Chinese Physics Letters, 2021, Vol. 38, No. 9, Article code 097101 Manipulation of Dirac Fermions in Nanochain-Structured Graphene Wen-Han Dong (董文翰)1†, De-Liang Bao (包德亮)1†, Jia-Tao Sun (孙家涛)2*, Feng Liu (刘峰)3, and Shixuan Du (杜世萱)1,4,5* Affiliations 1Institute of Physics and University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100190, China 2School of Information and Electronics, MIIT Key Laboratory for Low-Dimensional Quantum Structure and Devices, Beijing Institute of Technology, Beijing 100081, China 3Department of Materials Science and Engineering, University of Utah, Salt Lake City, Utah 84112, USA 4CAS Center for Excellence in Topological Quantum Computation, Beijing 100190, China 5Songshan Lake Materials Laboratory, Dongguan 523808, China Received 5 July 2021; accepted 13 August 2021; published online 2 September 2021 Supported by the National Key Research and Development Program of China (Grant Nos. 2020YFA0308800 and 2016YFA0202300), the National Natural Science Foundation of China (Grant Nos. 11974045 and 61888102), Chinese Academy of Sciences (Grant No. XDB30000000), Beijing Institute of Technology Research Fund Program for Young Scholars (Grant No. 3050011181909), and China Postdoctoral Science Foundation (Grant No. 2018M641511). F. L. was supported by U.S. DOE-BES (Grant No. DE-FG02-04ER46148).
These authors contributed equally to this work.
*Corresponding authors. Email:;
Citation Text: Dong W H, Bao D L, Sun J T, Liu F, and Du S X 2021 Chin. Phys. Lett. 38 097101    Abstract Graphene has afforded an ideal 2D platform for investigating a rich and fascinating behavior of Dirac fermions. Here, we develop a theoretical mechanism for manipulating the Dirac fermions in graphene, such as from type-I to type-II and type-III, by a top-down nanopatterning approach. We demonstrate that by selective chemical adsorption to pattern the 2D graphene into coupled 1D armchair chains (ACs), the intrinsic isotropic upright Dirac cone becomes anisotropic and strongly tilted. Based on model analyses and first-principles calculations, we show that both the shape and tilt of Dirac cone can be tuned by the species of chemisorption, e.g., halogen vs hydrogen, which modifies the strength of inter-AC coupling. Furthermore, the topological edge states and transport properties of the engineered Dirac fermions are investigated. Our work sheds lights on understanding the Dirac fermions in a nanopatterned graphene platform, and provides guidance for designing nanostructures with novel functionality. DOI:10.1088/0256-307X/38/9/097101 © 2021 Chinese Physics Society Article Text The discovery of graphene[1,2] ushered a blossoming of topological materials hosting quasiparticle-like fermions and bosons.[3–9] Especially, Dirac fermions in graphene exhibit a rich spectra of phenomena, such as quantum Hall effect,[10,11] Klein tunneling[12] and quantum transport,[13,14] which have also been extended to artificial acoustic[15] and photonic[16] Dirac systems. Recent discovery of type-II and type-III Dirac fermions in topological semimetals[17–19] further enriches the Dirac physics with novel Fermi surface and band topology, as manifested in gapless plasmon mode[20] and nodeless superconductivity.[21] The varying types of Dirac states[22,23] mean different transport properties[24] and magnetic response,[25] offering new opportunity for quantum and spintronics device applications. Beyond graphene, several two-dimensional (2D) carbon- and boron-based nanostructures[26–28] have been predicted with Dirac fermions. Different types of Dirac Fermions have also been predicted in a few carbon allotropes with bipartite-symmetry breaking penta-rings.[29,30] However, each individual material hosts inherently one particular type of Dirac Fermions. It will be fundamentally interesting and practically useful to develop a scheme to manipulate Dirac Fermions into different types in one single material platform, to realize different device functionalities. Traditionally there are two distinct approaches towards nanofabrication of quantum materials: the bottom-up[31] or top-down approach.[32,33] Similarly, such approaches can be adopted in creating novel nanostructured topological materials. As an example of the bottom-up approach, recent studies have shown three-dimensional (3D) “nanowire-structured” bulk materials assembled from 1D bismuth-halides (Bi$_{4}$Br$_{4}$ and Bi$_{4}$I$_{4}$) chains, to display tunable topological phases of weak topological insulator (TI) and high order TI.[34–36] As an example of top-down approach, the van der Waals (vdW) layered 2H-NbTe$_{2}$ has been modulated into one-dimensional (1D) stripe-like patterns by silicon intercalation, to induce formation of directional Dirac fermions.[37,38] Researchers also observed the emergence of multiple Dirac cones in high mobility graphene devices with a gate-tunable 1D superlattice.[39] Moreover, the atomic-thick BeN$_{4}$ layers consisting of polyacetylene-like nitrogen chains were reported to host anisotropic Dirac fermions.[40] These progresses offer new insights into Dirac fermions based on nanochain-structured materials, i.e., nanostructures with distinct 1D building blocks or 1D electronic patterns. Noting that 2D graphene consists of basic building blocks of 1D armchair and zigzag chains, it is desirable to create nanochain-structured graphene, to tune the properties of Dirac fermions. In this Letter, we demonstrate theoretically a top-down approach towards constructing nanochain-structured graphene and manipulating the tilting degree and anisotropy of Dirac fermions. By selective chemisorption, 2D graphene is nanopatterned into coupled 1D armchair chains (ACs), which by themselves in isolation represent the 1D Su–Schrieffer–Heeger (SSH) model.[41,42] Using $k$$\cdot $$p$ analysis and a tight-binding model, we show that collectively the coupled ACs can host strongly tilted Dirac cones depending on the strength of interchain coupling. Using first-principles calculations, we confirm the transition of Dirac fermions from type-I to type-II and type-III, by tuning the interchain coupling with different chemisorption species. Furthermore, we discuss potential applications of such nanochain-structured graphene as anisotropic quantum devices, based on calculations and analyses of their topological edge states and transport properties.
Fig. 1. Illustration of the top-down patterning of 2D graphene. (a) Armchair chain (AC) as the 1D building unit of graphene. (b) Zigzag chain as the 1D building unit of graphene. (c) Complete blocking the coupling between building units of graphene turns the relativistic Dirac fermions into ordinary fermions. (d) Schematic of partially blocking 2D graphene into patterned armchair-to-reversed-armchair (A–RA) chains. The tunable inter-AC coupling drives the system into different Dirac states without gap opening (see Fig. 3).
The concept of the proposed top-down patterning approach to create a nanostructured graphene platform is illustrated in Fig. 1. From reductionism, if one imagines graphene is made of 1D building blocks, then the smallest 1D unit is either an armchair, zigzag or chiral atomic chain in different orientations. It is well known that the AC [Fig. 1(a)], which represents a 1D SSH model, is a semiconductor, while the zigzag chain [Fig. 1(b)] is an antiferromagnetic semiconductor.[13,43] Accordingly, it has been proposed before to nanopatterning graphene into a semiconductor. For example, by patterning graphene into a nanohole superlattice with both armchair and zigzag edges, a dilute magnetic semiconductor can be designed.[44] It is important to note that previous nanopatterning approach has generally assumed to completely block the hopping between the nanoscale building blocks,[45,46] which converts the relativistic Dirac fermions into ordinary non-relativistic fermions by opening a gap at the Dirac point [Fig. 1(c)]. Differently, using the 1D SSH ACs as the building blocks, here we propose to only partially block the inter-chain coupling [Fig. 1(d)], which retains the Dirac fermions without opening a gap but convert them into different types (Fig. 3). We achieve this partial blocking by selective surface adsorption, which makes it possible to tune the inter-chain coupling by using different adsorption species, as demonstrated below.
Fig. 2. Schematic illustration of Dirac cone tilting evolution by symmetry breaking. (a) Pristine folded Dirac cone of the honeycomb lattice. (b) Formation of high-symmetry line (HSL) Dirac cone upon introducing anisotropy, with $\delta_{1}>m_{1}$, $\delta_{2}=0$, ${{\vert \tilde{v}}_{y}\vert }=0$. (c) Type-II Dirac cone upon chiral (sublattice) symmetry breaking, with $\delta_{1}>m_{1}$, $\delta_{2}=0$, ${{\vert \tilde{v}}_{y}\vert }\mathrm{>\vert }v_{y}\vert$.
We first discuss the general working principle underlying our idea, based on continuum model and symmetry analysis. Without losing generality, we start from the effective Hamiltonian of graphene, $$ H({\boldsymbol K}_{\pm }+{\boldsymbol k})=\pm v_{x}k_{x}\sigma_{x}+{v_{y}k}_{y}\sigma_{y},~~ \tag {1} $$ where $v_{x,y}$ is the Fermi velocity; $k_{x,y}$ is the wave vector; $\sigma_{i}$ are Pauli matrices of sublattices. For convenience, one can fold Dirac points ($K_{\pm}$ valleys) from the corners of Brillouin zone to $\varGamma$ point, as shown in Fig. 2(a), considering a hypothetical superlattice consisting of a periodic array of 1D ACs. In this stage, we neglect the inter-valley hybridization, and the superlattice Hamiltonian at $\varGamma$ is $$\begin{align} H_{\varGamma }^{0}({\boldsymbol q}\mathrm)={}&H({\boldsymbol K}_{+}+{\boldsymbol q})\oplus H({\boldsymbol K}_{-}+{\boldsymbol q})\\ ={}&v_{x}q_{x}\tau_{z}\sigma_{x}+{v_{y}q}_{y}\tau_{0}\sigma_{y},~~ \tag {2} \end{align} $$ where $\tau_{i}$ are Pauli matrices describing the valley pseudospin, $q_{x,y}$ is the wavevector measured from $\varGamma$ point. Next, considering the coupled 1D system, it supports a two-fold rotational symmetry $C_{2y}:\tau_{z}\sigma_{y}$, and also preserves time-reversal $T:\tau_{x}K$ ($K$ is a complex conjugation operator), inversion $P:\tau_{x}\sigma_{x}$, and mirror $M_{xz}$: $\tau_{y}\sigma_{z}$ symmetries, as inherited from the original honeycomb lattice. The terms that respect all these symmetries include $m_{1}\tau_{x}\sigma_{x}$, ${\delta_{1}\tau }_{z}\sigma_{y}$ and $\delta_{2}\tau_{y}\sigma_{z}$. Here, $m_{1}\tau_{x}\sigma_{x}$ denotes the mass term arising from the $K_{\pm}$ valley hybridization when they are approaching, which would open a gap. We introduce ${\delta_{1}\tau }_{z}\sigma_{y}$ and $\delta_{2}\tau_{y}\sigma_{z}$ to account for the apparent anisotropy in the patterned lattice, which denote the different separations of the $K_{\pm}$ valleys in the $q_{x}$ and $q_{y}$ directions, respectively, due to the inter-valley interaction. The anisotropy requires $\delta_{1}\ne \delta_{2}$ and their total strength acts as $\sqrt {\vert \delta_{1}^{2}-\delta_{2}^{2}\vert}$ [see Fig. S8(a)]. For simplicity, we take $\delta_{2}=0$ so that $\delta_{1}$ quantifies the degree of anisotropy. The greater the $\delta_{1}$, the larger the anisotropy and separation of $K_{\pm}$ valleys. Combining $m_{1}\tau_{x}\sigma_{x}$ and ${\delta_{1}\tau }_{z}\sigma_{y}$, in order to preserve the Dirac cone, $\delta_{1}>m_{1}$ must be satisfied, then the type-I Dirac cone appears along the high-symmetry line (HSL) $q_{x} = 0$ with the $K_{\pm}$ valleys locating at ($0, \pm \sqrt {\delta_{1}^{2}-m_{1}^{2}} /v_{y}$), as shown in Fig. 2(b). Moreover, tilting of Dirac cone occurs because the Dirac point is not located at a high-symmetry point.[47] One can consider another $\tilde{v_{y}}\vert q_{y}\vert \tau_{0}\sigma_{0}$ term to break chiral (sublattice) symmetry ${C:\tau }_{o}\sigma_{z}$ while preserving $T$, $P$, $C_{2y}$ and $M_{xz}$ symmetries. Then the full $k$$\cdot $$p$ Hamiltonian is $$\begin{align} H_{\varGamma}({\boldsymbol q})={}&v_{x}q_{x}\tau_{z}\sigma_{x}+{v_{y}q}_{y}{\tau_{0}\sigma}_{y}+m_{1}\tau_{x}\sigma_{x}\\ &+\delta_{1}\tau_{z}\sigma_{y}+\tilde{v_{y}}\vert q_{y}\vert \tau_{0}\sigma_{0}.~~ \tag {3} \end{align} $$ Equation (3) contains a critical point ${{\vert \tilde{v}}_{y}\vert }=\vert v_{y}\vert$ for the existence of type-III Dirac state. Therefore, the type-II Dirac cone emerges once ${{\vert \tilde{v}}_{y}\vert }>\vert v_{y}\vert$ is satisfied, as illustrated in Fig. 2(c).
Fig. 3. Engineering Dirac cone in coupled 1D ACs. (a) Schematic of the A–RA-type coupled-chain model (CCM). The gray and black sites belong to different sublattices. (c) The first Brillouin zone. (c)–(e) Band structures of tilted type-I, type-III and type-II Dirac states, respectively, with fixed $t_{1}=-1.90$, $t_{1}'=-1.96$, $t_{3}=0.3$: (c) $t_{2}=-0.15$, ${\varDelta =-0.03}$; (d) $t_{2}=-0.3$, ${\varDelta =-0.06}$; (e) $t_{2}=-0.5$, ${\varDelta =-0.14}$. The red (green) bands share $+$1 ($-1$) eigenvalues under $C_{2y}$ operation. In (c)–(e), only two bands of interest are shown and all parameters are in units of eV. (f) Phase diagram with fixed $t_{1}'=-1.96$, $t_{3}=0.27$ and ${\varDelta =0}$. Here, the gray lines denote the boundary between Dirac and gapped states, the gray dashed line denotes type-III Dirac state. The Dirac point position $\kappa$ is marked in (c) and $\kappa = 0$ (1) represents the gap location at $\varGamma$ ($M_{2}$).
We then investigate the specific parametric conditions for engineering the Dirac phases in the nanochain-structured graphene by developing a coupled-chain model (CCM). We focus on the symmorphic armchair-to-reversed-armchair (A–RA) configuration, as shown in Fig. 3(a), and leave the results of the A–A configuration in Figs. S1 and S2 (see the Supplementary Material). The Wannier orbitals are $p_{z}$-like and the gray (black) dots mark the sublattice B (C) sites. Then the four-site Hamiltonian is given by $$\begin{align} H_{0}={}&\sum\nolimits_{\alpha i} {\varDelta_{\alpha i}c_{\alpha i}^{† }c_{\alpha i}} +\sum\nolimits_{\langle \alpha i,\beta j \rangle _{x}} {t_{\alpha i,\beta j}^{x}c_{\alpha i}^{† }c_{\beta j}} \\ &+\sum\nolimits_{ {\mathop{ \langle \alpha i,\beta j \rangle _{y} } \limits_{\langle \langle \alpha i,\beta j\rangle \rangle _{y}}}}{t_{\alpha i,\beta j}^{y}c_{\alpha i}^{† }c_{\beta j}} +{\rm H.c.},~~ \tag {4} \end{align} $$ where $\langle ~\rangle$ and $\langle \langle ~\rangle \rangle$ represent the nearest and next-nearest hopping, respectively; $\alpha, \beta$ = B, C and $i,j = 1, 2$; on-site energies $\varDelta _{\alpha i}$ are uniformly set as $\varDelta$; $t_{\alpha i,\beta j}^{x}$ denotes intra-AC coupling $t_{1}$ and $t_{1}'$ along $x$ direction, and $t_{\alpha i,\beta j}^{y}$ represents inter-AC coupling $t_{2}$ and $t_{3}$ along $y$ direction. Note that the relatively weak inter-AC coupling is considered up to the next-nearest neighbor. CCM obeys time-reversal symmetry and $C_{2h}$ point group symmetry. The spatial operators include two-fold rotation $C_{2y}$ (B1$\to$C1, B2$\to$C2), mirror $M_{xz}$ (B1$\to$B2, C1$\to$C2) and inversion $P$ (B2$\to$C1, B1$\to$C2). Figures 3(c)–3(e) show the band structures of CCM, which illustrate the tilting of Dirac cone into tilted type-I, type-III and type-II Dirac in the coupled 1D ACs. The tilted Dirac cone elongates along the $C_{2y}$ invariant $\varGamma$–$M_{2}$ path, which is characterized by a nontrivial time-reversal invariant[48,49] $\mathbb{Z}_{2} = 1$ (see Table S1). We have constructed a phase diagram of the resulting Dirac fermions in the parameter space of relative hopping strength, as shown in Fig. 3(f). At the limit of vanishing inter-AC coupling, i.e., $t_{2}=t_{3}=0$, the CCM model reduces simply to the conventional SSH model, exhibiting a universal Peierls transition.[50] Given the intra-AC hopping $t_{1}\ne t_{1}'$, there exists a 1D gap $\varDelta_{1D} =2\vert t_{1}-t_{1}'\vert$.[51] With the inter-AC coupling, if $\vert t_{3}\vert >\varDelta_{1D}/2$, the 2D Dirac state retains, albeit with an anisotropic and tilted Dirac cone; otherwise, the Dirac state disappears with a gap opening. Thus, the conditions $t_{1}-t_{1}'=\pm {\vert t}_{3}\vert$ set up two horizontal phase boundaries for the existence of Dirac fermions, as shown in Fig. 3(f). Inside these two boundaries, another vertical boundary conditioned with $| t_{2} |=\vert t_{3}\vert$ divides different types of Dirac fermions. Note that the inter-AC coupling $t_{2}$ breaks chiral (or bipartite lattice) symmetry. Since the Fermi velocity of one Dirac branch is proportional to $| t_{2} |{-\vert t}_{3}\vert$, it goes to zero when $| t_{2} |=\vert t_{3}\vert$, which gives rise to the type-III Dirac fermions and divides the type-I ($| t_{2} |\mathrm{ < \vert }t_{3}\vert$) and type-II ($| t_{2} |\mathrm{>\vert }t_{3}\vert$) regions. The effect of spin-orbit coupling (SOC) and edge states of CCM are discussed in Figs. S3, S6 and S7.
Fig. 4. Highly tilted Dirac fermions in the AC-patterned graphene. (a) Atomic structure of the AC-patterned graphene. The $sp^2$ carbons (orange) constitute the A–RA type chains. Bond lengths of C–X ($d_{\rm H}$) are 1.11, 1.44, 1.87, 2.22 Å and $sp^3$ carbon buckling heights ($\Delta z$) are 0.67, 0.63, 0.63, 0.53 Å for X = H, F, Cl, Br, respectively. [(b), (c)] Spinless band structures with density of states (DOS) of C$_{6}$Cl$_{2}$ and C$_{6}$F$_{2}$, respectively. In (b) and (c), the red (green) bands share $+$1 ($-1$) eigenvalues under $C_{2y}$ operation and total DOS (gray filled) near the Fermi level are mainly from $p_{z}$ orbitals of $sp^2$ carbons (blue filled). The left insets are real-space charge distribution near the Fermi level. VHS is short for van Hove singularity. (d)–(g) Spinless semi-infinite spectra of C$_{6}$Cl$_{2}$ periodic along $[11]$, C$_{6}$F$_{2}$ periodic along $[11]$, C$_{6}$Cl$_{2}$ periodic along $[\bar{1}1]$ and C$_{6}$F$_{2}$ periodic along $[\bar{1}1]$, respectively. In (f) and (g), two helical edge states are degenerate.
Now, we propose material realization of the above CCM, namely a top-down nanopatterning process of graphene. Inspired by the fabrication of one-third hydrogenated graphene with long-range 1D AC patterns,[52] we devise a form of AC-patterned graphene in Fig. 4(a). One notices that the A–RA stacked $sp^2$ carbon ACs are staggered by $sp^3$ carbons. The unit formula of such AC-patterned graphene is C$_{6}$X$_{2}$ (X = H, F, Cl, Br). The lattice constants are 4.31, 4.30, 4.30, 4.31 Å for X = H, F, Cl, Br, respectively, and the lattice belongs to the layer group $C2/m$ with $C_{2y}$, $M_{xz}$ and $P$ operations. The phonon calculations confirm their dynamical stability (Fig. S10). Figures 4(b) and 4(c) (see also Fig. S9) show the band structures of AC-patterned graphene, obtained from first-principles calculations, which clearly illustrate the phase transition of Dirac fermions. In particular, Fig. 4(b) displays the spinless type-III Dirac fermion along the $\varGamma$–$M_{2}$ path for C$_{6}$Cl$_{2}$. One Dirac band branch is flat with a 13-meV bandwidth, leading to a van Hove singularity (VHS) in the density of states (DOS). The SOC opens a negligible gap of 2.0 meV. Figure 4(c) shows the type-II Dirac fermion for C$_{6}$F$_{2}$, and the Dirac point is $\sim $0.2 eV above the Fermi level. The manipulated Dirac states in the AC-patterned graphene are found to be robust when placed on an insulating substrates, such as BN (0001) (see Figs. S13 and S14). We also discovered the existence of type-II Dirac fermions in the AC-patterned silicene and strained germanene (Fig. S12). In addition, the first-principles results of AC-patterned graphene are well fitted by both CCM in Fig. S5 and maximally localized Wannier functions[53] in Fig. S11. The fitting parameters confirm quantitatively the phase diagram of CCM. For instance, the fitted parameters are $t_{2}=-0.54$ eV and $t_{3}=0.42$ eV for C$_{6}$F$_{2}$, which satisfy the $|t_{2}|>|t_{3}|$ condition for the type-II Dirac fermion. Next, we study the edge states of AC-patterned graphene. We first evaluate the topological descriptors (see Table S2). The spinless semimetal phase of AC-patterned graphene is featured with crossings of bands with opposite $C_{2y}$ eigenvalues. This gives rise to a nontrivial $\mathbb{Z}_{2} = 1$ and rotation invariant $\delta_{\rm r} = 1$ (as defined in Ref. [54]). Also, the 2D Zak phase[55] or electrical polarization[56] is calculated as $(p_{1}, p_{2}) = (0,1/2)$, indicating a fractional topological charge $e$/2 along $\varGamma$–$M_{2}$ path. The directional edge band structures and semi-infinite edge spectra of AC-patterned graphene are shown in Figs. 4(d)–4(g) and Figs. S17–S21. We have considered the periodic directions of $[10]$, $[11]$ and $[\bar{1}1]$ with different ribbon terminations. Here, we focus on the centrosymmetric ribbons along the $C_{2h}$ preserving $[11]$ and $[\bar{1}1]$ directions. Along the $[11]$ direction, two bulk Dirac cones of AC-patterned graphene are folded into the $\varGamma$ point. In Figs. 4(d) and 4(e) of edge spectra, the conduction band dispersion of C$_{6}$F$_{2}$ and C$_{6}$Cl$_{2}$ becomes parabolic with a U shape. Due to the existence of $e$/2 topological charge along $[\bar{1}1]$ direction, we observe two degenerate helical modes within the bulk states of C$_{6}$F$_{2}$ and C$_{6}$Cl$_{2}$ [Figs. 4(f) and 4(g)]. The helical modes come in pairs subjected to the $C_{2y}$ rotation symmetry. To further explore the transport properties of AC-patterned graphene, we calculate the static conductivity tensor using the Kubo–Bastin formula based on linear response theory:[57,58] $$\begin{alignat}{1} \sigma_{\alpha \beta }&(\mu,T)=\frac{ie^{2}\hslash }{\varOmega }\int_{\mathrm{-\infty }}^\infty d\varepsilon f(\varepsilon)\mathrm{\times Tr}\Big[ \nu_{\alpha }\delta (\varepsilon -H)\nu_{\beta}\\ &\cdot\frac{dG^{+}(\varepsilon,H)}{d\varepsilon} -\nu_{\alpha }\frac{dG^{-}(\varepsilon,H)}{d\varepsilon }\nu_{\beta }\delta (\varepsilon -H)\Big],~~ \tag {5} \end{alignat} $$ where $T$ is temperature, $\mu$ is chemical potential, $\varOmega$ is the volume of primitive cell, $\nu_{\alpha,\beta}$ is the velocity operator, $f(\varepsilon)$ is the Fermi–Dirac distribution and $G^{\pm} (\varepsilon,H) = [(\varepsilon - H \pm i\eta)]^{-1}$ are the advanced ($+$) and retarded ($-$) Green's functions. In Fig. 5, we compare the conductivities of AC-patterned graphene with graphene at 0 K. First, the coupled ACs display a larger longitudinal conductivity $\sigma_{xx}$ along the chain direction than graphene, while the “barrier” from $sp^3$ carbons in between the ACs limits the electron transport in the $y$-direction. The high anisotropy manifests a distinct quasi 1D electrical response in AC-patterned graphene. Second, there exist significant conductivity differences between type-I (graphene), tilted type-I (C$_{6}$H$_{2}$), type-III (C$_{6}$Cl$_{2}$) and type-II (C$_{6}$F$_{2}$) Dirac fermions. In particular, $\sigma_{xx}$ of C$_{6}$F$_{2}$ (C$_{6}$Cl$_{2}$) is as high as 21.9$\,e^{2}/h$ $(17.8\,e^{2}/h)$ at Fermi level, indicating that the highly tilted Dirac fermions endow C$_{6}$F$_{2}$ (C$_{6}$Cl$_{2}$) with metallic behavior, distinct from the semimetal graphene or C$_{6}$H$_{2}$. Consequently, C$_{6}$F$_{2}$ and C$_{6}$Cl$_{2}$ are promising for designing highly anisotropic quantum devices with $x$-directional transport and $y$-directional topological edge conductance.
Fig. 5. Kubo–Bastin static conductivity of tilted type-I C$_{6}$H$_{2}$ (red), type-II C$_{6}$F$_{2}$ (green), type-III C$_{6}$Cl$_{2}$ (blue) and graphene (black) at 0 K. The solid and dashed lines are longitudinal $\sigma_{xx}$ and $\sigma_{yy}$ tensor components, respectively, which are degenerate for graphene.
In conclusion, we have theoretically investigated a top-down nanopatterning approach of 2D graphene to manipulate the Dirac fermions, such as changing from isotropic type-I to anisotropic type-II and type-III. Our model analyses and material calculations extend the study of topological states to coupled 1D systems, which links the 1D SSH model with the 2D Kane–Mele model.[48,59] Furthermore, we demonstrate the concept of nanopatterning in reconstructing a pristine system with novel functionality by selective chemisorption. It is worth noting that one of our proposed nanochain-structured graphene, namely C$_{6}$H$_{2}$ monolayer, has already been successfully fabricated in millimeter scale using ultrahigh vacuum radio frequency plasma.[52] Similar plasma methods are promising for synthesizing halogenated graphene C$_{6}$F$_{2}$, C$_{6}$Cl$_{2}$ via halogenating agents such as XeF$_{2}$[60] and Cl$_{2}$[61] under suitable conditions of plasma concentration, treatment time, and substrate, etc. These results enrich the fundamental understanding of 2D Dirac fermions and provide guidance for designing functional nanostructures based on graphene, and therefore, are expected to draw immediate attention from experimentalists.
References Electric Field Effect in Atomically Thin Carbon FilmsTwo-dimensional gas of massless Dirac fermions in grapheneDiscovery of a Three-Dimensional Topological Dirac Semimetal, Na3BiExperimental Discovery of Weyl Semimetal TaAsType-II Weyl semimetalsNodal-chain metalsHourglass fermionsTriple Point Topological MetalsTopological Phonons and Weyl Lines in Three DimensionsUnconventional Integer Quantum Hall Effect in GrapheneExperimental observation of the quantum Hall effect and Berry's phase in grapheneColloquium : Andreev reflection and Klein tunneling in grapheneHalf-metallic graphene nanoribbonsQuantum Transport of Massless Dirac FermionsDeterministic Scheme for Two-Dimensional Type-II Dirac Points and Experimental Realization in AcousticsTopological photonicsExperimental Realization of Type-II Dirac Fermions in a PdTe 2 SuperconductorLorentz-violating type-II Dirac fermions in transition metal dichalcogenide PtTe2Black-hole horizon in the Dirac semimetal Zn 2 In 2 S 5 Novel Undamped Gapless Plasmon Mode in a Tilted Type-II Dirac SemimetalNodeless superconductivity in the type-II Dirac semimetal PdTe 2 : London penetration depth and pairing-symmetry analysisRealization of low-energy type-II Dirac fermions in (Ir 1− x Pt x )Te 2 superconductorsFermionic Analogue of High Temperature Hawking Radiation in Black PhosphorusNontrivial Berry phase and type-II Dirac transport in the layered material PdT e 2 Mirror Anomaly in Dirac SemimetalsStructural and Electronic Properties of T Graphene: A Two-Dimensional Carbon Allotrope with TetraringsCompetition for Graphene: Graphynes with Direction-Dependent Dirac ConesSemimetallic Two-Dimensional Boron Allotrope with Massless Dirac FermionsEmergence of Type-I and Type-II Dirac line nodes in penta-octa-grapheneTheoretical prediction of low-energy Stone-Wales graphene with an intrinsic type-III Dirac coneAtomically precise bottom-up fabrication of graphene nanoribbonsNanomechanical Architecture of Strained Bilayer Thin Films: From Design Principles to Experimental FabricationA family of zeolites with controlled pore size prepared using a top-down methodA novel quasi-one-dimensional topological insulator in bismuth iodide β-Bi4I4A weak topological insulator state in quasi-one-dimensional bismuth iodideEvidence for a higher-order topological insulator in a three-dimensional material built from van der Waals stacking of bismuth-halide chainsDirectional massless Dirac fermions in a layered van der Waals material with one-dimensional long-range orderOne-Dimensional Metal Embedded in Two-Dimensional Semiconductor in Nb 2 Si x –1 Te 4Anisotropic band flattening in graphene with one-dimensional superlatticesHigh-Pressure Synthesis of Dirac Materials: Layered van der Waals Bonded BeN 4 PolymorphSolitons in PolyacetyleneSoliton excitations in polyacetyleneEnergy Gaps in Graphene NanoribbonsCollective magnetic behavior of graphene nanohole superlatticesBandgap opening in graphene induced by patterned hydrogen adsorptionControl of Graphene's Properties by Reversible Hydrogenation: Evidence for GraphaneTilted anisotropic Dirac cones in quinoid-type graphene and α ( BEDT-TTF ) 2 I 3 Z 2 Topological Order and the Quantum Spin Hall EffectSpin Nernst effect and Nernst effect in two-dimensional electron systemsFluctuation Effects at a Peierls TransitionEngineering of robust topological quantum phases in graphene nanoribbonsFabrication of Millimeter‐Scale, Single‐Crystal One‐Third‐Hydrogenated Graphene with Anisotropic Electronic PropertiesMaximally localized generalized Wannier functions for composite energy bandsQuantitative mappings between symmetry and topology in solidsNovel Topological Phase with a Zero Berry CurvatureHelical Topological Edge States in a Quadrupole PhaseQuantum oscillations of the hall effect of a fermion gas with random impurity scatteringReal-Space Calculation of the Conductivity Tensor for Disordered Topological MatterQuantum Spin Hall Effect in GrapheneProperties of Fluorinated Graphene FilmsControlled Chlorine Plasma Reaction for Noninvasive Graphene Doping
[1] Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V, and Firsov A A 2004 Science 306 666
[2] Novoselov K S, Geim A K, Morozov S V, Jiang D, Katsnelson M I, Grigorieva I V, Dubonos S V, and Firsov A A 2005 Nature 438 197
[3] Liu Z K, Zhou B, Zhang Y, Wang Z J, Weng H M, Prabhakaran D, Mo S- K, Shen Z X, Fang Z, Dai X, Hussain Z, and Chen Y L 2014 Science 343 864
[4] Lv B Q, Weng H, Fu B B, Wang X P, Miao H, Ma J, Richard P, Huang X, Zhao L, and Chen G 2015 Phys. Rev. X 5 031013
[5] Soluyanov A A, Gresch D, Wang Z J, Wu Q S, Troyer M, Dai X, and Bernevig B A 2015 Nature 527 495
[6] Bzdušek T, Wu Q, Rüegg A, Sigrist M, and Soluyanov A A 2016 Nature 538 75
[7] Wang Z, Alexandradinata A, Cava R J, Bernevig B A 2016 Nature 532 189
[8] Zhu Z, Winkler G W, Wu Q, Li J, and Soluyanov A A 2016 Phys. Rev. X 6 031003
[9] Stenull O, Kane C, and Lubensky T 2016 Phys. Rev. Lett. 117 068001
[10] Gusynin V and Sharapov S 2005 Phys. Rev. Lett. 95 146801
[11] Zhang Y, Tan Y W, Stormer H L, and Kim P 2005 Nature 438 201
[12] Beenakker C 2008 Rev. Mod. Phys. 80 1337
[13] Son Y W, Cohen M L, and Louie S G 2006 Nature 444 347
[14] Nomura K and MacDonald A H 2007 Phys. Rev. Lett. 98 076602
[15] Wu X, Li X, Zhang R Y, Xiang X, Tian J, Huang Y, Wang S, Hou B, Chan C T, and Wen W 2020 Phys. Rev. Lett. 124 075501
[16] Ozawa T, Price H M, Amo A, Goldman N, Hafezi M, Lu L, Rechtsman M C, Schuster D, Simon J, Zilberberg O, and Carusotto I 2019 Rev. Mod. Phys. 91 015006
[17] Noh H J, Jeong J, Cho E J, Kim K, Min B, and Park B G 2017 Phys. Rev. Lett. 119 016401
[18] Yan M, Huang H, Zhang K, Wang E, Yao W, Deng K, Wan G, Zhang H, Arita M, and Yang H 2017 Nat. Commun. 8 257
[19] Huang H Q, Jin Y W, and Liu F 2018 Phys. Rev. B 98 121110(R)
[20] Sadhukhan K, Politano A, and Agarwal A 2020 Phys. Rev. Lett. 124 046803
[21] Teknowijoyo S, Jo N H, Scheurer M S, Tanatar M A, Cho K, Bud'ko S L, Orth P P, Canfield P C, and Prozorov R 2018 Phys. Rev. B 98 024508
[22] Fu B B, Yi C J, Wang Z J, Yang M, Lv B Q, Gao X, Li M, Huang Y B, Weng H M, and Shi Y G 2019 Chin. Phys. B 28 037103
[23] Liu H, Sun J T, Song C, Huang H, Liu F, and Meng S 2020 Chin. Phys. Lett. 37 067101
[24] Fei F, Bo X, Wang R, Wu B, Jiang J, Fu D, Gao M, Zheng H, Chen Y, and Wang X 2017 Phys. Rev. B 96 041201
[25] Burkov A 2018 Phys. Rev. Lett. 120 016603
[26] Liu Y, Wang G, Huang Q, Guo L, and Chen X 2012 Phys. Rev. Lett. 108 225505
[27] Malko D, Neiss C, Vines F, and Görling A 2012 Phys. Rev. Lett. 108 086804
[28] Zhou X F, Dong X, Oganov A R, Zhu Q, Tian Y, and Wang H T 2014 Phys. Rev. Lett. 112 085502
[29] Gao H and Ren W 2020 Carbon 158 210
[30] Gong Z, Shi X, Li J, Li S, He C, Ouyang T, Zhang C, Tang C, and Zhong J 2020 Phys. Rev. B 101 155427
[31] Cai J, Ruffieux P, Jaafar R, Bieri M, Braun T, Blankenburg S, Muoth M, Seitsonen A P, Saleh M, and Feng X 2010 Nature 466 470
[32] Huang M, Boone C, Roberts M, Savage D E, Lagally M G, Shaji N, Qin H, Blick R, Nairn J A, and Liu F 2005 Adv. Mater. 17 2860
[33] Roth W J, Nachtigall P, Morris R E, Wheatley P S, Seymour V R, Ashbrook S E, Chlubná P, Grajciar L, Položij M, and Zukal A 2013 Nat. Chem. 5 628
[34] Autès G, Isaeva A, Moreschini L, Johannsen J C, Pisoni A, Mori R, Zhang W, Filatova T G, Kuznetsov A N, and Forró L 2016 Nat. Mater. 15 154
[35] Noguchi R, Takahashi T, Kuroda K, Ochi M, Shirasawa T, Sakano M, Bareille C, Nakayama M, Watson M, and Yaji K 2019 Nature 566 518
[36] Noguchi R, Kobayashi M, Jiang Z, Kuroda K, Takahashi T, Xu Z, Lee D, Hirayama M, Ochi M, and Shirasawa T 2021 Nat. Mater. 20 473
[37] Yang T, Wan Q, Yan D, Zhu Z, Wang Z, Peng C, Huang Y, Yu R, Hu J, and Mao Z 2020 Nat. Mater. 19 27
[38] Wang B, Xia W, Li S, Wang K, Yang S A, Guo Y, and Xue J 2021 ACS Nano 15 7149
[39] Li Y, Dietrich S, Forsythe C, Taniguchi T, Watanabe K, Moon P, and Dean C R 2021 Nat. Nanotechnol. 16 525
[40] Bykov M, Fedotenko T, Chariton S, Laniel D, Glazyrin K, Hanfland M, Smith J S, Prakapenka V B, Mahmood M F, and Goncharov A F 2021 Phys. Rev. Lett. 126 175501
[41] Su W P, Schrieffer J, and Heeger A J 1979 Phys. Rev. Lett. 42 1698
[42] Su W P, Schrieffer J, and Heeger A 1980 Phys. Rev. B 22 2099
[43] Son Y W, Cohen M L, and Louie S G 2006 Phys. Rev. Lett. 97 216803
[44] Yu D, Lupton E M, Liu M, Liu W, and Liu F 2008 Nano Res. 1 56
[45] Balog R, Jørgensen B, Nilsson L, Andersen M, Rienks E, Bianchi M, Fanetti M, Lægsgaard E, Baraldi A, and Lizzit S 2010 Nat. Mater. 9 315
[46] Elias D C, Nair R R, Mohiuddin T, Morozov S, Blake P, Halsall M, Ferrari A C, Boukhvalov D, Katsnelson M, and Geim A 2009 Science 323 610
[47] Goerbig M O, Fuchs J N, Montambaux G, and Piechon F 2008 Phys. Rev. B 78 045415
[48] Kane C L and Mele E J 2005 Phys. Rev. Lett. 95 146802
[49] Qi X L, Hughes T L, and Zhang S C 2008 Phys. Rev. B 78 045302
[50] Lee P A, Rice T, and Anderson P 1973 Phys. Rev. Lett. 31 462
[51] Gröning O, Wang S, Yao X, Pignedoli C A, Barin G B, Daniels C, Cupo A, Meunier V, Feng X, and Narita A 2018 Nature 560 209
[52] Chen H, Bao D L, Wang D F, Que Y D, Xiao W D, Qian G J, Guo H, Sun J T, Zhang Y Y, Du S X, Pantelides S T, and Gao H J 2018 Adv. Mater. 30 1801838
[53] Marzari N and Vanderbilt D 1997 Phys. Rev. B 56 12847
[54] Song Z, Zhang T, Fang Z, and Fang C 2018 Nat. Commun. 9 3530
[55] Liu F and Wakabayashi K 2017 Phys. Rev. Lett. 118 076803
[56] Liu F, Deng H Y, and Wakabayashi K 2019 Phys. Rev. Lett. 122 086804
[57] Bastin A, Lewiner C, Betbeder-Matibet O, and Nozieres P 1971 J. Phys. Chem. Solids 32 1811
[58] García J H, Covaci L, and Rappoport T G 2015 Phys. Rev. Lett. 114 116602
[59] Kane C L and Mele E J 2005 Phys. Rev. Lett. 95 226801
[60] Robinson J T, Burgess J S, Junkermeier C E, Badescu S C, Reinecke T L, Perkins F K, Zalalutdniov M K, Baldwin J W, Culbertson J C, and Sheehan P E 2010 Nano Lett. 10 3001
[61] Wu J, Xie L, Li Y, Wang H, Ouyang Y, Guo J, and Dai H 2011 J. Am. Chem. Soc. 133 19668