molecules

Article
Electron Transport Properties of Graphene/WS2 Van Der
Waals Heterojunctions
Junnan Guo 1 , Xinyue Dai 2 , Lishu Zhang 3 and Hui Li 1, *

1 Key Laboratory for Liquid-Solid Structural Evolution and Processing of Materials, Ministry of Education,
Shandong University, Jinan 250061, China; guojunnan113005@hotmail.com
2 Materdicine Lab, School of Life Sciences, Shanghai University, Shanghai 200444, China; dxy1120@shu.edu.cn
3 Peter Grünberg Institut (PGI-1) and Institute for Advanced Simulation (IAS-1), Forschungszentrum Jülich,
Jülich 52428, Germany; lis.zhang@fz-juelich.de
* Correspondence: lihuilmy@sdu.edu.cn

Abstract: Van der Waals heterojunctions of two-dimensional atomic crystals are widely used to build
functional devices due to their excellent optoelectronic properties, which are attracting more and
more attention, and various methods have been developed to study their structure and properties.
Here, density functional theory combined with the nonequilibrium Green’s function technique has
been used to calculate the transport properties of graphene/WS2 heterojunctions. It is observed that
the formation of heterojunctions does not lead to the opening of the Dirac point of graphene. Instead,
the respective band structures of both graphene and WS2 are preserved. Therefore, the heterojunction
follows a unique Ohm’s law at low bias voltages, despite the presence of a certain rotation angle
between the two surfaces within the heterojunction. The transmission spectra, the density of states,
and the transmission eigenstate are used to investigate the origin and mechanism of unique linear
I–V characteristics. This study provides a theoretical framework for designing mixed-dimensional
heterojunction nanoelectronic devices.

Keywords: graphene/WS2 heterojunctions; electronic transport; first-principles calculation

Citation: Guo, J.; Dai, X.; Zhang, L.;

1. Introduction
Li, H. Electron Transport Properties Two-dimensional (2D) layered materials have always been a cutting-edge field in
of Graphene/WS2 Van Der Waals condensed matter physics and materials research [1]. Various 2D layers can be combined
Heterojunctions. Molecules 2023, 28, using van der Waals (vdW) forces to build heterostructures with diverse functionalities [2].
6866. https://doi.org/10.3390/ These heterostructures exhibit a range of excellent properties and are applied to optoelec-
molecules28196866 tronic devices [3], providing unprecedented opportunities for the development of advanced
Academic Editor: Bryan M. Wong nanoelectronics devices [4].
As one of the atomically thin 2D materials, graphene [5] has attracted worldwide
Received: 30 August 2023 attention due to its excellent optical [6], electrical [7], and mechanical properties [8], and it
Revised: 26 September 2023
is expected to be used to build a new generation of miniaturized and intelligent electronic
Accepted: 27 September 2023
devices [9]. However, the absence of a band gap has limited the application of graphene,
Published: 29 September 2023
particularly in the semiconductor industry [10]. Significant efforts have been devoted
to addressing this issue in the gap-opening of graphene, like functionalization [11], dop-
ing [12], and the construction of heterostructures [13]. Recently, many graphene-based
Copyright: © 2023 by the authors.
vdW heterostructures have been investigated theoretically and experimentally [14]. For
Licensee MDPI, Basel, Switzerland. instance, Lan et al. transferred graphene grown on a copper foil to a sapphire substrate with
This article is an open access article Bi2 Te3 crystals via low-pressure chemical vapor deposition (CVD). The crystallized Bi2 Te3
distributed under the terms and was synthesized directly using spin-coated coring (SCCA). This procedure avoided any
conditions of the Creative Commons degradation of the nanoplates and significantly improved the quality of the heterojunction
Attribution (CC BY) license (https:// sample [15]. Hu et al. utilized a polymethyl methacrylate (PMMA)/polydimethylsiloxane
creativecommons.org/licenses/by/ (PDMS) blend to transfer metal-catalyzed CVD-fabricated graphene/SiNWS heterojunc-
4.0/). tions onto stretchable polytetrafluoroethylene (PTFE) substrates. The high preparation

Molecules 2023, 28, 6866. https://doi.org/10.3390/molecules28196866 https://www.mdpi.com/journal/molecules

Molecules 2023, 28, 6866 2 of 12

efficiency and outstanding quality were extremely encouraging for daily industrial pro-
duction and life [16]. Ren et al. developed a novel flexible self-powered photodetector
that transfers electrons through a solid electrolyte. The developed flexible WS2 /graphene
photodetector displayed a quick photo response time and high photosensitivity [17]. Liu
et al. fabricated Bi2 Se3 /graphene heterojunctions using molecular beam epitaxy and ob-
served a spiral growth mechanism during the growth process [18]. By vertically stacking
single-layer MoS2 /h-BN/graphene, Lee’s team created random access memory with tun-
neling. It had excellent stretchability, long retention times, and highly dependable memory
performance [19]. Additionally, Liu et al. investigated different conceivable atomic configu-
rations of phosphorene/graphene in-plane heterojunctions and their effects on interfacial
heat conductivity by using density functional theory calculations and molecular dynamics
simulations [20]. Gao et al. simulated the heat transfer properties of graphene/MoS2 hetero-
junctions using nonequilibrium molecular dynamics simulations and found that the degree
of lattice matching of graphene and MoS2 had an effect on phonon thermal transport [21].
However, the majority of these studies on graphene heterojunctions primarily focused on
their electronic structures [22], preparation methods [23], and applications [24]. Little re-
search has been conducted on their electron transport properties and intrinsic mechanisms.
In this context, constructing new graphene heterojunctions and studying their electron
transport properties are essential if one wants to realize the practical application of graphene
heterojunctions in nanoelectronic devices. With excellent electron mobility and a large
direct band gap, monolayer WS2 has a lot of potential uses in nanodevices [25]. In particular,
in recent years, there have been significant breakthroughs in its synthesis and applications.
For example, Prof. Feng’s group produced monolayer triangular WS2 single crystal wafers
with excellent uniformity, large size, and high quality by controlling the nucleation density
by changing the time of the introduction of the sulfur precursor and the distance between
the tungsten source and the growth substrate [26]. Furthermore, some researchers have
used chemical doping to significantly improve the optoelectronic performance of WS2
field-effect transistors [27]. Inspired by these advancements, we selected monolayer WS2 to
create a series of graphene/WS2 heterojunction models and design nanoelectronic devices.
We systematically investigated their electronic structures and transport properties using
first-principles methods based on the density functional theory (DFT) and nonequilibrium
Green’s function (NEGF) [28].

2. Results and Discussions

The hexagonal unit cell of WS2 was the same as that of graphene. For graphene and
WS2 , the optimized lattice parameters were 2.45 Å and 3.15 Å, respectively. The unit cell
parameters we calculated closely matched experimental results [29,30].
To construct the graphene/WS2 heterojunctions, we used a 3 × 3 × 1 supercell of
WS2 and a 4 × 4 × 1 supercell of graphene with 68 total atom numbers, and a 4 × 4 × 1
supercell of WS2 and a 5 × 5 × 1 supercell of graphene with 109 total atom numbers. In this
orientation, both components maintained their original hexagonal lattices without surface
rotation and exhibited slight lattice mismatches of 3.1% and 2.4%, respectively. The inter-
layer spacings of the equilibrium geometries of these two heterojunctions were 3.41 Å and
3.46 Å, respectively, which are typical distances in graphene-based vdW heterostructures
with weak interactions.
However, the devices built from the above two heterojunctions contained 366 and
603 atoms, separately. Due to the limitations of quantum-mechanics-based calculations
used in this study, we continued to construct a series of heterojunctions with specific
rotation angles between each surface to reduce the models’ sizes. In these heterojunctions,
the interatomic distances were consistently around 3.4 Å, indicating weak vdW interactions.
At the same time, we could also analyze the electron transport properties of devices that
had different rotation angles. The equilibrium geometries of heterojunctions and their
related parameters are shown in Figure 1 and Table 1.
Molecules 2023, 28, x FOR PEER REVIEW 3 of 13

Molecules 2023, 28, 6866 that had different rotation angles. The equilibrium geometries of heterojunctions and3 their
of 12
related parameters are shown in Figure 1 and Table 1.

Figure 1. Top views of (a) Gr/WS2 -1, (b) Gr/WS2 -2, (e) Gr/WS2 -3, (f) Gr/WS2 -4, (g) Gr/WS2 -5,
Figure
(h) Gr/WS1. Top views
2 -6, (i) Gr/WSof (a)
2 -7,Gr/WS
and (j)2-1, (b) Gr/WS
Gr/WS 2-2, (e) Gr/WS
2 -8 ball-and-stick 2-3, (f) Side
models. Gr/WS 2-4, (g)
views Gr/WS
of (c) Gr/WS2-5, 2(h)
-1
Gr/WS
and -6, (i) Gr/WS
(d)2Gr/WS 2 -2. 2 -7, and (j) Gr/WS 2 -8 ball-and-stick models. Side views of (c) Gr/WS 2 -1 and (d)
Gr/WS2-2.
Table 1. The related parameters of heterojunctions.
Table 1. The related parameters of heterojunctions.
Lattice Parameters of Rotation Angle of Lattice Parameters of Rotation Angle of WS2
Heterojunction
Lattice Parameters
Graphene (Å) Rotation
Graphene Angle
(◦ ) of Lattice WSParame-
2 (Å) Rotation(◦An- )
Lattice Mismatch (%)
Heterojunction Lattice Mismatch (%)
Gr/WS2 -1 of Graphene
a = b = 9.8 (Å) Graphene
0.0 (°) ters aof
= bWS= 9.52 (Å) gle of WS 0.02 (°) 3.1
Gr/WS2 -2 a = b = 12.3 0.0 a = b = 12.6 0.0 2.4
Gr/WS
Gr/WS2 -32-1 aa ==bb= =6.59.8 0.0
21.8 a a==bb == 6.3
9.5 0.0 60.0 3.13.1
Gr/WS 2-2 aa== bb ==6.5
12.3 0.0 a =a =bb== 12.6 0.0180.0 2.43.1
Gr/WS2 -4 a = b = 6.5 141.8 a = b = 6.3 60.0 3.1
Gr/WS2 -5 21.8 6.3
Gr/WS2-3
Gr/WS 2 -6 aa ==bb= =6.56.5 21.8
141.8 a a==bb == 6.3
6.3 60.0180.0 3.13.1
Gr/WS2 -7 a = b = 8.5 0.0 a = b = 8.3 21.8 2.1
Gr/WS
Gr/WS2 -82-4 aa ==bb= =8.56.5 141.8
120.0 a a==bb == 8.3
6.3 60.021.8 3.12.1
Gr/WS2-5 a = b = 6.5 21.8 a = b = 6.3 180.0 3.1
Gr/WS2-6 a = b = 6.5In order to prove
141.8the thermodynamic a = b = 6.3stability of180.0
these heterojunctions, 3.1the binding
Gr/WS2-7 a = benergies
= 8.5 of the graphene/WS
0.0 a = b = 8.3 21.8 2.1
2 vdW heterojunctions were calculated to assess the system
Gr/WS2-8 a = bstability,
= 8.5 as follows: 120.0 a = b = 8.3 21.8 2.1

In order to proveEb =theE(thermodynamic

heterojunction) − stability of these
E(graphene ) − heterojunctions,
E(WS2 ) the binding
energies of the graphene/WS2 vdW heterojunctions were calculated to assess the system
stability,
where as follows:
E(heterojunction), E(graphene), and E(WS2 ) represent the total energy of the hetero-
junctions, graphene layers, and WS2 layers, respectively. The calculated binding energies
are presented in Table𝐸𝐸2.𝑏𝑏 = The𝐸𝐸(heterojunction)
negative binding − energies
𝐸𝐸(graphene) − table
in the 𝐸𝐸(WSindicate
2) the stability
of
where E(heterojunction), E(graphene), and E(WS2) represent the total energy of the was
these systems. Upon comparison, we observed that the most stable heterojunction het-
Gr/WS 2 -1. Another
erojunctions, regularity
graphene layers,we found
and WS2 was thatrespectively.
layers, smaller heterojunctions
The calculatedwere more stable
binding ener-
when thepresented
gies are two layersinwere
Tablenot2. rotated. However,
The negative bindingwhen there exist
energies rotation
in the angles between
table indicate the sta-
the two
bility oflayers, the stability
these systems. Upon of comparison,
the heterojunctions decreased
we observed that and the larger
the most stableheterojunctions
heterojunction
were more stable.
was Gr/WS 2-1. Another regularity we found was that smaller heterojunctions were more

stable when the two layers were not rotated. However, when there exist rotation angles
between the two layers, the stability of the heterojunctions decreased and the larger het-
erojunctions were more stable.
Molecules 2023, 28, x FOR PEER REVIEW 4 of 13

Table 2. The binding energy of heterojunctions.

Molecules 2023, 28, 6866 4 of 12
Energy of Heterojunc- Binding Energy
Heterojunction Energy of Graphene (eV) Energy of WS2 (eV)
tion (eV) (eV)
Gr/WS2-1 Table −5038.3 −10,206.0
2. The binding energy of heterojunctions. −15,246.9 −2.6
Gr/WS2-2 −7874.7 −18,145.4 −26,022.2 −2.1
Energy of Heterojunction
Gr/WS2-3
Heterojunction −2204.4(eV)
Energy of Graphene −4563.7
Energy of WS 2 (eV) −6741.6
(eV) −0.6(eV)
Binding Energy

Gr/WS
Gr/WS 2 -12-4 −2204.4
−5038.3 −4563.7
−10,206.0 −6741.6
−15,246.9 −0.6
−2.6
Gr/WS
Gr/WS2 -2
Gr/WS2 -3
2-5 −2204.4
−7874.7
−2204.4
−4563.7
−18,145.4
−4563.7
−6741.6
−26,022.2
−6741.6
−0.6
−2.1
−0.6
Gr/WS
Gr/WS 2 -42-6 −2204.4
−2204.4 −4563.7
−4563.7 −6741.6
−6741.6 −0.6
−0.6
Gr/WS2 -5 −2204.4 −4563.7 −6741.6 −0.6
Gr/WS
Gr/WS 2 -62-7 −3779.3
−2204.4 −7939.2
−4563.7 −11,719.4
−6741.6 −0.9
−0.6

Gr/WS2 -82-8
Gr/WS −3379.3 −7939.2 −11,719.4 −0.9
Gr/WS2 -7 −3779.3 −7939.2 −11,719.4 −0.9
−3379.3 −7939.2 −11,719.4 −0.9

We initially investigated the electronic properties of Gr/WS2-1 and Gr/WS2-2 to deter-

We initially investigated the electronic properties of Gr/WS2 -1 and Gr/WS2 -2 to
mine whether they can be transported as electronic devices. As plotted in Figure 2a, gra-
determine whether they can be transported as electronic devices. As plotted in Figure 2a,
phene exhibits metallic properties with a zero bandgap semiconductor, where the top va-
graphene exhibits metallic properties with a zero bandgap semiconductor, where the top
lence band and bottom conduction band intersect at the K point. In contrast, WS2 is a sem-
valence band and bottom conduction band intersect at the K point. In contrast, WS2 is a
iconductor with a direct band gap of 1.95 eV, as shown in Figure 2b. It is worth noting that
semiconductor with a direct band gap of 1.95 eV, as shown in Figure 2b. It is worth noting
our calculations closely aligned with other theoretical predictions and were slightly lower
that our calculations closely aligned with other theoretical predictions and were slightly
than experimental values [31]. This discrepancy can be attributed to the inherent limita-
lower than experimental values [31]. This discrepancy can be attributed to the inherent
tions of the GGA-PBE method, which tends to overestimate lattice constants and under-
limitations of the GGA-PBE method, which tends to overestimate lattice constants and
estimate band gaps. Hybrid functionals, such as meta-GGA, HSE06, etc., are known to
underestimate band gaps. Hybrid functionals, such as meta-GGA, HSE06, etc., are known
provide more accurate bandgap calculations [32]. However, the WS2 bandgap calculated
to provide more accurate bandgap calculations [32]. However, the WS2 bandgap calculated
by meta-GGA was 2.13 eV, which was only slightly higher than the PBE value (1.95 eV).
by meta-GGA was 2.13 eV, which was only slightly higher than the PBE value (1.95 eV).
Thus, webelieve
Thus, we believethat
thatthe
theGGA-PBE
GGA-PBE approach
approach waswas accurate
accurate enough
enough for calculation
for our our calculation
and
and did not significantly impact other aspects of the analysis, such as energy
did not significantly impact other aspects of the analysis, such as energy band band struc-
structure
tureelectron
and and electron transport.
transport.

Figure 2.
Figure Band structures
2. Band structures of
of the
thestand-alone
stand-alone(a)
(a)graphene
grapheneandand(b)
(b)WS
WS ; (c,d) are band structures
2; 2(c,d) are band structures of

Gr/WS2-1 2and Gr/WS2-2. The

of Gr/WS -1 and Gr/WS 2 red and blue lines represent the top of the valence
-2. The red and blue lines represent the top of band and
the valence theand
band bottom
the
of the conduction
bottom band. (e)
of the conduction The(e)
band. PDOS and DOS
The PDOS andofDOS
the of
graphene and WS
the graphene and 2 components in theinvdW
WS2 components the
Gr/WS
vdW 2-1.
Gr/WS 2 -1.

Figure 2c and d display the band structures of Gr/WS2 -1 and Gr/WS2 -2, which are
simple superpositions of graphene and WS2 and preserve their electronic systems. Notably,
the valence band’s top and the conduction band’s bottom still intersected at the K point in
the Brillouin zone, indicating that the Dirac point still exists in the heterojunction. Gr/WS2 -
Molecules 2023, 28, 6866 5 of 12

1 behaved as an N-type semiconductor, with the Ec and Ev of WS2 shifting downwards.

Additionally, the Fermi energy level turned from near the top of the valence band to near
the bottom of the conduction band. Conversely, Gr/WS2 -2 exhibited P-type semiconductor
properties, with the Fermi energy level still close to the top of the valence band, but
the conduction band bottom and valence band top shifted from the original G to the K
point. This indicates that factors such as layer spacing, the degree of mismatch, and lattice
parameters within the heterojunction influence its electronic energy band.
Next, we calculated the density of states (DOS) and the projected calculation density
of states (PDOS). Due to the similarity in the calculation results, we present the results for
Gr/WS2 -1 as an example. According to Figure 2e, near the Fermi level, the 2p orbital of
the carbon atom in graphene plays a vital role in the density of states. The 5d orbital of
the W atom also makes a contribution. Contributions from other valence electron orbitals
can be disregarded. The absence of resonance peaks indicates that there was no bonding
between WS2 and C. Instead, weak van der Waals forces maintained the interlayer stability
between the heterojunctions, corresponding to optimized interlayer spacing of around
3.4 Å. This weak hybridization between the graphene and WS2 is another indication of
why the graphene’s Dirac points are still present in the heterojunctions.
As depicted in Figure 3, when there is a certain rotation angle between the two surfaces,
no matter the change in the lattice constants or the rotation angle of graphene or WS2 , its
effect on the energy band is little. But when the lattice parameter of heterojunctions is
increased to around 8 Å, the Dirac cone of graphene shifts from K to G point due to the
inequivalent K and K’ points being folded and coupled into the same G-point (Figure 3e,f).
However, the Dirac cone does not open. We predicted that these six heterojunctions had
comparable electronic transport properties. Consequently, nanoelectronic devices could be
built using heterostructures with rotation angles to reduce device size while maintaining
their high transport properties.
With the Gr/WS2 -3 and Gr-1 (composed of graphene, with the same lattice parameter
and rotation angle as Gr/WS2 -3), we built two devices, as depicted in Figure 4. As seen in
the enlarged area, the rotation angle between graphene and WS2 was still maintained. The
poles of the device formed by themselves, the current transport direction was along the
Z-axis, and the surface was perpendicular to the X-axis.
The I–V characteristics of the devices in a bias zone [0.0 V, 2.0 V] were calculated
to explore the transport characteristics of these two devices, and the findings are shown
in Figure 4. We can see from the current-voltage (I–V) characteristic curves (Figure 4c)
that the heterojunction had comparable transport properties to graphene, unlike some
typical heterojunction semiconductor devices. Interestingly, the Ohmic behavior of linear
I–V curves was found in the 0–1.2 V bias voltage. After 1.2 V, the slope of the I–V curve
gradually increased, leading to nonlinear transport properties. This was caused by a certain
degree of rotation in the graphene and heterojunction, while the transport direction was
primarily along the armchair direction of the graphene. Simultaneously, it became evident
that the transport properties of both devices changed gradually as the voltage increased,
signifying a weakened coupling between WS2 and graphene. A nonlinear relationship only
began to emerge at high bias voltages. Compared to other graphene-based heterojunctions,
the transport current of graphene/WS2 was nearly one order of magnitude higher than
that of graphene/MoS2 in-plane heterojunctions [33,34], graphene/BN heterojunctions [35],
and so on. In addition, when compared to other WS2 -based heterojunctions, such het-
erojunctions could behave up to two orders of magnitude higher than that of WS2 /WSe2
heterojunctions [36], with greater performance than that of MoS2 /WS2 heterojunctions [37].
Thus, we can conclude that such heterojunctions can greatly enhance the transport current
and decrease the contact resistance, which will be very important for achieving superior
optoelectronic devices such as vertical field-effect transistors (FETs). Our calculations can
reveal why graphene/WS2 heterojunctions are widely used to build FETs and have superior
behavioral properties [38–41]. In addition, the heterojunction used in our calculations not
 Molecules 2023, 28, 6866 6 of 12

FOR PEER REVIEW 6 of 13

only maintained the perfect transport properties but also largely reduced the size of the
electronic devices, which is very important in the post-Moore era.

Figure 3. Band structures of (a) Gr/WS2 -3, (b) Gr/WS2 -4, (c) Gr/WS2 -5, (d) Gr/WS2 -6, (e) Gr/WS2 -7,
Figure 3. Band structures of (a) 2Gr/WS
and (f) Gr/WS 2-3, (b)
-8. The orange Gr/WS
and 2-4, represent
blue lines (c) Gr/WS -5, of
the2top (d)theGr/WS
valence2-6, (e)and
band Gr/WS 2-7, of
the bottom
and (f) Gr/WS2-8. The orange and blue lines represent the top of the valence band and the bottom of
the conduction band.
the conduction band.

With the Gr/WS2-3 and Gr-1 (composed of graphene, with the same lattice parameter
and rotation angle as Gr/WS2-3), we built two devices, as depicted in Figure 4. As seen in
the enlarged area, the rotation angle between graphene and WS2 was still maintained. The
poles of the device formed by themselves, the current transport direction was along the
Z-axis, and the surface was perpendicular to the X-axis.
Molecules 2023,28,
Molecules2023, 28,x6866
FOR PEER REVIEW 7 7ofof13
12

Figure 4. The device configuration with (a) Gr/WS2 -3 and (b) Gr-1. (c) I–V characteristics of devices.
Figure 4. The device configuration with (a) Gr/WS2-3 and (b) Gr-1. (c) I–V characteristics of devices.
Although the differences in transport properties between these two devices were
The
slight, theI–V characteristics
transport mechanism of the devicesdifferent
exhibited in a biasphenomena
zone [0.0 V,due 2.0toV]thewere
weak calculated
vdW forces to
explore the transport characteristics of these two devices,
between the WS2 and graphene. The most understandable depiction of the behavior of and the findings are shown in
Figure
electron4. transport
We can see wasfrom thethe current-voltage
transmission (I–V)T(E),
spectrum characteristic curves (Figure
and the transmission 4c) that
coefficient
the heterojunction
of each energy point hadwas comparable
determined transport properties to
by diagonalizing thegraphene,
transmission unlike somefrom
matrix typicalthe
heterojunction
eigenvalues of semiconductor
electron transmission. devices.Therefore,
Interestingly, the Ohmic
we calculated thebehavior
transmission of linear
spectra I–V of
curves was found in the 0–1.2 V bias voltage.
the above devices to further study their transport properties. After 1.2 V, the slope of the I–V curve grad-
ually Generally
increased,speaking,
leading to thenonlinear
magnitude transport properties. This
of the transmission was caused
coefficient near the byFermi
a certain
level
degree of rotation
represents in the graphene
the transport capabilityand of theheterojunction,
device, especiallywhileatthe thetransport
Fermi level. direction was
The larger
primarily along thecoefficient
the transmission armchair direction
at the Fermi of the graphene.
level, Simultaneously,
the stronger the transport it became evident
capability. As
that
shownthe in
transport properties
Figure 5a,d, these of two both devices
devices changed
exhibited gradually
metallic as the voltage
properties, increased,
corresponding to
signifying a weakened
the current–voltage coupling
curves. The between WS2 and graphene.
electron transmission spectraAofnonlinear
graphene relationship
devices and
only
Gr/WS began to emerge
2 -3 devices at high bias
displayed voltages.
quantum stepsCompared
between to −1other
eV and graphene-based
1 eV, resembling hetero-the
junctions, the transportnanowires.
ideal one-dimensional current ofAnd graphene/WS
the electron 2 was nearly one
transmission order of at
probability magnitude
the Fermi
higher
energythan
levelthatwasofalmost
graphene/MoS
zero, which 2 in-plane
shows aheterojunctions
band gap feature [33,34],
between graphene/BN
the conduction hetero-
and
junctions [35], and
valence bands, so on. In addition,
corresponding when
to a Dirac compared
cone to otherband
in the energy WS2-based
structure.heterojunctions,
Although the
such
systemheterojunctions
had almost nocould electronsbehave
passingup to two orders
through at this of magnitude
energy, at higherhigher
energiesthanelectrons
that of
WS2/WSe2 heterojunctions [36], with greater performance than that of MoS2/WS2 hetero-
could easily tunnel through the potential barrier, increasing their mobility and the step
junctions [37]. Thus, we can conclude that such heterojunctions can greatly enhance the
transmission coefficient, which indicates that there were several electron transmission
transport
channels in current and decrease
these devices. After the contact resistance,
the formation which will be
of the heterojunction, manyveryspikes
important
appearedfor
achieving
away from superior
the Fermi optoelectronic
energy level,devicesshowing such
thatasthevertical
coupling field-effect
betweentransistors
the graphene (FETs).
and
Our
WS2calculations
was weak. can Thereveal
band gap whyof graphene/WS
graphene was 2 heterojunctions
not open, although are widely
it tends usedto to
bebuild
open,
FETs and have superior behavioral properties [38–41]. In addition, the heterojunction used
which does not have a significant influence on its transport properties.
in ourTocalculations
further shed notlight
onlyon the inherent
maintained themechanisms of these
perfect transport two devices,
properties but alsowe largely
discuss
DOS around the Fermi level for these devices. Figure 5a,d illustrate
reduced the size of the electronic devices, which is very important in the post-Moore that the DOS of theera.
two
devices were zero at the Fermi energy level, corresponding
Although the differences in transport properties between these two devices were to their electron transmission
spectrum.
slight, Before themechanism
the transport constructionexhibited
of the heterojunction,
different phenomenathe contribution
due to of theDOSweak near
vdW the
Fermi energy level originated mainly from the 2p orbitals
forces between the WS2 and graphene. The most understandable depiction of the behavior of the graphene carbon atoms.
After
of the formation
electron transport of was thethe
heterojunction,
transmission spectrumthe contribution
T(E), and wasthe mainly from thecoefficient
transmission 2p orbital
of the graphene carbon atom and the 5d of the W atom. We can see clearly that several
of each energy point was determined by diagonalizing the transmission matrix from the
peaks exceeded 100 in the Gr/WS2 -3, more than twice that of the Gr-1. The highest peaks
eigenvalues of electron transmission. Therefore, we calculated the transmission spectra of
in the valence band region were observed at −1.92 eV, while those in the conduction band
the above devices to further study their transport properties.
region were found at 1.44 eV. These peaks serve to protect fewer delocalized states near the
Generally speaking, the magnitude of the transmission coefficient near the Fermi
Fermi level.
level represents the transport capability of the device, especially at the Fermi level. The
larger the transmission coefficient at the Fermi level, the stronger the transport capability.
 trons could easily tunnel through the potential barrier, increasing their mobility and the
step transmission coefficient, which indicates that there were several electron transmis-
sion channels in these devices. After the formation of the heterojunction, many spikes ap-
peared away from the Fermi energy level, showing that the coupling between the gra-
Molecules 2023, 28, 6866 phene and WS2 was weak. The band gap of graphene was not open, although it tends 8 ofto
12
be open, which does not have a significant influence on its transport properties.

Figure
Figure 5.
5. Transmission
Transmissionspectrum
spectrumand
andDOS
DOSofof(a)(a)Gr-1
Gr-1 device and
device (d)(d)
and Gr/WS
Gr/WS 2-3 device at a at
2 -3 device free ap-
a free
plied biasbias
applied (0.0(0.0
V). V).
Transmission eigenstate
Transmission of Gr-1
eigenstate (b,c)(b,c)
of Gr-1 andand
Gr/WS 2-3 (e,f)
Gr/WS 2 -3 around FermiFermi
(e,f) around level with
level
an isovalue
with of 0.21.
an isovalue of 0.21.

To further
Here, shed light
the dominant on the inherent
transmission mechanisms
eigenstates near theof theseenergy
Fermi two devices, we discuss
level at equilibrium
DOS
were around
calculatedthetoFermi
explorelevel
thefor these devices.
physical Figure
roots of their 5a,d illustrate
transport that the
phenomena. TheDOS of the
calculated
two devices
results were5b,c,e,
in Figure zero at
and thef showed
Fermi energy level,
that the corresponding
transmission of twotodevices
their electron
around transmis-
the Fermi
sion
levelspectrum.
was providedBeforebythe
two construction of thechannels,
major transport heterojunction, the transmission
both with contribution of DOS near
eigenvalues
the Fermi energy level originated mainly from the 2p orbitals of the graphene carbon at-
of nearly 1.000. The transmission eigenstates of both devices exhibited delocalization
oms. After the
throughout theformation of the region,
whole central heterojunction,
resultingthe
in contribution was mainly
significant transport from thenear
capability 2p
orbital of the graphene carbon atom and the 5d of the W atom. We can see clearly that
the Fermi energy level. We can see that the electronic states were evenly distributed in
the diffusion region between the left and right electrodes, along the graphene
several peaks exceeded 100 in the Gr/WS2-3, more than twice that of the Gr-1. The highest armchair
direction.
peaks in theThis indicates
valence band that
regionthese
werestates were at
observed all−1.92
π-orbitals of the
eV, while C atom
those in theofconduction
graphene,
leading to their metallic characteristic. However, the contribution of WS2 in Gr/WS2 -3 was
almost negligible.
It is well known that the study of transmission spectra at non-zero bias voltages can
provide useful information for the study of I–V characteristics. This is because the current is
defined by the integrated area of the transmission curve within the bias window, as shown
by the Landauer–Buttiker formula. As a result, we calculated the transmission spectra
of the Gr-1 and Gr/WS2 -3 devices under 0.4, 0.8, 1.2, 1.6, and 2.0 to further reveal their
transport phenomena (Figure 6a,b). The bias window’s perimeter is represented by the
colored parts. The effective integral area of the transmission curve within the bias window
grew with increased bias, producing a linear I–V characteristic, as we can see from both
devices. However, when the bias window increased to 1.2 V, the step transmission spectrum
started to change shape and expand in an arc, so the I–V curve began to show non-linear
features, and the slope subsequently increased. It is evident from the transmission spectrum
that quantum steps are always present within the bias window at different bias voltages
and that the steps shift as the bias window expands. The movement tendency of the steps
Molecules 2023, 28, 6866 9 of 12

in the conduction and valence band regions was indicated by the arrows, respectively. The
number of wave valleys within the bias window in the Gr/WS2 -3 devices progressively
increased. Spikes far from the Fermi energy level moved in the opposite direction and were
unable to move inside the bias window, so the contribution of these spikes to the transport
properties was almost negligible. Interestingly, the lowest transmission probability was
Molecules 2023, 28, x FOR PEER REVIEW 10 of 13
always located at the boundary of the bias window, and as the bias increased from 0 V to
2 V, the gap shifted to the boundary of the bias window.

Figure
Figure 6.
6. Transmission
Transmission spectra
spectra of (a) Gr-1
of (a) Gr-1 device
device and
and (b)(b) Gr/WS
Gr/WS22-3
-3device
deviceat
atnonzero
nonzerobiasbiasvoltage.
voltage.
Transmission eigenstates of
Transmission eigenstates of Gr-1
Gr-1 (c)
(c) and
and Gr/WS
Gr/WS22-3-3 (d)
(d) at
at 2.0
2.0 V
V with
with an
an isovalue
isovalue of
of 0.21.
0.21. The
The colored
colored
arrows
arrows represent
represent the
the movement
movement of the steps
of the steps in
in the
the conduction
conduction and and valence
valence band
band regions.
regions.

It
It is
is worth
worth noting
noting that
that the
the transmission
transmission eigenstates
eigenstates ofof electrons
electrons may
may change
change under
under
external
external biasbias after
afterforming
formingthe theheterojunction.
heterojunction.Therefore, thethe
Therefore, electronic transmission
electronic transmission ei-
genstates
eigenstatesofofGr-1Gr-1and
andGr/WS
Gr/WS 2-3 devices
2 -3 deviceswere
werecalculated
calculatedatatdifferent
differentbias
biasvoltages.
voltages. Before
Before
2.0
2.0 V, the transmission
V, the transmission eigenstates
eigenstates of of both
both devices
devices were
were mainly
mainly contributed
contributed by by the
the two
two
transmission channels of graphene. However, when the bias voltage
transmission channels of graphene. However, when the bias voltage increased to 2.0 V, increased to 2.0 V,
the transmission channels at the Fermi level of Gr/WS -3 changed from
the transmission channels at the Fermi level of Gr/WS22-3 changed from two to multiple two to multiple
channels, as
channels, asshown
shownininFigure
Figure6c,d.
6c,d.WSWS 2 started
2 started
to participate
to participate in transport,
in the the transport, but
but its its
elec-
electronic state was localized at the left electrode and the transmission eigenvalue
tronic state was localized at the left electrode and the transmission eigenvalue was so small was so
smallit that
that it can
can still bestill be disregarded.
disregarded.
3. Computational Method
3. Computational Method
DFT implemented in the Atomistix ToolKit (ATK) package was used to optimize the ge-
DFT implemented in the Atomistix ToolKit (ATK) package was used to optimize the
ometry structures and calculate the electronic structures of graphene/WS2 heterostructures.
geometry structures and calculate the electronic structures of graphene/WS2 heterostruc-
The exchange–correlation function is a generalized gradient approximation (GGA) [42]
tures. The exchange–correlation function is a generalized gradient approximation (GGA)
of Perdew–Burke–Ernzerhof (PBE) [43]. The selected valence electron configurations in
[42] of Perdew–Burke–Ernzerhof
our calculation were W 5d4 6s2 , S(PBE)
3s2 3p[43].
4 , C The selected
2s2 2p valence
2 . In order electron
to meet configurations
the computational
in our calculation
precision, the linearwere W 5d 6s of
4
combination
2 , Satomic
3s 3p orbitals
2 4 , C 2s 2p
2 2 . In order
(LCAO) basistowas
meet the computational
selected for all atoms.
precision, the linear combination of atomic orbitals (LCAO) basis was selected for all at-
oms. Double-ζplus polarization (DZP) basis sets were adopted for the local atomic numer-
ical orbitals, and norm-conserving pseudo-potentials were employed. The Monkhorst–
Pack k-points of 5 × 5 × 1 were used to sample the Brillouin zone [44]. The cut-off energy
for the density mesh and the electron temperature were set to 75 Ha and 300 K, accord-
Molecules 2023, 28, 6866 10 of 12

Double-ζplus polarization (DZP) basis sets were adopted for the local atomic numerical
orbitals, and norm-conserving pseudo-potentials were employed. The Monkhorst–Pack
k-points of 5 × 5 × 1 were used to sample the Brillouin zone [44]. The cut-off energy for
the density mesh and the electron temperature were set to 75 Ha and 300 K, accordingly.
The device’s performances were studied with the DFT coupled with the NEGF method,
using the ATK package. A 15 Å vacuum layer along the X-direction was used to avoid
interactions between periodic images that were nearest neighbors. For self-consistent
calculation, the k-points of 5 × 5 × 100 were used for device models. The other parameters
of the DFT calculation remained unchanged, and the energy convergence criterion was
set to 10−4 eV. Before analyses, the devices were fully optimized by the quasi-Newton
approach until all residual stresses on each atom were less than 0.05 eV. The devices’
electronic properties were investigated by computing their currents, the density of states,
and the transmission spectra, and the current I through the device was calculated using the
Landauer–Buttiker equation [45]:
Z +∞
2e
I= dE( T ( E, V )( f 1 ( E) − f 2 ( E)))
h −∞

The quantity T ( E) is the transmission function, which expresses the likelihood that
electrons will go through the device from source to drain; f 1,2 ( E) denote the Fermi
functions of the source and drain electrodes; and e and h are the electron charge and
Planck’s constant, respectively.

4. Conclusions
In this work, we systematically studied the electronic transport properties and intrinsic
mechanisms of graphene/WS2 heterojunctions using first-principal calculations. Unique
linear I–V characteristics were found among the devices. Even though there was an
angle between the two surfaces, the heterojunction continued to exhibit this intriguing
Ohm’s law behavior. The transmission spectra, the density of states, and the transmission
eigenstate were calculated to explain this phenomenon. After forming the heterojunctions,
the quantum steps near the Fermi level approximated an ideal one-dimensional nanowire.
The DOS shows that the vdW heterojunctions significantly increased the number of peaks
and improved the maximum value of peaks, which protected less delocalized states near
the Fermi level. The transmission eigenstates showed that the high transport properties
came from the π orbitals of the C atoms in the graphene armchair direction. This study
provides valuable insights into the transport properties of graphene heterojunctions and
the potential fabrication of mixed-dimensional heterojunctions.

Author Contributions: Data curation, investigation, writing—original draft, writing—review &

editing, J.G.; writing—review & editing, X.D., L.Z. and H.L. The manuscript was written through
contributions of all authors. These authors contributed equally. All authors have read and agreed to
the published version of the manuscript.
Funding: This research was funded by the National Natural Science Foundation of China (NNSFC)
(Grant No. 52171038). This work was also supported by Special Funding in the Project of the
Taishan Scholar Construction Engineering and the program of Jinan Science and Technology Bureau
(2020GXRC019), as well as the new material demonstration platform construction project from the
Ministry of Industry and Information Technology (2020-370104-34-03-043952-01-11) and the Key
Research and Development Plan of Shandong Province (2021SFGC1001).
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Data will be made available on request.
Acknowledgments: The authors would like to thank Kangwei Wu and Jie Li for their help.
Conflicts of Interest: The authors declare no conflict of interest.
Molecules 2023, 28, 6866 11 of 12

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