Here, we
show that this is possible with a careful distribution of top/bottom gate
pairs, enabling the formation of a lateral constriction over two parallel
kinks, as sketched in Fig. 1a. The two kinks have symmetric changes in the
field polarity and opposite propagation directions for a given valley, thus
one kink being the antikink of the other Martin  _et al._ (2008); Zarenia  _et
al._ (2011). We below demonstrate that a kink-antikink constriction is able to
modulate the transmission electrostatically, what paves the way for the
fabrication of topological quantum point contacts and wires.

Figure 1:  (a) Schematic of a bilayer graphene kink-antikink system with lead
regions ($L$ and $R$) and a central scatterer ($C$). The electric field
direction on the graphene layers (gray sheets) is controlled by the voltages
applied to the nine top and bottom gates (blue and orange regions). The values
of the $V_{a}$ applied potentials to the lower gates (not shown) are reversed
with respect to the top gates. The field inversion between the blue and orange
regions creates an interface (white region) where topological modes emerge.
Additionally, a voltage $V_{s}^{(C)}$ is applied only to the central region.
Red lines represent these propagating topological modes for the valley $K$.
$K^{\prime}$ modes are obtained by reversing the arrows. Current is generated
from the left (source) terminal to the right (drain) terminal. Dimensions and
position of the central scatterer are given by $L_{x}$, $L_{y}$ and $y_{0}$,
while the asymptotic separation of the two kinks is $L^{\prime}_{y}$. (b,c)
Selected configurations representing a particular constriction (b) and side
loop (c), the latter formed when $y_{0}>L^{\prime}_{y}$. (d) Energy bands of a
translationally invariant kink-antikink wire of width
$L^{\prime}_{y}=300\,{\rm nm}$, kink potential height $V_{a}=10\,{\rm meV}$
and potential smoothness $s=38\,{\rm nm}$. Each band is fourfold degenerate
(spin and valley) in the absence of magnetic fields.

Importantly, the setup can be slightly rearranged to also furnish bound states
when the kink-antikink constriction (Fig. 1b) transforms into a side loop
(Fig. 1c). Previous works on similar BLG islands assumed a sharp-potential
kink Xavier  _et al._ (2010) or infinite-mass boundary conditions da Costa
_et al._ (2014) that create rings exhibiting Aharonov-Bohm energy levels.
However, these are closed systems and as such their properties would be
difficult to examine in an experiment. In contrast, our loops are weakly
coupled to the external (side) kinks, topological as well. We show that the
measured conductance peak pattern is caused by the system level distribution.
Therefore, the setup is most suitable for doing spectroscopy of chiral bound
states.

Let us discuss in more detail our proposal, as illustrated in Fig. 1a, and
highlight our main findings. The system consists of a BLG with the same gate
distribution in both the upper and lower graphene layers (gray sheets). The
applied potentials to the lower gates, not shown in Fig. 1a, reverse the
values for $V_{a}$ with respect to the top gates but keep the same value for
$V_{s}^{(C)}$, as detailed below in Sec. II. The changes in electric field
orientation occur in the white interfaces defining, respectively, the
topological kink and antikink that form the quasi-one dimensional (1D)
propagating channels. The arrows in Fig. 1a qualitatively indicate electron
propagation for a given valley on the BLG planes when the kink and antikink
are well separated. $x$ is the transport direction, the edge states are
confined along $y$ and the direction perpendicular to the graphene layers is
denoted with $z$ (not shown here). Electronic motion is determined by
chirality due to valley-momentum locking (we only depict states from valley
$K$).

A narrow constriction in the central region allows for a controlled
transmission of the injected beams. In our parametrization (Fig. 1a), a
constriction corresponds to having $L_{y}<L^{\prime}_{y}$ and $y_{0}\leq
L^{\prime}_{y}-L_{y}$. A particular example for a constriction is shown in
Fig. 1b for $y_{0}=L^{\prime}_{y}-L_{y}$, although other configurations are
possible. We find that for narrow constrictions the precise value of $y_{0}$
is not relevant and the conductance shows anomalous steps as the central
potential is varied. If $y_{0}\geq L^{\prime}_{y}$ or $y_{0}\leq-L_{y}$ the
constriction becomes a loop that detaches from the left and right channels;
Fig. 1c shows a case for $y_{0}>L^{\prime}_{y}$. For narrow loops the
conductance displays resonant peaks as a function of central potential, their
location giving information about the energy levels inside the loop.
Altogether, the structure is a remarkable playground for electrical transport
studies of both propagating and localized topological valley states.

##  II Model

We use an effective eight-component model, valid for low energies near the
Dirac points of the BLG crystalline band structure. The Hamiltonian reads
McCann and Koshino (2013); Rozhkov  _et al._ (2016)

$\displaystyle H$ $\displaystyle=$ $\displaystyle
v_{F}\left(p_{x}-\hbar\frac{y}{l_{z}^{2}}\right)\tau_{z}\sigma_{x}+v_{F}\,p_{y}\sigma_{y}$
(1) $\displaystyle+$
$\displaystyle\frac{t}{2}\,\left(\,\lambda_{x}\sigma_{x}+\lambda_{y}\sigma_{y}\,\right)+V_{s}+V_{a}\,\lambda_{z}\;,$

with three characteristic pseudospins (valley $\tau_{xyz}$, sublattice
$\sigma_{xyz}$ and layer $\lambda_{xyz}$) described by corresponding Pauli
matrices while $p_{x}$ and $p_{y}$ are momentum operators. Two of the model
paremeters are intrinsic of BLG, namely, the graphene Fermi velocity $\hbar
v_{F}=660\,{\rm meV}\,{\rm nm}$ and the interlayer coupling $t=380\,{\rm
meV}$. Then, $l_{z}=\sqrt{\hbar/eB}$ is the magnetic length for an external
magnetic field $B$ described in the Landau gauge whereas $V_{s}$ and $V_{a}$
are respectively the symmetric and asymmetric potentials applied to the
layers. For uniform potentials, $V_{s}$ is just a global energy shift while
$V_{a}$ is a displacement energy that opens a gap in the BLG spectrum. The
inhomogenous system of Fig. 1a has position dependent potentials $V_{a}(y)$
and $V_{s}(x)$, with transitions between plateau values defined by the gates.

The time reversal $\Theta$ and chiral-symmetry ${\cal C}$ operators and
transformations relevant to our system read

$\begin{array}[]{llll}\Theta=i\tau_{y}{\cal
K}&\Rightarrow&\Theta^{2}=-1\;,&\Theta H(B)\Theta=-H(-B)\;,\\\ {\cal
C}=\sigma_{x}\tau_{x}\lambda_{y}&\Rightarrow&{\cal C}^{2}=1\;,&{\cal C}H{\cal
C}=-H\;,\end{array}$ (2)

where ${\cal K}$ refers to complex conjugation. The symmetry transformations
on a state $|Ekv\rangle$ with a given energy, momentum and valley are given by
$\Theta|Ekv\rangle\propto|E\bar{k}\bar{v}\rangle$ and ${\cal
C}|Ekv\rangle\propto|\bar{E}k\bar{v}\rangle$.

We first discuss the spectrum that arises from Eq. (1) for a translationally
invariant kink-antinkink system at $B=0$ and $V_{s}=0$. The absence of a
central region in Fig. 1a can be represented by $L_{x}=0$ or, alternatively,
by $y_{0}=0$ and $L^{\prime}_{y}=L_{y}$. In this case, states propagate along
$x$ and are characterized by a real wave number $k$, i.e., $p_{x}\to\hbar k$
in Eq. (1). Whenever $V_{a}=V_{a}(y)$ changes its sign the gap is inverted and
as a consequence four topological states per valley appear at each kink Martin
_et al._ (2008); Zarenia  _et al._ (2011). These correspond to the branches
seen around zero energy in Fig. 1d. The states above $E=4.2$ meV are extended
states that do not remain attached to the kinks in contrast to the topological
states. Further, the energy bands in Fig. 1d are not bounded either from below
or from above since Eq. (1) describes Dirac fermions. We also note that $H$ is
both valley diagonal (so that each valley can be independently treated in a
four-component subspace) and diagonal in the real spin basis. However, whereas
all states are hereafter degenerate for spins up and down, the spectrum is not
valley degenerate but obeys
$E(k,\tau_{z}\\!\to\\!1)=E(-k,\tau_{z}\\!\to\\!-1)$ due to time reversal
symmetry. As a consequence, kink-antikink currents are valley unpolarized.
Later, we will remark that a magnetic field breaks time reversal symmetry and
thus valley polarizations can be observed in the measured conductance.

In our calculations, the kink potentials vary smoothly in $y$ by means of a
diffusivity $s$ (see App. A for details of the potential modeling). This
smoothness becomes important when the kink-antikink separation is small, i.e.,
the constriction in Fig. 1b or the loop in Fig. 1c. Then, $s$ couples the kink
states running on the two sides, a mechanism that is eventually responsible
for the transmission modulation. Along the transport direction $x$, the
potential interfaces are considered sharp. This assumption is well justified
since the mode wavelength $\lambda$ is much larger than the characteristic
length $l_{a}$ for inversion of the static potentials. Electrostatic modeling
in bilayer grapheneLi  _et al._ (2016); Chen  _et al._ (2020) yields an
estimate $l_{a}<50$ nm, while in our calculations we typically have
$\lambda\gtrsim 300$ nm.

We next consider the inhomogenous situation with $L$, $C$ and $R$ regions
along the transport direction $x$, sketched in Fig. 1a. The distribution of
applied potentials is seen in Fig. 1a for the top layer. Gates on the bottom
layer have reversed $V_{a}$ and the same central shift $V_{s}^{(C)}$. We solve
the scattering problem in the presence of either the constriction or the loop
using complex band structure methods Serra (2013); Osca and Serra (2019). A
survey of this method is given in App. C. The technique is especially well
suited to describe piecewise homogenous potentials in topological systems. For
each region $a=L,C,R$ in Fig. 1a, a large set of complex wavenumbers and
eigenstates $\\{k^{(a)},\phi_{k}^{(a)}\\}$ is determined by exact
diagonalization Lehoucq  _et al._ (1998). These sets of solutions are then
properly matched at the interfaces between central ($C$) and side regions
($L,R$). The ensuing linear system of equations yields the transmission
amplitudes $t_{n^{\prime}n}$ from input mode $n$ to output mode $n^{\prime}$.
The electric conductance is then determined by the two-terminal formula
$G=(2e^{2}/h)\sum_{nn^{\prime}}{|t_{n^{\prime}n}|^{2}}$, where spin degeneracy
is already taken into account and we assume zero temperature (the experiments
in Ref. Overweg  _et al._ , 2018b are done at a very low temperature of 1.7
K). The set of complex wavenumbers and wavefunctions of each region is
obtained with a finite difference discretization of a 1D equation depending
only on $y$ since the $x$ dependence disappears thanks to the homogeneity of
each region along the transport direction. This 1D character enables an
accurate numerical resolution for large numbers of $y$ grid points, while no
grid in $x$ is needed.

The use of grid discretization methods for Dirac-like problems leads to the
infamous Fermion doubling problem Susskind (1977); Nielsen and Ninomiya
(1981); Hernández and Lewenkopf (2012), which introduces spurious replica
states. These are characterized by very short wavelength oscillations,
strongly fluctuating from one grid point to the next. Similar replicas are
obtained in our approach when calculating the complex band structure of each
region $\\{k^{(a)},\phi_{k}^{(a)}\\}$. We filter out the replicas by coarse
graining, performing an average with the right or left neighboring point and
neglecting those states whose norm is affected by coarse graining. For dense
grids, we easily arrive at an unambiguous identification of the physical
states, which need to be smooth on the grid by definition. Thus, the Fermion
doubling problem does not affect the linear system that determines the
conductance since $G$ is based only on the sets of previously filtered
solutions $\\{k^{(a)},\phi_{k}^{(a)}\\}$ and no further spatial grid is
required near the interfaces.

##  III Results

We study two systems formed with the gate distribution and geometry depicted
in Fig. 1a: (i) when $0<y_{0}\leq L_{y}^{\prime}-L_{y}$ propagating modes can
exist within the central area and a quantum point contact behavior is expected
(Fig. 1b); (ii) when $y_{0}>L_{y}^{\prime}$ a loop detaches from the left and
right leads (Fig. 1c) and we will consequently find quantum resonance effects.
In both devices, a key parameter is the symmetric potential $V_{s}^{(C)}$ in
the central region, which acts as an effective local probe allowing energy
spectroscopy of the constriction.

Figure 2:  Results for a narrow constriction (inset in panel a) with
$L_{y}=100\,{\rm nm}$, $y_{0}=200\,{\rm nm}$ and a Fermi energy $E=0.02\,{\rm
meV}$. (a) Conductance for $L_{x}=1\,{\mu}{\rm m}$ as a function of the
central potential $V_{s}^{(C)}$. (b) Energy bands for a kink-antikink wire
having the same parameters of the constriction. The capital letters and
horizontal lines indicate the correspondence with the conductance ranges of
panel a. (c,d) Same as panels a and b, respectively, with a magnetic field of
$B=50\,{\rm mT}$. The two colors mark the two different valleys.

###  III.1 Quantum point contacts

We first present results for a constriction corresponding to a narrow point
contact with a kink-antikink separation of $100\,{\rm nm}$. This value is
compatible with the width of presently available BLG point contacts Overweg
_et al._ (2018b); Kraft  _et al._ (2018). We set the kink diffusivity to
$s=38\,{\rm nm}$, which is taken from the electric potential distribution in
dual split gate BLG devices Li  _et al._ (2016); Chen  _et al._ (2020), and
assume an almost vanishing Fermi energy $E=0.02\,{\rm meV}$, close to the
charge neutrality point. The results are not strongly affected by changes
around this value, as long as higher energy modes in the asymptotic kink-
antikink are not activated and remain far from the Fermi energy (cf. Fig. 1d).

The conductance for a $1\,\mu{\rm m}$-long constriction (hereafter the wire)
as a function of the central potential is shown in Fig. 2a. We observe that
$G$ is strongly suppressed around $V_{s}^{(C)}=0$ (region A). This is in
principle a surprise since the presence of the edge states at $E=0$ in Fig. 1d
would imply a fully transparent constriction. However, when we plot in Fig. 2b
the wire band structure we notice that the topological bands display an
absolute gap (for any $k$) in region A (details of this gap are discussed in
App. B). The kink potentials in the constriction couple the edge states,
leading to an almost complete backscattering and hence a reduction of the
conductance. $G$ does not reach zero because the electrons can traverse the
constriction by tunnel effect, which yields in any case a tiny value for $G$.
Then, as $V_{s}^{(C)}$ increases the energy exceeds the gap and we find in
region B two propagating states with positive velocity, per valley and spin.
It follows that $G$ quickly reaches the quantized value of $8e^{2}/h$. If
$V_{s}^{(C)}$ is further enhanced we enter region C, where a single mode is
only allowed, thus bringing $G$ down to an anomalous step of $4e^{2}/h$.
Finally, larger values of $V_{s}^{(C)}$ approach us into region D, where
another mode starts to contribute and $G$ grows again. The oscillations seen
in Fig. 2a are due to quantum interference of several modes coexisting in the
wire. We can thus conclude that there is a remarkable correspondence between
$G$ and the wire energy bands. In fact, the particle-hole symmetry of Fig. 2b
implies that $G(V_{s}^{(C)})=G(-V_{s}^{(C)})$, as found in Fig. 2a. We also
point out that for nonzero temperatures the conductance curves will be thermal
smeared.

Figure 3: Density distribution, in arbitrary units, for the scattering states
corresponding to the conductance curves of Fig. 2c. Panels a and b are for the
two different valleys $K$ and $K^{\prime}$, respectively. Parameters:
$L_{x}=1000\;{\rm nm}$, $L_{y}=100\;{\rm nm}$, $L^{\prime}_{y}=300\;{\rm nm}$,
$y_{0}=200\;{\rm nm}$, $B=50\;{\rm mT}$, $E=0.02\;{\rm meV}$ and
$V_{s}^{(C)}=0.2\;{\rm meV}$.

A small magnetic field, in the mT range, suffices to yield large effects on
the constriction conductance even if the valley splitting of the energy bands
is small (Fig. 2c). Here, we choose to separately display each valley
contribution to the conductance. Accordingly, the scale of $G$ reduces a
factor $2e^{2}/h$ as compared with Fig. 2a. We find that the conductance
shifts in opposite directions for the two valleys $\tau_{z}\to\pm 1$, making
it possible the creation of highly polarized valley currents, where one valley
component is essentially blocked while the other is transmitted. The valley
split bands are shown in Fig. 2d. This behavior can be also seen with a single
kink due to valley-momentum locking. However, if we wish to invert the current
valley polarization with a kink we would need to revert the extended lateral
gates defining the kink whereas Fig. 2c shows the interesting possibility of
switching the valley polarization by simply changing $V_{s}^{(C)}$, leaving
both the lateral gates defining the kinks and the magnetic field fixed.
Therefore, our system would work as an electrically tunable, fully reversible
valley filter using tiny magnetic fields. Notice that BLG valley filters based
on nontopological states require much larger fields, in the tesla range Park
(2019).

This is better seen in Fig. 3, where we plot the density distribution of the
Fermi-energy scattering states when electrons are injected from the source
terminal (left side). The two valleys (Fig. 3a and Fig. 3b) contribute
differently since the magnetic field is finite. While for valley $K$ electrons
impinge from the bottom left kink (Fig. 3a), the opposite valley $K^{\prime}$
electrons (Fig. 3b) enter from the top left kink. The former (latter) are
mostly transmitted (reflected), giving rise to a valley polarized current in
the drain terminal (right side).

Figure 4:  Dependence of the conductance on the constriction length $L_{x}$
for the results of Fig. 2a and selected values of $V_{s}^{(C)}$, as given in
parenthesis. A-D labels are used to indicate the same regions of Fig. 2a.

The dependence of $G$ on the constriction length $L_{x}$ (see Fig. 4) further
supports our interpretation. We display the conductance for $V_{s}^{(C)}$
corresponding to the four regions indicated in Fig. 2a. In the gapped region A
the conductance decays exponentially for large values of $L_{x}$, which agrees
with a transport mechanism based on tunnel effect. In region B the conductance
shows an oscillatory behavior up to arbitrarily large distances, implying a
Fabry-Perot interference between propagating modes in the central area. The
conductance becomes quantized at $4e^{2}/h$ in region C, which occurs when the
interfaces between the leads and the constriction becomes transparent.
Finally, in region D we recover the oscillatory behavior due to the activation
of a new transport channel. In all cases the role of quantum tunneling for
small $L_{x}$ is clearly seen because the conductance increases as $L_{x}$
shrinks to zero and scattering thus disappears.

###  III.2 Side loops

Let us turn to the loops created as the gate position $y_{0}$ shown in Fig. 1a
increases. Then, the edge states in the central region detach as illustrated
in Fig. 1c. Figure 5a shows in this case a conductance pattern that strongly
differs from the wire system of Fig. 2a. $G$ is characterized by resonant
peaks that reach values of the order of $4e^{2}/h$ (we plot $G/2$ for
convenience). Interestingly, these peaks are causally correlated with the
discrete levels in the closed loop. To see this, we plot in Fig. 5d the loop
energy spectrum. We find that the position of the conductance peaks agree,
apart from a slight renormalization due to the coupling with to external edge
states, with the level positions. The particular peak structure is highly
sensitive to the loop dimensions $(L_{x},L_{y})$ due to quantum confinement.

Figure 5: Results for a side loop (inset in panel a) with $L_{y}=150\,{\rm
nm}$, $y_{0}=310\,{\rm nm}$ and Fermi energy $E=0.02\;{\rm meV}$. (a,b)
Conductance for $L_{x}=1\,\mu{\rm m}$ as a function of the central potential
$V_{s}^{(C)}$ for $B=0$ (a) and $B=50\,{\rm mT}$ (b). (c) Conductance as a
function of the magnetic field for $V_{s}{(C)}=0$. (d) Energy levels for the
finite loop $(L_{x},L_{y})=(1\,\mu{\rm m},150\,{\rm nm})$ as a function of the
field. The two colors in panels b-d indicate the two different valleys. The
peak separation [around $0.04\,{\rm meV}$ in (b)] could be resolved at low
temperatures $T\lesssim 0.5\,{\rm K}$.

A small magnetic field splits the conductance peaks, as shown in Fig. 5b where
we plot the valley resolved $G$ for $B=50$ mT. It is noticeable that the peak
widths are significantly reduced in the presence of $B$, thus leading to
smaller conductance minima; cf. Figs. 5a and 5c. The conductance splitting is
explained with the level behavior as a function of $B$ as shown in Fig. 5d.
The field acts differently on the two valleys, thus raising (lowering) the
energy for $\tau_{z}\to 1$ ($\tau_{z}\to-1$). $B$-splitting of the two valleys
is also present for the case of non topological bound states in graphene
circular quantum dots, discussed in Ref. Recher  _et al._ , 2009, where states
of the same angular momentum and opposite valleys show opposite dispersions at
low fields. Besides the splitting, the spectrum in Fig. 5d for topological
loops shows a pattern of almost parallel lines for each valley, reflecting a
quantization condition of the topological states along the perimeter of the
loop Xavier  _et al._ (2010). Our results obey reciprocity, i.e., $G$ is
unchanged when both $B$ and the valley index are simultaneously reversed, as
can be seen in Fig. 5c. It is also worth stressing that the valley and gate
sensitivity allows, as in the constriction, switching the valley polarization
of the current by soley tuning the gate potential, only that a finer tuning is
needed in the detached loop in order to hit the narrower peak maxima.
Resolving the narrow peaks needs low temperatures to avoid thermal broadening,
which we can estimate below $500\,{\rm mK}$ based on Fig. 5.

Figure 6: Density distribution corresponding to the results of a side loop
depicted in Fig. 5b. The central gate potential is $V_{s}^{(C)}=0.3\,{\rm
meV}$ and it corresponds to a $K^{\prime}$ valley peak (light color) in Fig.
5b. Panels a and b are for the two different valleys $K$ and $K^{\prime}$,
respectively. Parameters: $L_{x}=1000\;{\rm nm}$, $L_{y}=150\;{\rm nm}$,
$L^{\prime}_{y}=300\;{\rm nm}$, $y_{0}=310\;{\rm nm}$, $B=50\;{\rm mT}$,
$E=0.02\;{\rm meV}$ and $V_{s}^{(C)}=0.3\;{\rm meV}$.

Probability density distributions in space provide a more visual support for
this valley switch effect (see Fig. 6). We note that just one valley is
populating the loop (Fig. 6b) while the other one is reflected (Fig. 6a).
Thus, the valley-split resonant conductances of side loops in small magnetic
fields imply high valley accumulations on the loop for specific gate
potentials.

Figure 7: (a) $L_{x}$ dependence of the conductance for two selected values of
$V_{s}^{(C)}$ for the parameters of the $L_{y}=150\;{\rm nm}$ side loop of
Fig. 5a. (b) Energy levels of the closed rectangular loop with
$L_{y}=150\;{\rm nm}$ as a function of $L_{x}$.

We now briefly discuss the dependence on $L_{x}$. In Fig. 7 we show results
for a wide loop of $L_{y}=150\;{\rm nm}$ as in Fig. 5a. For $V_{s}^{(C)}$
corresponding to a conductance valley in Fig. 5a the conductance curve in Fig.
7a (black line) shows a single-mode regular spacing, which is in agreement
with the degeneracy due to level crossing for $E=0$ of Fig. 7b. In contrast,
for a $V_{s}^{(C)}$ value that generates a $G$ peak in Fig. 5a the conductance
curve shows multiple-mode spacings in Fig. 7a (light green line). Further, we
get accidental crossings at values of $L_{x}$ that lead to additional factor-2
degeneracies in Fig. 7b. The case of a narrow loop ($L_{y}=100\;{\rm nm}$) is
shown in Fig. 8. In contrast to the previous case, the conductance valley
curve is quenched as $L_{x}$ increases (black line in Fig. 8a) whereas the
peak curve exhibits a beating pattern overimposed to the peak sequence (green
line). Figure 8b shows the energy levels when the narrow loop is closed. We
observe regions with a bunching of levels separated from others with regularly
spaced levels. This is consistent with the conductance peaks obtained in Fig.
8a. Despite the fact that the conductance patterns depend on the specific
geometry of the loop (via $L_{x}$ and $L_{y}$), in all cases the conductance
peaks are correlated with the energy levels of the closed loop. Thus, the
conductance serves as an excellent tool to probe the internal structure of
topologically bound states.

Figure 8: (a) $L_{x}$ dependence of the conductance for a narrow loop
($L_{y}=100\;{\rm nm}$) and two selected values of $V_{s}^{(C)}$ with the rest
of the parameters as in Fig. 7. (b) Energy levels of the closed loop with the
same $L_{y}=100\;{\rm nm}$ as a function of $L_{x}$.

##  IV Conclusions

We have proposed a versatile nanodevice for topological studies in quantum
valley transport. Transmission manipulation is achieved by means of a kink-
antikink local potential that allows the formation of (i) point contacts with
anomalous quantized conductance and (ii) side loops with chiral quasi-bound
states. The obtained conductance curves provide information on the system
energy spectrum. For tiny magnetic fields we obtain a valley polarization
effect and this polarization is tunable with the gate potential.

Possible drawbacks of our proposal might be the effect of imperfections and
misalignments of top and bottom gates in Fig. 1a as well as the presence of
disorder due to impurities. Our results, however, suggest robustness against
small gate displacements since we take into account smooth transition profiles
of the asymmetric potential with diffusivity values as large as $s\approx
40\,{\rm nm}$. On the other hand, the chiral character of the low-energy
valley-momentum-locked states offers protected transmission against
backscattering by impurities that conserve the valley degree of freedom Li
_et al._ (2011).

The illustrative examples considered in this work do not exhaust the
capabilities of the system, and more sophisticated setups could be envisaged.
Further, our model could be straightforwardly extended to multivalley
materials other than bilayer graphene such as silicene Pan  _et al._ (2014),
sonic crystals Lu  _et al._ (2017) and photonic platforms Gao  _et al._
(2017).

###### Acknowledgements.

We acknowledge support from MINECO (Spain) Grant No. MAT2017-82639, No.
PID2020-117347GB-100, MINECO/AEI/FEDER María de Maeztu Program for Units of
Excellence MDM2017-0711.

##  Appendix A Potential smoothness

Figure 9:  Smooth asymmetric potentials in a single kink and a double kink
(also called kink-antikink in the main text). Parameters: $V_{a}=10\,{\rm
meV}$, $s=38\,{\rm nm}$, single kink $y_{1}=0$, double kink $y_{1}=-150\,{\rm
nm}$ and $y_{2}=150\,{\rm nm}$.

Smoothness in the space variation of the asymmetric potential $V_{a}(y)$ is
described with a logistic function. A smooth step at position $y_{1}$ and
diffusivity $s$ (equivalently, the steepness inverse) is represented by

${\cal F}(y,y_{1},s)=\frac{1}{1+e^{(y-y_{1})/s}}\;.$ (3)

In detail, the case of a single kink at $y_{1}$ reads

$\displaystyle V_{a}^{(sk)}(y)$ $\displaystyle=$ $\displaystyle
V_{a}\,\left[1-2{\cal F}(y,y_{1},s)\right]\;,$ (4)

with the asymptotic values $V_{a}^{(sk)}(\pm\infty)=\pm V_{a}$. In a
straightforward extension, the double kink forming a kink-antikink system at
$y_{1}$ and $y_{2}$ reads

$\displaystyle V_{a}^{(dk)}(y)$ $\displaystyle=$ $\displaystyle
V_{a}\,\left[1+2{\cal F}(y,y_{1},s)-2{\cal F}(y,y_{2},s)\right]\;,$

Plots of single and double kinks with the above parametrizations are shown in
Fig. 9.

Figure 10:  a) Energy gap of the kink-antikink band structure as a function of
the separation $L_{y}$ and for two potential diffusivities $s$. The data
points are obtained from the numerical band structure while the joining lines
are a guide to the eye. The inset shows the wave function overlap ${\cal O}$
of independent kink and antinkink as a function of distance with the same
color code. b) Energy gap as a function of the diffusivity for a fixed
separation. Parameters: $V_{a}=10\,{\rm meV}$, $V_{s}=0$.

##  Appendix B Kink-antikink gap

We address here how the gap of the kink-antinkink band structure (see Fig. 2b)
varies with the separation $L_{y}$ and the potential diffusivity $s$. The
results are shown in Fig. 10a and b, respectively, and they confirm, as was
anticipated, that the gap strongly increases when $L_{y}$ decreases. However,
it is remarkable that this dependence is non monotonic, with oscillations and
with particular values of $L_{y}$ and $s$ for which the gap vanishes. This
behavior can be attributed to the oscillations of the wave functions Martin
_et al._ (2008); Zarenia  _et al._ (2011), as can be seen from the overlap of
two displaced kink states

${\cal
O}(L_{y})=\left|\int{dy\;{\bm{\phi}}^{T*}(y)\,{\bm{\phi}}(y-L_{y})}\right|\;,$
(6)

shown in the inset to Fig. 10a. In Eq. (6) ${\bm{\phi}}$ is the eight-
component wave function (and ${\bm{\phi}}^{T}$ its transpose) for a real $k$
near the branch crossing at zero energy of the single kink. For the steep
potential in Fig. 10 ($s=12\,{\rm nm}$) the $L_{y}$ of minimum gap and the
$L_{y}$ of vanishing overlap are in good agreement, while for the smooth
potential ($s=38\,{\rm nm}$) the agreement is only qualitative.

##  Appendix C Scattering with complex band structure

Having mentioned in Sec. II the overall idea of the complex-band-structure
approach, we here present the details of the method. The double junction, Fig.
1, is formed by three regions $a=L,C,R$, each one homogenous along $x$.
Introducing a local wave number $p_{x}\to\hbar k^{(a)}$ and a factorizing wave
function
$\phi_{k}^{(a)}(y\eta_{\sigma}\eta_{\tau}\eta_{\lambda})e^{ik^{(a)}x}$, with
$\eta_{\sigma\tau\lambda}=1,2$ representing the pseudospin discrete
components, the energy eigenmode equation from Hamiltonian given by Eq. (1)
for each region can be recast as

$\displaystyle\left[\frac{E}{v_{F}}\,\tau_{z}\sigma_{x}+\hbar\frac{y}{l_{z}^{2}}-ip_{y}\tau_{z}\sigma_{z}-\frac{t}{2v_{F}}\,\tau_{z}(\lambda_{x}+i\lambda_{y}\sigma_{z})\right.$
$\displaystyle\left.-\frac{V_{s}}{v_{F}}\,\tau_{z}\sigma_{x}-\frac{V_{a}(y)}{v_{F}}\,\lambda_{z}\tau_{z}\sigma_{x}\right]\phi_{k}^{(a)}=\hbar
k^{(a)}\,\phi_{k}^{(a)}\;,$ (7)

where we left-multiplied all terms by $\sigma_{x}\tau_{z}/v_{F}$, included a
global sign change and moved the $k^{(a)}$ linear term to the right-hand side.

Equation (7) is an eigenvalue equation for the mode sets
$\\{k^{(a)},\phi_{k}^{(a)}\\}$ corresponding to a given energy $E$. The
problem is non-Hermitian due to the $ip_{y}\tau_{z}\sigma_{z}$ and
$i\lambda_{y}\sigma_{z}$ contributions, physically allowing the possibility of
complex wave numbers $k^{(a)}$. We solve Eq. (7) as a matrix eigenvalue
problem, introducing a uniformly spaced 1D grid for the $y$ coordinate,
determining the topological modes smoothly vanishing at the $y$ boundaries
with numerical methods well suited to large sparse matrices Lehoucq  _et al._
(1998). The local complex wavenumbers and modes are our complex band structure
basis sets in term of which we represent the solution in each region of the
double junction.

The wave function is given by a set of amplitudes $\\{C_{k}^{(a)}\\}$ in each
$x$-homogenous region as

$\Psi^{(a)}(xy\eta_{\sigma}\eta_{\tau}\eta_{\lambda})=\sum_{k^{(a)}}{C_{k}^{(a)}\,\phi_{k}^{(a)}(y\eta_{\sigma}\eta_{\tau}\eta_{\lambda})\,e^{ik^{(a)}(x-x_{k}^{(a)})}\;.}$
(8)

The $x_{k}^{(a)}$ values in Eq. (8) are a gauge-like choice for the amplitudes
$C_{k}^{(a)}$. This choice has its importance to avoid numerical instabilities
due to exponentially large numbers with complex wave numbers. For $a=L(R)$,
the sum in Eq. (8) only includes wavenumbers with either zero or negative
(positive) imaginary parts and we set $x_{k}^{(L,R)}=x_{1,2}$, with $x_{1,2}$
the positions of the two interfaces. In the central region, however, Eq. (8)
includes wave numbers with both $\leq 0$ and $>0$ imaginary parts, and we set
$x_{k}^{(C)}=x_{2}$ and $x_{1}$ for those two cases, respectively.

At the two interfaces $x_{1,2}$ the wave function fulfills continuity. This is
the only requirement with a Dirac-like Hamiltonian like Eq. (1), as opposed to
a Schrödinger problem where the kinetic term also requires continuity of the
wave function first derivative. The continuity conditions yield the amplitudes
of outgoing modes $C_{k}^{(L,C,R)}$ in terms of those for input modes. The
latter are assumed nonvanishing in the left lead only. Thus, we obtain a
closed set of linear equations by projecting the continuity conditions at the
two interfaces on the total set of outgoing complex modes. In detail, the
linear system reads

$\left\\{\begin{array}[]{rrll}\displaystyle\sum_{k^{(L)}\,{\rm out}}{{\cal
M}_{k^{\prime}k}^{(aL)}\,{C_{k}^{(L)}}}&-\displaystyle\sum_{k^{(C)}\,{\rm
out}}{{\cal
M}_{k^{\prime}k}^{(aC)}\,e^{ik^{(C)}(x_{1}-x_{k}^{(C)})}\,{C_{k}^{(C)}}}&=-\displaystyle\sum_{k^{(L)}\,{\rm
in}\,{\rm prop.}}{{\cal M}_{k^{\prime}k}^{(aL)}\,{C_{k}^{(L)}}}\;,&\quad{\rm
if}\;x_{k^{\prime}}^{(a)}=x_{1}\;,\\\
\rule{0.0pt}{22.76228pt}\displaystyle\sum_{k^{(R)\,{\rm out}}}{{\cal
M}_{k^{\prime}k}^{(aR)}\,{C_{k}^{(R)}}}&-\displaystyle\sum_{k^{(C)}\,{\rm
out}}{{\cal
M}_{k^{\prime}k}^{(aC)}\,e^{ik^{(C)}(x_{2}-x_{k}^{(C)})}\,{C_{k}^{(C)}}}&=0\;,&\quad{\rm
if}\;x_{k^{\prime}}^{(a)}=x_{2}\;,\end{array}\right.$ (9)

where we have defined the overlap matrices

${\cal
M}_{k^{\prime}k}^{(ab)}=\sum_{\eta_{\sigma}\eta_{\tau}\eta_{\lambda}}\int{dy\,\phi_{k^{\prime}}^{(a)*}\phi_{k}^{(b)}\;,}$
(10)

and, as mentioned, the sum on the right-hand side of Eq. (9) is restricted to
the $k^{(L)}$ input propagating modes (with vanishing imaginary wavenumber)
while those on the left-hand side contain output modes of all types, i.e.,
propagating and evanescent.

The input/output character of the modes is summarized in Tab. 1. Notice that
in the central region all modes are output modes, while in left and right
regions there can be input modes of propagating or evanescent character, in
our notation. In $L$ and $R$, the propagating modes are classified as input or
output depending on their quasiparticle flux, defined as

$I_{k}^{(a)}=\langle\phi_{k}^{(a)}|v_{F}\tau_{z}\sigma_{x}|\phi_{k}^{(a)}\rangle\;,$
(11)

where $\partial H/\partial p_{x}=v_{F}\tau_{z}\sigma_{x}$ is the $x$-velocity
operator.

Setting $C_{k}^{(L)}=1$ for a particular input propagating mode in Eq. (9),
with all other inputs vanishing, we obtain a particular input/output pair
transmission $t_{n^{\prime}n}\propto C_{k^{\prime}}^{(L,R)}$ by solving the
linear system. We have numerically checked that flux conservation is fulfilled
in our method when the number of modes in each region is large enough.
Typically, with 300 modes in each region we obtain a flux conservation with an
error of 0.1% or even smaller.

region | mode character | condition
---|---|---
$C$ | output | none
$L$ | input/output prop. |  $I_{k}^{(a)}>0(<0)$; ${\rm Im}(k^{(a)})=0$
$L$ | input/output evan. | ${\rm Im}(k^{(a)})>0(<0)$
$R$ | input/output prop. |  $I_{k}^{(a)}<0(>0)$; ${\rm Im}(k^{(a)})=0$
$R$ | input/output evan. | ${\rm Im}(k^{(a)})<0(>0)$
Table 1: Summary of the conditions for mode clasification as input or output
in the three device regions. $I_{k}^{(a)}$ is the mode flux defined in Eq.
(11).

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