# Nanowrinkle Waveguide in Graphene for Enabling Secure Dirac Fermion
Transport

Seunghyun Jun  Department of Physics Education, Chosun University, Gwangju
61452, Republic of Korea  Myung-Chul Jung  Department of Physics Education,
Chosun University, Gwangju 61452, Republic of Korea  Institute of Well-Aging
Medicare & Chosun University G-LAMP Project Group, Chosun University, Gwangju,
61452, Republic of Korea  Nojoon Myoung<EMAIL_ADDRESS>Department of
Physics Education, Chosun University, Gwangju 61452, Republic of Korea
Institute of Well-Aging Medicare & Chosun University G-LAMP Project Group,
Chosun University, Gwangju, 61452, Republic of Korea

###### Abstract

Localized states in graphene have garnered significant attention in quantum
information science due to their potential applications. Despite graphene’s
superior transport and electronic properties compared to other semiconductors,
achieving nanoscale confinement remains challenging due to its gapless nature.
In this study, we explore the unique transport properties along nanowrinkles
in monolayer graphene. We demonstrate the creation of a one-dimensional
conduction channel by alternating pseudo-magnetic fields along the
nanowrinkle, enabling ballistic Dirac fermion transport without leakage. This
suggests a feasible method for secure quantum information transfer over long
distances. Furthermore, we extend our analysis to bent nanowrinkles,
showcasing well-guided Dirac fermion propagation unless the bent angle is
sufficiently large. Our demonstration of the nanowrinkle waveguide in graphene
introduces a novel approach to controlling Dirac fermion transport through
strain engineering, for quantum information technology applications.

##  I Introduction

Graphene, a two-dimensional material composed of a single layer of carbon
atoms arranged in a hexagonal lattice, has attracted significant attention
since its discovery due to its remarkable electronic, mechanical, and thermal
properties [1, 2]. Graphene exhibits high charge carrier mobility, exceptional
mechanical strength, and superior thermal conductivity [1, 2, 3, 4, 5, 6].
These properties make graphene an ideal candidate for various applications,
from flexible electronics to high-frequency transistors . In addition, the
distinctive electronic structure of graphene, characterized by its Dirac
fermions, makes it a promising material for quantum information science and
valleytronics [1, 2, 3, 7, 4, 5, 8, 6].

However, contrary to expectations, the energy gap-free nature of graphene
meant that the current could not be controlled in the same way as in many
conventional semiconductor devices. Various methods were proposed to open an
energy gap in graphene, but the electrical properties of graphene get worse
than those of pristine graphene [4, 5]. To overcome this, mechanical
deformation of graphene was utilized. In actual synthesis, graphene is
mechanically deformed in a variety of ways. New quantum states can be tuned by
structural modifications, it could lead to a variety of quantum devices using
graphene with superior transport properties [9, 10, 11, 6]. Recently, many
efforts have been made to create quantum dots on graphene and utilize them for
applications such as quantum information devices or quantum information
transportation [12, 13, 14, 15, 16, 17, 18, 19]. In this paper, we focus on
the quantum information transportation in strained graphene.

Lattice deformation of graphene to enhance or manipulate its electrical
properties has been widely studied [20, 21, 22, 23, 24, 25, 26, 27]. As a
result of this lattice deformation, it has been found that the Dirac fermions
behave as if they feel a magnetic field, which is called pseudomagnetic field
(PMF) [28, 29, 30, 31, 32, 33]. The PMF in nanobubble or nanowinkle can affect
the momentum of electrons or holes in graphene, leading to interesting
physical phenomena such as the formation of the zero-field Landau level or the
valley Hall effect [29, 34, 35, 36, 37].

These strain-induced fields create new pathways for electronic transport,
especially in nanowinkle, allowing the formation of one-dimensional (1D)
conducting channels [38, 36, 33]. These channels are particularly interesting
for quantum information applications because they can guide Dirac fermions
with minimal scattering and decoherence . The ability to control the
propagation of Dirac fermions through strain engineering opens new avenues for
the development of graphene-based quantum devices, including waveguides and
logic gates, which are essential for building scalable quantum networks.

In this paper, we demonstrate a strain-engineered graphene waveguide for use
as a quantum information device, and identify the waveguide mode by the PMF
due to a nanowrinkle using the tight-binding model. Our results show that the
waveguide enables quantum information transfer through 1D conduction channels.
We further show that Dirac fermions are well guided along a bent nanowrinkle,
examining the effects of the bent angle on its transport properties.

The manuscript is organized as follows. In Sec. II, we introduce the
theoretical frameworks of the system, addressing the characteristics of the
strain-induced PMF and methodologies for both analytic analysis and numerical
calculations. In Sec. II, we demonstrate the occurrence of the waveguide mode
by presenting the band structures and transport properties through the
nanowrinkle. We show that Dirac fermions are indeed well-guided along the
nanowrinkle waveguide, exhibiting perfect transmission. Furthermore, in Sec.
II, we explore the effects of waveguide modulation on Dirac fermion transport
by considering bent nanowrinkles. We reveal that bent nanowrinkles can also
host waveguide modes, indicating the potential application of the nanowrinkle
waveguide as a building block for quantum devices or circuits. Finally, in
Sec. III, we summarize our findings and discuss the implications of our
results for future research and potential applications in quantum information
technology.

##  II Results

### Strain-induced pseudo-magnetic fields

Figure 1: (a) Schematic of a nanowrinkle along a graphene nanoribbon with
width $W$. (b) Side view of the nanowrinkle modeled as a Gaussian function
with maximum height $h_{0}$. (c) Pseudo-magnetic field (PMF) profile
calculated from the low-energy effective Hamiltonian. (d-f) Energy bands of
the graphene nanoribbon subjected to the PMF due to the nanowrinkle, shown for
different ribbon widths. The strain-induced waveguide modes in the band
structures are highlighted in red, indicating their invariance with respect to
ribbonwidth variation. The nanowrinkle parameters used are
$h_{0}=3.69~{}\mathrm{nm}$ and $\sigma=2.46~{}\mathrm{nm}$.

Figure 1(a) and (b) depict a theoretical model of the nanowrinkle, defined by
a Gaussian function[39]:

$\displaystyle
z\left(\vec{r}\right)=h_{0}e^{-\frac{\left(y-y_{0}\right)^{2}}{2\sigma^{2}}},$
(1)

where $h_{0}$ represents the maximum vertical deformation of the nanowrinkle
and $\sqrt{2}\sigma$ denotes the standard deviation of the Gaussian profile,
indicating the lateral size of the nanowrinkle. These nanowrinkles can form
accidentally during the transfer process onto a substrate, or can be
intentionally created via various strain engineering techniques[40, 38, 41].
Due to the local strain accompanied by the lattice deformation of the
nanowrinkle, strain-induced gauge fields emerge in the strained region:

$\displaystyle\vec{A}_{ps}\left(\vec{r}\right)=-\frac{\sqrt{3}\beta\hbar}{ea_{0}}\frac{h_{0}^{2}}{2\sigma^{4}}y^{2}e^{-\left(y-y_{0}\right)^{2}/\sigma^{2}}\hat{x},$
(2)

assuming the zigzag-edge orientation is along the $x$-axis. Here, $\beta=3.37$
and $a_{0}=0.246~{}\mathrm{nm}$. Consequently, the PMF due to the nanowrinkle
is obtained by $\vec{B}_{ps}=\vec{\nabla}\times\vec{A}_{ps}$:

$\displaystyle\vec{B}_{ps}=\frac{\beta\hbar}{\sqrt{3}ea_{0}}\frac{h_{0}^{2}}{\sigma^{6}}y\left(\sigma^{2}-y^{2}\right)e^{-\left(y-y_{0}\right)^{2}/\sigma}\hat{z}.$
(3)

As depicted in Fig. 1(c), the nanowrinkle can host three ‘snake-orbit’
conducting channels[29, 32] along the ‘zero’-PMF lines at $y=0$ and
$\pm\sigma$. To confirm the existence of the strain-induced conducting
channels in the nanowrinkle, we examine the band structures of zigzag graphene
nanoribbons subjected to nanowrinkles, varing the ribbonwidth. Figures 1(d-f)
clearly show a robust band, highlighted in red, which remains unchanged with
respect to the system size. This band is purely associated with the presence
of nanowrinkle. Notably, the degeneracy of the strain-induced waveguide mode
is two-fold due to the preserved pseudospin degeneracy of graphene.

To examine the transport properties of the nanowrinke waveguide in graphene,
we investigate transport phenomena through graphene subjected to a
nanowrinkle-induced PMF using numerical calculation with the S-matrix approach
based on the tight-binding Hamiltonian of graphene:

$\displaystyle
H=\sum_{\left<i,j\right>}\left(t_{ij}a^{\dagger}_{i}b_{j}+\mathrm{h.c.}\right),$
(4)

with the modified hopping term due to lattice deformation:

$\displaystyle t_{ij}=t_{0}e^{-\beta\left(\frac{d_{ij}}{a_{0}}-1\right)},$ (5)

where $t_{0}=2.7~{}\mathrm{eV}$, and $d_{ij}$ is the distance between adjacent
atomic sites in the strained graphene lattice. The quantum conductance through
the system is obtained using the Landauer-Büttiker formalism:

$\displaystyle
G\left(E\right)=\frac{2e^{2}}{h}\sum_{\alpha,\beta}T_{\alpha\beta}\left(E\right),$
(6)

where $T_{\alpha,\beta}$ is the transmission function from a lead$\beta$ to a
lead$\alpha$ at a given energy $E$, numerically acquired from the S-matrix of
the systems. We take into account the spin degeneracy as a factor 2 in Eq.
(6).

### Graphene nanowrinkle waveguide

Figure 2: (a) Zoom-in band structure from Fig. 1(f). (b-d) Probability current
density maps through the system. Each current map corresponds to the
left/right-propagating mode in the case of electron/hole transport,
respectively denoted as open/solid color circles in (a). Insets in (b-d) are
the schematic diagrams of the allowed snake-orbit channels, respectively
corresponding to the current maps. Note that all maps are normalized to the
same values.

As aforementioned, the snake-orbit channels can be formed at $y=0$ and
$\pm\sigma$, with the propagation directions along the channel at $y=0$ being
opposite to those at $y=\pm\sigma$. This spatial distinction of the right- and
left-propagating waveguide modes arises from the opposite signs of the PMF
profiles near the interfaces $y=0$ and $\pm\sigma$. Figure 1(c) displays the
PMF profile obtained from the low-energy effective Hamiltonian model in
continuum limit:

$\displaystyle H$
$\displaystyle=v_{F}\hbar\vec{\sigma}\cdot\left(\vec{k}+se\vec{A}_{ps}\right)$
$\displaystyle=v_{F}\hbar\left[\sigma_{x}\left(k_{x}+seA_{ps,x}\right)+\sigma_{y}k_{y}\right],$
(7)

obeying the Dirac equation, $H\psi=E\psi$, with its eigenenergy $E=\hbar
v_{F}\sqrt{\left(k_{x}+seA_{ps,x}\right)^{2}+k_{y}^{2}}$, where $v_{F}\simeq
10^{6}~{}\mathrm{ms}^{-1}$ is the Fermi velocity of graphene. As shown in Fig.
1(d-f), the energy of the system is preserved with momentum inversion in the
$x$ direction, i.e., $E\left(-k_{x}\right)=E\left(k_{x}\right)$. Thus, the
strain-induced PMF $\vec{B}_{ps}$ should be accordingly reversed as the
propagation direction changes, i.e., $s=\mathrm{sign}\left(k_{x}\right)$ from
Eq. (7). Indeed, as displayed in Fig. 2(b) and (c), the right-propagating
electron moves along the channels at $y=\pm\sigma$, and the left-propagating
electron also moves along the same channels, since it feels the reversed PMF
compared to that for the left-propagating electron. On the other hands, for
hole transport, the conducting channels in the nanowrinkle waveguide are
formed at $y=0$, as shown in Fig. 2(d) and (e).

Figure 3: (a) Schematics of the system with four leads. The shaded region
indicates the existence of a nanowrinkle connecting two leads, lead1 and
lead2. (b) and (c) Conductance spectra $G_{21}$ and $G_{41}$ versus Dirac
fermion energy $E$ and the nanowrinkle height $h_{0}$. The dashed lines
represent the critical nanowrinkle height to secure the waveguide mode,
$2.13~{}\mathrm{nm}$. (d) Probability current density maps at a given energy
$E=0.27~{}\mathrm{eV}$, for different $h_{0}$ values, $1.476~{}\mathrm{nm}$
and $3.69~{}\mathrm{nm}$, respectively denoted by colored circles in (b). A
parameter used for the calculations is $\sigma=2.46~{}\mathrm{nm}$.

We now examine the transport properties of the graphene nanowrinkle waveguide
to consider its potential application for quantum information transfer. First,
we need to be confirm whether Dirac fermions transport is truly guided by the
PMF due to the nanowrinkle. To investigate this, we introduce a four-terminal
graphene device, where a straight nanowrinkle is placed between two leads,
labeled as lead1 and lead2, as depicted in Fig. 3(a). If the nanowrinkle
indeed plays a role of guiding Dirac fermions, incident Dirac fermions from
the lead1 will not be transmitted to the lead3 and the lead4, but will be
entirely collected by the lead2.

The calculated results of the conductance, $G_{21}$ and $G_{41}$, are shown in
Fig. 3(b) and (c). For small $h_{0}$, i.e., for a relatively week PMF, the
conductance spectra $G_{21}$ exhibit the characteristic step-wise increments
as a function of Dirac fermion energy, reflecting the quasi-1D nature of the
zigzag graphene nanoribbon. In this case, it is clear that $G_{41}$ spectra do
not vanish as shown in Fig. 3(c), because Dirac fermions are not guided but
spread out over the system. On the other hand, as $h_{0}$ increases, $G_{41}$
spectra become substantially suppressed, and the step-wise behavior of
$G_{21}$ spectra changes compared to the $h_{0}=0$ case. Interestingly, the
new conductance plateaus for large $h_{0}$ values are in good agreement with
the band structure displayed in Fig. 1(e).

Although the contribution of the waveguide mode to the conductance plateau in
$G_{21}$ cannot be perfectly resolved, we can directly confirm the formation
of the nanowrinkle waveguide by observing Dirac fermion transport phenomena
for a large $h_{0}$ value. Figure 3(d) shows a comparison of the current
densities between two cases; non-guided and guided transport of Dirac
fermions, which are respectively denoted by different color circles in Fig.
3(b). Indeed, for the small $h_{0}$ case, Dirac fermions are spread out over
the entire system, whereas for the large $h_{0}$ case, Dirac fermions are
perfectly guided inside the strained region where the nanowrinkle is created.

The waveguide mode from the band structures depicted in Fig. 1(e) is doubly
degenerate, leading to a conductance contribution as much as $2\times
2e^{2}/h$. However, the magnitude of the conductance plateau in Fig. 3(b) is
found to be $3\times 2e^{2}/h$, because the zero-energy mode of the zigzag
graphene nanoribbon contributes an additional $2e^{2}/h$ to the waveguide
mode.

Moreover, it is worth considering how large $h_{0}$ needs to be to create a
waveguide. To address this question, we now conduct a theoretical analysis
using a semiclassical approach based on the Dirac equation. Though the PMF
profile due to the nanowrinkle is non-uniform in the $y$ direction, we can
simplify the system by considering a staggered PMF profile, where each region
is under a uniform PMF. In this study, the reduced PMF strengths for a given
nanowrinkle with $h_{0}=3.69~{}\mathrm{nm}$ and $\sigma=2.46~{}\mathrm{nm}$
are calculated to be $2600~{}\mathrm{T}$ and $1310~{}\mathrm{T}$,
respectively. The corresponding magnetic lengths are found to be
$0.5~{}\mathrm{nm}$ and $0.7~{}\mathrm{nm}$, respectively. Since the magnetic
length indicates the radius of the snake-orbit trajectory, these length scales
correspond to the spectral width of the interface channels quantum
mechanically. The distance between channels in the nanowrinkle waveguide is
given by $\sigma=2.46~{}\mathrm{nm}$, which is much larger than the magnetic
lengths calculated for our nanowrinkle waveguide.

Similarly, we can evaluate the corresponding magnetic lengths for
$h_{0}=0.5~{}\mathrm{nm}$, $3.75~{}\mathrm{nm}$ and $5.24~{}\mathrm{nm}$,
which are larger than $\sigma/2$. Thus, for a small $h_{0}$ value, the
resulting PMF strength is insufficient to distinguish two waveguide channels,
resulting in the significant admixture of two snake-orbit states. This
analysis provides a critical value of $h_{0}$ for a nanowrinkle to be
considered a good waveguide, determined by finding the required $h_{0}$ value
for the magnetic length is smaller than $\sigma/2$. After some algebra, we
obtain the critical $h_{0}$ value of $2.13~{}\mathrm{nm}$ for the given
$\sigma=2.46~{}\mathrm{nm}$. As shown in Fig. 3(b) and (c), the $G_{41}$
spectra indeed vanish for $h_{0}>2.13~{}\mathrm{nm}$, as the conducting
channels start being spatially resolved by the sufficiently strong PMF.

### Bent waveguide for strain engineering

Figure 4: (a) Schematic view of the system with a bent nanowrinkle. The shaded
region indicates the existence of the bent nanowrinkle connecting two leads,
lead1 and lead2. (b) Conductance spectra versus Dirac fermion energy $E$ and
the separation between two leads $d$, for a given $h_{0}=3.69~{}\mathrm{nm}$.
(c) Conductance spectra versus Dirac fermion energy $E$ and the nanowrinkle
height $h_{0}$, for a given $d=0.984~{}\mathrm{nm}$. (d) Probability current
density maps at a given energy $E=0.27~{}\mathrm{eV}$, for different $d$
values, $0.984~{}\mathrm{nm}$ and $4.428~{}\mathrm{nm}$, respectively denoted
by colored circles in (b). A parameter used for the calculations is
$\sigma=2.46~{}\mathrm{nm}$.

So far, we have examine the properties of the nanowrinkle waveguide, which is
straight. In terms of strain engineering, the effects of bending on the
nanowrinkle waveguide can be important, especially when designing
sophisticated waveguide devices. For instance, electronic waveguides made of
semiconductor have been used to fabricate quantum devices like quantum
interferometry or flying qubit devices[13, 42, 43, 44]. The bent nanowrinkle
waveguide in our study is displayed in Fig. 4(a), which its conformal shape
formulated by

$\displaystyle
z\left(\vec{r}\right)=h_{0}e^{-\left[y-f\left(x\right)\right]^{2}/2\sigma^{2}},$
(8)

with a given functional landscape:

$\displaystyle f\left(x\right)=d\tanh{\left(\frac{x}{\xi}\right)},$ (9)

where $d$ is the separation between the centers of the two leads, and $\xi$
indicates the steepness of the bending. The bent angle is defined by using $d$
and $\xi$:

$\displaystyle\theta=\tan^{-1}{\left(\frac{d}{\xi}\right)}.$ (10)

Here, we consider an appropriate value of $\xi=24.6~{}\mathrm{nm}$, which is
within a practical device scale.

First, we examine the transport properties of the bent waveguide to determine
whether Dirac fermion transport is also well guided along the bent waveguide.
The results are plotted in Fig. 4(b). For small $d$, the conductance spectra
exhibit a clear plateau at $G=3\times 2e^{2}/h$, indicating the bent
nanowrinke also hosts the waveguide mode, as expected. As $d$ increases,
however, the conductance plateau is reduced to $G=2e^{2}/h$. This reduction
can be understood by examining how the current densities flow throughout the
system, as shown in Fig. 4(d). For the small bent angle $2.3^{\circ}$ (denoted
by the magenta circle), Dirac fermions propagate well along the bent
waveguide, similary to the straight nanowrinkle case. On the other hand, for
the sufficiently large bent angle $10.2^{\circ}$, the net flux of the
probability current via the waveguide modes is cancelled out due to the
coexistence of thr right- and left-propagating current flows. (Note that the
right-propagating mode delivers the same conductance contribution as the left-
propagating mode, even if the right-propagating mode is spatially separated at
$y=\pm\sigma$, as aforementioned.) Despite of the washed-out waveguide modes,
there still remains the $2e^{2}/h$ conductance contribution from the zero-
energy mode of the zigzag graphene nanoribbon.

To confirm the existence of the bent waveguide, we also examine the
conductance spectra through the given bent nanowrinkle with the given bent
angle $2.3^{\circ}$. Figure 4(c) presents the resulting conductance map
through the bent nanowrinkle as functions of $E$ and $h_{0}$. It is evident
that the transport properties of the bent nanowrikle are almost identical to
the straight nanowrinkle, clearly exhibiting the $3\times 2e^{2}/h$
conductance plateau.

In addition, one can observe that the conductance spectra shown in Fig. 4(b)
and (c) exhibit a fluctuating nature as a function of $E$. It is also worth
noting that the reduction in the conductance through the bent waveguide is due
to the emergent backscattering. The presence of the left-propagating mode in
Fig. 4(d) is allowed if the right-propagating electron from the left lead
(lead1) is backscattered. Such back-and-forth propagation along the
nanowrinkle waveguide can lead to Fabry-Perot interference in the conductance
spectra, resulting in the fluctuation in the conductance.

##  III Discussion

In this work, we have demonstrated that a nanowrinkle in graphene can be used
for a Dirac fermion waveguide due to the strain-induced pseudo-magnetic field
(PMF). The waveguide mode in the nanowrinkle is formed at the zero-PMF line,
contributing a conductance of $4e^{2}/h$. Interestingly, we have further
revealed that the nanowrinkle waveguide can be engineered by modifying its
shape. Dirac fermion transport remains well-guided along the bent waveguide,
if the bent angle is less than a certain value.

Our graphene nanowrinkle waveguide offers several advantageous for
applications in quantum information technologies. First, we can realize the
control of Dirac fermion transport in monolayer graphene without opening a
gap, thereby preserving its transport properties. With the rapid progress in
strain engineering techniques for graphene and 2D materials, such strain-
induced nanostructures in graphene are now feasible for experimental
studies[39, 29, 45, 41, 46, 47, 35, 36]. Second, our nanowrinkle waveguide
inherently minimize information loss due to leackage, as the snake-orbit
interface channel is basically topologically protected[48, 49]. Unless the PMF
profile of the nanowrinkle is significanlty altered, the existence of the
conducting channels between regions of opposite-sign PMF is robust against
common disorders such as charge inhomogeneity.

In conclusion, our results suggest that nanowrinkles in graphene can serve as
a building block for graphene-based quantum information technology, such as
flying qubits.

##  IV Methods

Finite-size tight-binding Hamiltonian construction and numerical calculations
of the transport properties of the graphene nanowrinkle waveguide are
performed using the kwant package with the mumps routines[50, 51].

## Data availability

The data supporting the findings of this study are available from the
corresponding authors upon reasonable request.

## Code availability

The codes used in this study are available from the corresponding author on
reasonable request.

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###### Acknowledgements.

This work was supported from the National Research Foundation of Korea(NRF)
grant funded by the Ministry of Education(No. RS-2023-00285353) and Ministry
of Science and ICT(No. NRF-2022R1F1A1065365). The authors are grateful to
Prof. Hee Chul Park for his insightful discussions and contributions to this
work.

## Author information

### Contributions

SJ performed the numerical calculations. NM designed and oversaw the project.
MCJ and NM contributed to writing the manuscript.

### Corresponding author

Correspond to Nojoon Myoung.

## Ethics declarations

### Competing interests

The authors declare no competing interests.