Tunable Dirac points and zero-energy modes in periodic curved graphene superlattices

TL;DR

This paper explores how curvature and electrostatic potentials in graphene superlattices can be used to tune Dirac points, energy gaps, and the emergence of new Dirac points, offering new ways to control electronic properties.

Contribution

It introduces the concept of using curvature in graphene superlattices to tune Dirac points and energy gaps, revealing new Dirac points and tunable electronic features.

Findings

Dirac points can be shifted by asymmetric potentials and curvature.

Curvature enables the appearance of extra tunable energy gaps.

New Dirac points can emerge under smaller potentials with curvature.

Abstract

We combined periodic ripples and electrostatic potentials to form curved graphene superlattices and studied the effects of space-dependent Fermi velocity induced from curvature on their electronic properties. With equal periods and symmetric potentials, the Dirac points do not move, but their locations shift under asymmetric potentials. This shift can be tuned by curvature and potentials. Tunable extra gaps in band structures can appear with unequal periods. The existence of new Dirac points is proposed, such that these new Dirac points can appear under smaller potentials with curvature, and their locations can be changed even under a fixed potential by adjusting the curvature. Our results suggest that curvature provides a new possible dimension to tune the electronic properties in graphene superlattices and a platform to more easier study physics near new Dirac points.

Equations

$$H_m=\hslash v_f(k_x{\sigma }_x+k_y{\sigma }_y),$$

$$g_{\mu \nu }=\text{diag}(1,-(1+g^2(x)),-1).$$

$$H_c=-i\hslash v_f(\frac{{\sigma }_x{\partial }_x}{\sqrt{1+g^2(x)}}+{\sigma }_y{\partial }_y).$$

$$H_c$$

$$H=H_c+V(x)I,$$

$${M_{j}}(\Delta x,E,{k_{y}})=\left({\begin{array}[]{*{20}{c}}{\frac{{\cos({q_{j}}\Delta x-{\theta_{j}})}}{{\cos{\theta_{j}}}}}&{i\frac{{\sin({q_{j}}\Delta x)}}{{\cos{\theta_{j}}}}}\\ {i\frac{{\sin({q_{j}}\Delta x)}}{{\cos{\theta_{j}}}}}&{\frac{{\cos({q_{j}}\Delta x+{\theta_{j}})}}{{\cos{\theta_{j}}}}}\end{array}}\right).$$

$$q_j=\{\begin{array}{l}\text{sgn}(k_j)\sqrt{(f_j{k}_j{)}^2-(f_j{k}_y{)}^2},k_j^2>k_y^2\\ i\sqrt{(f_j{k}_y{)}^2-(f_j{k}_j{)}^2},\text{otherwise}\end{array}.$$

$$M_j(\Delta x,E,k_y)=\text{diag}(e^{f_j{k}_y\Delta x},e^{-f_j{k}_y\Delta x})$$

$$M_m(w_m,E,k_y)=\prod _{j=1}^n{M}_j(w_j,E,k_y),$$

$$\cos ({\beta }_{x}T)=\frac{1}{2}\text{Tr}\prod _{i=1}^{T/\Lambda }(M_A{M}_B),$$

$$r(E,k_y)=\frac{x_{22}{e}^{i{\theta }_0}-x_{11}{e}^{i{\theta }_e}-x_{12}{e}^{i({\theta }_0+{\theta }_e)}+x_{21}}{x_{22}{e}^{-i{\theta }_0}+x_{11}{e}^{i{\theta }_e}-x_{12}{e}^{i({\theta }_e-{\theta }_0)}-x_{21}},$$

$$t(E,k_y)=\frac{2\cos {\theta }_0}{x_{22}{e}^{-i{\theta }_0}+x_{11}{e}^{i{\theta }_e}-x_{12}{e}^{i({\theta }_e-{\theta }_0)}-x_{21}},$$

$$\cos ({\beta }_x\Lambda )=$$

$$+\frac{\cos ({\theta }_A-{\theta }_B)-1}{\cos {\theta }_A\cos {\theta }_B}\sin (q_A{w}_A)\sin (q_B{w}_B),$$

$$E=\frac{f_A{w}_A{V}_A+f_B{w}_B{V}_B}{f_A{w}_A+f_B{w}_B}.$$

$$\Delta =\frac{(f_A-f_B)(V_A-V_B)w_A{w}_B}{(f_A{w}_A+f_B{w}_B)(w_A+w_B)}.$$

$$q_A{w}_A=-q_B{w}_B=m\pi ,m=1,2,\mathrm{3...}$$

$$k_{y,m}=\pm \sqrt{(\frac{V}{\hslash v_f}{)}^2-(\frac{2m\pi }{f\Lambda }{)}^2},m=1,2,\mathrm{3...},$$

$$k_{y,m}=\pm \frac{2\pi }{\Lambda }\sqrt{l^2-(\frac{m}{f}{)}^2},m=1,2,\mathrm{3...},$$

$$i\hslash {\bar{\gamma }}^{\mu }{\partial }_{\mu }\psi =0,$$

$$i\hslash {\gamma }^{\mu }{D}_{\mu }\psi =0.$$

$${\Omega }_{\mu }=\frac{1}{4}{e}^{\nu a}({\partial }_{\mu }{e}_{\nu }^b-{\Gamma }_{\mu \nu }^{\lambda }{e}_{\lambda }^b){\bar{\gamma }}_a{\bar{\gamma }}_b,$$

$$d{s}^2$$

$$=(1+g^2(x))d{x}^2+d{y}^2,$$

$$g_{\mu \nu }=\text{diag}(1,-(1+g^2(x)),-1).$$

$$e_{\mu }^a=\text{diag}(1,\sqrt{1+g^2(x)},1),$$

$$\{\begin{array}{l}\frac{1}{f(x)}\frac{d}{dx}{\psi }_A-k_y{\psi }_A=ik{\psi }_B\\ \frac{1}{f(x)}\frac{d}{dx}{\psi }_B+k_y{\psi }_B=ik{\psi }_A\end{array},$$

$$\{\begin{array}{l}\frac{d^2}{d{x}^2}{\psi }_A+f_j^2(k_j^2-k_y^2){\psi }_A=0\\ \frac{d^2}{d{x}^2}{\psi }_B+f_j^2(k_j^2-k_y^2){\psi }_B=0\end{array}.$$

Full text

I Introduction

The Dirac fermion in curved two-dimensional space is an intriguing field of research due to its considerable applications in condensed matter physicsGeim and Novoselov (2007); Lee (2009); Morimoto et al. (2015); Castro et al. (2018), materials scienceLiu et al. (2010); Martins and Galvão (2010), quantum field theoryBirrel and Davies (1982); Parker and Toms (2009); Iorio and Lambiase (2014) and astrophysicsHawking (1976). This topic has attracted interest for decadesBirrel and Davies (1982); Parker and Toms (2009) but has received a particular boost due to the recent development of curved configuration of graphene superlattices (GSLs)Castro Neto et al. (2009); Vozmediano et al. (2010); Rusponi et al. (2010); Yankowitz et al. (2012); Dubey et al. (2013); Ponomarenko et al. (2013); Ni et al. (2015); Amorim et al. (2016), in which the valance and conductance band touch at Dirac points (DPs)Castro Neto et al. (2009). We can also confine massless Dirac particles and measure a quantum spin-Hall current by implementing a curved Dirac equation solver based on the quantum lattice Boltzmann method.Flouris et al. (2018a, 2019) Near the DPs in such curved graphene superlattices (CGSLs), energy bands have linear dispersion and electrons can be described by massless Dirac equations in a curved background. In addition to the CGSLs, there are various methods to induce different GSLs. In these structures, it has been predicted that the chirality of the charge carriers prevents the opening of a band gap and that, instead, new DPs appearPark et al. (2008a, 2009); Amorim et al. (2016); Brey and Fertig (2009); Barbier et al. (2010); Wang and Zhu (2010); Dubey et al. (2013); Ponomarenko et al. (2013), leading to a rich variety of remarkable results including the optical conductivity, anisotropic Fermi velocity Park et al. (2008a, 2009); Amorim et al. (2016); Brey and Fertig (2009); Barbier et al. (2010); Wang and Zhu (2010); Dubey et al. (2013); Ponomarenko et al. (2013); Park et al. (2008b); Barbier et al. (2010); Yankowitz et al. (2012), unconventional superconductivity Cao et al. (2018a); *Cao20182, surface plasmons Sunku et al. (2018); Ni et al. (2018); *Ni2015, etc. It has also been predicted that the energy gap may be modulated by scalar potentials.Low et al. (2011) Curvature generates electrochemical potential, which in turn stabilizes the curvature.Kim and Neto (2008) These novel properties not only stimulate the development of interesting physics in various fields but are also important for designing future electronic and optical devices.

In GSLs formed by periodic potentials, DPs and the associated band gaps are robust Wang and Zhu (2010), while the group velocity near the original DP only changes in one direction Brey and Fertig (2009); Barbier et al. (2010); the existence of new DPs is also limited by strict conditions, and their locations cannot be tuned with a fixed potential Park et al. (2009); Brey and Fertig (2009); Barbier et al. (2010). As a two-dimensional material, graphene exhibits intrinsic ripples to maintain its stability Fasolino et al. (2007). Recently, fabricating controllable periodic ripples Tapasztó et al. (2012); Vázquez de Parga et al. (2008); Bao et al. (2009); Yan et al. (2013a); Ni et al. (2012); Park et al. (2016) has provided new insight for realizing tunable Dirac points and zero-energy modes in CGSLs, as the curvature may play various roles in tuning electronic structures Castro Neto et al. (2009); Vozmediano et al. (2010); Amorim et al. (2016), in which the space-dependent Fermi velocity de Juan et al. (2007, 2012) and pseudomagnetic field Guinea et al. (2008); López-Sancho and Brey (2016); Levy et al. (2010); Kun et al. (2019) are the most prominent ones. The electronic transport can been studied through the condensed matter approach with non-equilibrium Green’s function (NEGF) method and general relativistic approach under a pseudo-magnetic field.Stegmann and Szpak (2016) Some more optical properties can also be tuned by curvature, such as optical conductivity Chaves et al. (2014), surface plasmons Smirnova et al. (2016) and the Wolf effect Xu et al. (2018). It is natural to introduce curvature to adjust GSLs’ properties due to its controllability and wide effects, as well as its intrinsic characteristics in real materials observed in experiments.

Inspired by these studies, we construct periodic curved graphene superlattices by applying electrostatic potentials on curved graphene and study their electronic band structures. As the space-dependent fermi velocity plays an important role in graphene, which can produce bound statesGhosh and Roy (2017), control resonant tunnelingLima et al. (2016), break the symmetry between the electron and hole minibandsLima (2015), influence the total conductance, electronic structure, the Fano factor, guided modes, localized current density and the Goos-Hanchen shiftLima et al. (2016, 2018); Lima (2015); Wang et al. (2013); Lima (2017), etc. Here, we mainly concentrated on the effects of space-dependent Fermi velocity on its band structure of the curved periodic super-lattices. Due to the realization of controllable curved graphene Tapasztó et al. (2012); Vázquez de Parga et al. (2008); Bao et al. (2009); Yan et al. (2013a); Ni et al. (2012); Park et al. (2016) and GSLs Dubey et al. (2013); Rusponi et al. (2010); Ponomarenko et al. (2013); Yankowitz et al. (2012); Ni et al. (2015), our proposal may be realizable experimentally.

CGSLs have two periodic structures (i.e., curved surface and potential). We first study the case of equal periods and symmetric potentials. The locations of original DPs are robust against curvature, but the band structures’ slope near the original DP (i.e., Fermi velocity) in directions both along and perpendicular to the potential wells decreases. That structure brings characteristics of new DPs Brey and Fertig (2009); Barbier et al. (2010) into the original DPs. Nevertheless, the DPs can shift in asymmetric potentials and their displacements can be changed by adjusting curvature and potentials. Since locations of Dirac points can be changed under fixed potentials by adjusting curvature in addition to adjusting potentials only Wang and Zhu (2010), it may be more easily to study physics near new Dirac points. Tunable extra gaps can also exist with larger periods of curved surfaces. In addition, zero-energy modes Brey and Fertig (2009); Barbier et al. (2010) and the existence of new DPs are studied systematically. New DPs can appear with a smaller potential due to the space-dependent Fermi velocity. Moreover, their locations are also tunable. With their prominent roles in tuning electronic properties, CGSLs have wide potential in applications.

II Model and method

The Hamiltonian of the low-energy electronic state in monolayer flat graphene reads Wallace (1947)

[TABLE]

where $v_{f}=10^{6}$m/s is Fermi velocity, $\sigma_{x}$ and $\sigma_{y}$ are Pauli matrices, and $\vec{k}=(-i\frac{\partial}{\partial x},-i\frac{\partial}{\partial y})$ is the wave vector from the $K$ point.

In the following, we turn to curved caseArias et al. (2015); Atanasov and Saxena (2015); Castro-Villarreal and Ruiz-Sánchez (2017); de Juan et al. (2013); Castro-Villarreal and Ruiz-Sánchez (2017); Pacheco Sanjuan et al. (2014); Yang (2015); Flouris et al. (2018b). There are two widely used theoretical approaches to model the Hamiltonian of curved graphene. One approach is quantum field theory in curved spacetime de Juan et al. (2007, 2012); Chaves et al. (2014); Arias et al. (2015); Atanasov and Saxena (2015); Castro-Villarreal and Ruiz-Sánchez (2017). This method rests on the basis that the low-energy electron in graphene can be described by a massless Dirac equation, and graphene is regarded as a continuum object. One can obtain the Hamiltonian by combining the Dirac equation and the metric of the curved surface Vozmediano et al. (2010). In this way, topological defects de Juan et al. (2007) and helicoidal graphene Atanasov and Saxena (2015) are investigated, and the space-dependent Fermi velocity is derived de Juan et al. (2007, 2012). The other way is the tight-binding model that takes into account the hopping’s change caused by displacements of carbon atoms and strain Isacsson et al. (2008); Guinea et al. (2008); López-Sancho and Brey (2016). It successfully predicts the pseudomagnetic field Guinea et al. (2008); López-Sancho and Brey (2016). However, both approaches have a weakness in roundly describing the curvature’s effects, so other theories were developed, such as considering strain in the metric de Juan et al. (2012, 2013); Castro-Villarreal and Ruiz-Sánchez (2017), using discrete differential geometry Pacheco Sanjuan et al. (2014), employing the metric in reciprocal space Yang (2015) and rewriting the Dirac equation to include strain Flouris et al. (2018b).

The space-dependent Fermi velocity and pseudomagnetic fields are two main effects of curvature. Here, we mainly focus on the former’s effects, so we use the Dirac equation in curved spacetime to obtain the Hamiltonian. This method does not need to consider ripples extending in the armchair or zigzag direction, which is a convenient approach to apply potentials and is in the view of experiments Atanasov and Saxena (2015).

We rewrite the Dirac equation into a covariant form to obtain it in curved spacetime and the Hamiltonian with the method proposed by previous researchers Vozmediano et al. (2010). In this paper, we consider a one-dimensional periodic curved surface, which refers to ripples that are only dependent on one coordinate and are written as $z=h(x)$. The metric is

[TABLE]

where $g(x)=\frac{dz}{dx}=h^{\prime}(x)$. It is worth noting that the flat space time metric is adjusted for 1D ripples in the z dimension with the resulting ”space” metric here. After calculations described in Appendix A, we obtain the Hamiltonian of graphene in this shape as

[TABLE]

Comparing Eq. (1) with (3), we find that graphene gets space-dependent decreased Fermi velocity in $x$-direction. The Hamiltonian above is non-Hermitian. However, it still has the real and positive eigenvalues while obeying the space-time reflection (PT) symmetryBender and Boettcher (1998) althought we cannot obtain analytic solution. One may modify Eq.(4) into the Hermite form $H^{\prime}=\sqrt{v_{F}(x)}\bm{\sigma\cdot p}\sqrt{v_{F}(x)}$ Peres (2009). However, in order to solve the problem within the frame of the original two cpmponent pseudospin wavefunctions $\psi_{A,B}$, we will solve the differential equations derived from Eq.(4) .

The form of Hamiltonian above has been mentioned in the previous studiesde Juan et al. (2007); Chaves et al. (2014).Next, we will systematically study the influence of the periodic potential field and the modulation of the DPs. Since we do not take into account the discreteness of the lattice and the strain induced by ripples and that the spin connection Vozmediano et al. (2010) is zero in the one-dimensional curved surface, our Hamiltonian cannot include the pseudomagnetic field in curved graphene. However, we are only concerned about the Fermi velocity and low-energy electronic states here, and we consider that GSLs are also continuum objects and that their lengths are larger than the deformation of lattices. This method is also appropriate for any ripple directions and more convenient in considering applied potentials. In addition, the tight-binding method cannot reveal the Fermi velocity’s variation Guinea et al. (2008). Although other modified methods can account for other effects and derive other forms of the Fermi velocity de Juan et al. (2012); Flouris et al. (2018b), they are only different from ours in specific values. Thus, our concise model can still reveal the influences of the renormalized Fermi velocity well and fits this paper’s concerns.

In previous experiments, controllable curved graphene has been realized Tapasztó et al. (2012); Vázquez de Parga et al. (2008); Bao et al. (2009); Yan et al. (2013a); Ni et al. (2012); Park et al. (2016) and can be described by one-dimensional sinusoidal functions well Bao et al. (2009). Hence, we use the one-dimensional function $z=h(x)=a_{0}\cos(\alpha x)$ to model curved graphene with $a_{0}$ and $\Lambda=\frac{2\pi}{\alpha}$ representing its amplitude and period, respectively. The Hamiltonian with curvature can be written as

[TABLE]

Here, we set $f(x)=\sqrt{1+a_{0}^{2}\alpha^{2}\sin^{2}(\alpha x)}$ to represent the effect of curvature. The period of $f(x)$ is $T_{f}=\frac{\lambda}{2}$. When $a_{0}=0$, then $f(x)=1$, and Eq. (4) degrades into Eq. (1). Comparing Eqs. (1) and (4), we find that graphene obtains a space-dependent decreased Fermi velocity in the $x$-direction.

Then, we apply one-dimensional periodic potentials $V(x)$ with square barriers on sinusoidal graphene and assume it is infinite in the $y$-direction to obtain a CGSL. The schematic of this CGSL is shown in Fig. 1. These potentials have the AB structure with a constant value inside each part, and widths are indicated by $w_{A}$ and $w_{B}$. Thus, there are two periods in our model (i.e., $T_{f}$ and the potentials’ period $\Lambda=w_{A}+w_{B}$); then, the lattice constant $T$ of a supercell in a CGSL is their lowest common multiple. Since scalar potentials in curved graphene can be added directly to the Hamiltonian Toyama and Nogami (1999), we have the total Hamiltonian

[TABLE]

with a 2$\times$2 unit matrix $I$. The Hamiltonian (5) acts on the two-component pseudospin wave function $\Psi=(\tilde{\psi}_{A},\tilde{\psi}_{B})^{T}$, and $\tilde{\psi}_{A,B}$ indicate smooth enveloping functions for the $A$ and $B$ sublattice in graphene Wang and Zhu (2010). $\tilde{\psi}_{A,B}$ are written as $\psi_{A,B}e^{ik_{y}y}$ because of translation invariance.

To solve the eigenequation of $H$, we need to obtain the transfer matrix according to the construction by Ref.Wang and Zhu (2010). The main problem of the model described by Eq. (4) is that $f(x)$ is a continuous function, so it is not constant in the $m$th potential. Therefore, we divided the $m$th potential into $n$ parts with an extremely small width. Then, we can regard $f(x)$ as constant in each small part and use the value at the midpoint of the $j$th part $f_{j}$ to represent it. After this approximation, the transfer matrix that connects the wave function from $x$ to $x+\Delta x$ in the $j$th part reads

[TABLE]

Although it has the same expression as flat graphene Wang and Zhu (2010), the parameters in it are totally different and reflect the effects of potentials and curvature (see below). In Eq. (6), $\theta_{j}$ represents the incident angle of wave functions and $\sin\theta_{j}=\frac{k_{y}}{k_{j}}$, $\cos\theta_{j}=\frac{q_{j}}{f_{j}k_{j}}$ with the wave vector $k_{j}=\frac{E-V_{j}}{\hbar v_{f}}$ and satisfy the relation $(f_{j}k_{j})^{2}=(q_{j})^{2}+(f_{j}k_{y})^{2}$. Hence, $q_{j}$ indicates $x$-component of wave vector and reads

[TABLE]

The $f_{j}$ reflects the influences of curvature, and Eq. (6) is also valid for the flat situation when $f_{j}=1$. For the case of $k_{j}=0$, Eq. (6) should be replaced by

[TABLE]

The determinants of the above matrices both fit det$[M_{j}]=1$. Detailed parameters and processes for deriving transfer matrix can also be found in Appendix B. Then, we can determine that the transfer matrix that connects the two terminals of the $m$th potential should be

[TABLE]

where $n$ is the total number of divided small parts in the $m$th potential.

For an infinite CGSL system (AB)N with $N\rightarrow\infty$, the electronic dispersion at any incident angle can be calculated from Bloch’s theorem

[TABLE]

where $T$ is the lattice constant of CGSL, $\Lambda$ is the period of potentials and $\beta_{x}$ is the $x$-component of the Bloch wave vector of the whole system. This relation is influenced by the two periods $\Lambda$ and $T_{f}$. If there is a real solution of $\beta_{x}$, an electron or hole state will exist in the band structure; otherwise, the band structure will show an energy gap. According to this, we can obtain band structures and find the locations of DPs.

The transport properties for a finite superlattice $(AB)^{N}$ system can also be calculated by Eq. (6). We obtain the electronic reflection and transmission amplitudes from the continuity of wave functions Wang and Zhu (2010) with the property of det[$M_{j}$]=1 as follows:

[TABLE]

[TABLE]

where $\theta_{0}$ and $\theta_{e}$ are the incident and exit angle through the superlattice, respectively. $x_{ij}$ are the elements of the entire transfer matrix $X=\prod_{m=1}^{N}M_{m}(w_{m},E,k_{y})=\prod_{m=1}^{N}\prod_{j=1}^{n}M_{j}(w_{j},E,k_{y})$. Then, the transmission probability reads $T=|t|^{2}$. These transport properties of finite systems are another reflection of band structures in infinite ones.

III results and discussion

In this section, we calculate the band structures of periodic CGSLs from the above model to discuss the effects of the space-dependent Fermi velocity induced by curvature. We focus on the location of DPs, effective Fermi velocity and appearance of new DPs. To make our results more realistic, we choose a realizable amplitude and period of curved graphene in the experiments. Moreover, there are two kinds of curved graphene. One of them has an amplitude and period of nanometer length Tapasztó et al. (2012), and the other has amplitudes of 0.7 nm-30 nm and periods of 370 nm-5 $\mu$m Bao et al. (2009). For convenience, we choose amplitude $a_{0}$ and period $\lambda$ that are in the same order of magnitude with the latter case (i.e., $a_{0}$ is of order 10 nm and $\lambda$ is of order 102 nm) in our calculation.

We first consider the case in which the potential and $f(x)$ have the same periods, which refers to $T_{f}=\Lambda$, and symmetric potentials with $w_{A}=w_{B}$. Here, we choose $V_{A}=50$ meV, $V_{B}=0$ and $w_{A}=w_{B}=30$ nm, so the periods of the curved surface and $f(x)$ are $\lambda=120$ nm and $T_{f}=\frac{\lambda}{2}=60$ nm, respectively.

Figs. 2 (a),(b) and (c) show band structures of CGSLs with $a_{0}=0,15,30$ nm, respectively. From these bands, one can find that the locations of DPs associated with the zero-$\bar{k}$ gapWang and Zhu (2010) are 25 meV and robust with different $a_{0}$ values. This observation originates from the fact that potentials are symmetric and $\Lambda=T_{f}$, so $f(x)$ has the same values in the $A$ or $B$ potential, which is the case shown in Fig. 1(b) and is discussed in detail by a simplified model in SM III. However, with increasing $a_{0}$, the locations of other touching points of subbands with higher or lower energies are shifted and closer to the one at 25 meV. The widths of gaps associated with them and all subbands also decrease simultaneously. This effect is due to the decreased slope of energy bands in the $k_{y}$-direction with increasing $a_{0}$. Meanwhile, the band structures’ slopes in the $k_{x}$-direction also decrease obviously when $a_{0}\neq 0$, despite being near the DP with the zero-$\bar{k}$ gap. This property means that the effective Fermi velocity will decrease in both the $x$- and $y$-direction near original DPs in CGSLs and is completely different from that in flat GSLs, since near the original DPs in flat GSLs, the $x$-direction Fermi velocity is unchangedBarbier et al. (2010); Brey and Fertig (2009). These results suggest that the space-dependent Fermi velocity induced by curvature works on band structures as a special potential and introduces the characters of new DPsBarbier et al. (2010); Brey and Fertig (2009) into CGSLs. Hence, CGSLs can be regarded as a platform to use new DPs’ propertiesLuan et al. (2018).

We also plot the ratio of the effective Fermi velocity between flat and curved GSLs in Fig. 2 (d). Here, we use $v_{x}$ and $v_{y}$ to indicate Fermi velocity in CGSLs. In flat GSLs, the $x$-direction Fermi velocity maintains $v_{f}$ and the $y$-direction Fermi velocity decreases to $v_{y0}$. It is demonstrated that with increasing $a_{0}$ and fixed $T_{f}$, the ratios $\frac{v_{x}}{v_{f}}$ and $\frac{v_{y}}{v_{y0}}$ are both decreased, and $\frac{v_{x}}{v_{f}}$ is approximately equal to $\frac{1}{\overline{f(x)}}$. The $\frac{v_{y}}{v_{y0}}$ is nearly equal to it initially, but the distinctions between them increase dramatically with large $a_{0}$. These distinctions can originate from the variation of $f(x)$, which increases with $a_{0}$. The continuous variation of the curved surface can produce an effective potential, and it has been derived in helicoidal graphene Atanasov and Saxena (2015). Fig. 2 (d) suggests that the above relations are also robust with $V_{A,B}$ and $w_{A,B}$. Therefore, the influences of curvature can sometimes be represented partially by $\overline{f(x)}$.

Then, we move to asymmetric potentials with $w_{A}\neq w_{B}$, and band structures are obtained in Fig. 3. The most obvious feature in these figures is that DPs are shifted. From previous research, the DPs’ locations in flat CGLs should be at 30 meV, 20 meV and 16.7 meV Wang and Zhu (2010) under the condition shown in Figs. 3 (a)-(c), (d) and (e), respectively. Thus, DPs move to higher energy when $w_{A}>w_{B}$ as in Figs. 3 (a)-(c). The opposite conclusion can be seen in Figs. 3 (d)(e) when $w_{A}<w_{B}$. This is because $f(x)$ has a larger value in the $A$ region than in the $B$ region with $w_{A}>w_{B}$ (shown in Fig. 1(c)) and is opposite that of $w_{A}<w_{B}$. Comparing Figs. 3 (b)(c) or (d)(e), one can also find that displacements of DPs increase with increasing $a_{0}$ and $w_{A(B)}$.

To understand the robustness and shift of DPs induced by the space-dependent Fermi velocity in different conditions, we propose a simplified theoretical model. We set $f(x)$ in each potential $A$ or $B$ as constant $f_{A(B)}$ and $T_{f}=\Lambda$. Since the average of $f(x)$ can reflect some effects of curvature, we set $f_{A(B)}$ as this average in the range of the potential. Then Eq. 10 reduces to

[TABLE]

with $q_{A(B)}^{2}=(f_{A(B)}k_{A(B)})^{2}-(f_{A(B)}k_{y})^{2}$, and the band structures under this simplified model should be calculated. From the previous analysis, when $V_{B}<E<V_{A}$, the DPs in $k_{y}=0$ should exist and the locations are decided by $q_{A}w_{A}=-q_{B}w_{B}$ Wang and Zhu (2010). With $k_{y}=0$, it should be $f_{A}k_{A}w_{A}=-f_{B}k_{B}w_{B}$. Substituting the expression of $k_{A(B)}$, we obtain the DP’s location:

[TABLE]

When $f_{A}=f_{B}=1$, Eq. (14) reduces to the flat situation. When $f_{A}\neq f_{B}$, the locations of DPs may shift, and the displacement from flat situation $\Delta$ is shown by

[TABLE]

In Fig. 2, $f(x)$ has the same value in potential $A$ and $B$ since $w_{A}=w_{B}$, which corresponds to $f_{A}=f_{B}$ and $\Delta=0$. In Figs. 3 (b) and (c), $f_{A}>f_{B}$ and $V_{A}>V_{B}$; then, $\Delta>0$ and DPs shift to higher energy. Figs. 3 (d) and (e) are the opposite. Meanwhile, Eq. (15) states that $\Delta$ is proportional to $w_{A,B}$, which is also consistent with the conclusion gained by comparing Figs. 3(d) and (e).

In Fig. 3(f), the changes in the DPs’ positions with different $a_{0}$ and $\Lambda$ values are demonstrated by the scattering diagrams. One can immediately find that with increasing $a_{0}$ and $\Lambda$, the DPs shift more, which agrees with Eq. (15) and suggests their tunability. We also computed the shift from the simplified model with $\overline{f(x)}$ and Eq. (15). These results are shown by the lines and fit the realistic results well. Therefore, constructing CGSLs can be a feasible way to tune the locations of DPs by adjusting both potentials and curved surfaces. Other ways that can change the Fermi velocity Pellegrino et al. (2012); Raoux et al. (2010); Krstajić and Vasilopoulos (2011) are also possible to tune DPs. Furthermore, one may need to concentrate on these effects in experiments since the intrinsic feature Fasolino et al. (2007) of ripples in graphene and potentials may not be strictly symmetric.

Next, we discuss the condition for $T_{f}>\Lambda$. Here, we plot band structures when $T_{f}=2\Lambda$, $1.5\Lambda$ and $1.25\Lambda$ in Figs. 4 (a) (c) and (e). To analyze the impacts of periods, we fix the amplitude of $f(x)$. Note that $f(x)$ has period-dependent amplitude, and we need to change $a_{0}$ and $T_{f}$ simultaneously to make $a_{0}\alpha$ fixed. We choose $a_{0}\alpha=\frac{\pi}{4}$, which is the same as that of Fig. 2(b). Therefore, the locations of DPs and slope of bands are the same in these figures but there are some new gaps in the band structures. In addition, the number of gaps increases from Fig. 4 (a)-(e). That means that two different periods construct aperiodic structures with various orders. The robustness of locations of DPs are also the same as aperiodic GSLs Ma et al. (2012). Then, we plot the electronic states versus different $T_{f}$ and find that with increasing $T_{f}$, the number of new gaps clearly increases, which means the order of aperiodic structures increases. These new gaps are also controllable by adjusting $T_{f}$. When $T_{f}$ is larger than a specific value, the energy bands become the discontinuous ones. Our results propose another way to construct aperiodic GSLs and acquire tunable band gaps by changing $T_{f}$. Meanwhile, curved graphene in experiments may not be exactly periodic Tapasztó et al. (2012), so there may be some gaps in realistic CGSLs.

Previous studies reveal that new DPs that locate at $k_{y}\neq 0$ could appear with some conditions in flat GSLs Park et al. (2008a, 2009); Brey and Fertig (2009); Barbier et al. (2010). Now, we discuss the condition for the appearance of new DPs in CGSLs. We still consider the simplest situation $w_{A}=w_{B}$ and $T_{f}=\Lambda$ with the simplified model proposed above.

We also increase the periods of potentials to obtain new DPs. In Figs. 5 (a)-(c), we plot band structures with $w_{A}=w_{B}=80$ nm and $a_{0}=0$, 40 nm and 50 nm. It is demonstrated that the slopes of band structures decrease such that they nearly vanish, and then, new DPs appear with increasing $a_{0}$. The $k_{y}$-direction coordinates of new DPs also increase with increasing $a_{0}$. These results illustrate that the space-dependent Fermi velocity makes the existence of new DPs easier. Thus, we plot electronic states with different $w_{A(B)}$ values. Comparing the results and those of Fig. 5(d) in Ref. Wang and Zhu (2010), it indicates that new DPs can exist with smaller $w$ when $a_{0}\neq 0$.

The condition for the existence of new DPs can also been discussed by the simplified model proposed above. For new DPs, it has been illustrated in the literature Barbier et al. (2010); Wang and Zhu (2010); Fan et al. (2016) and Eq. (13) that once the condition

[TABLE]

is satisfied under some specific $k_{y}$, $\sin(q_{A}w_{A})=\sin(q_{B}w_{B})=0$ and $\cos(q_{A}w_{A}+q_{B}w_{B})=1$; then, $\cos(\beta_{x}\Lambda)=1$ and $\beta_{x}$ always has a real solution with all energies. This condition leads to the closing of the zero-$\bar{k}$ gap, and a pair of new DPs will appear away from $k_{y}=0$. If we set $V_{A}=-V_{B}$ and $w_{A}=w_{B}$, then $f_{A}=f_{B}=f$, and the DPs should be located at zero energy, so we discuss $E=0$ next, which refers to zero-energy modes studied in previous worksBrey and Fertig (2009); Fan et al. (2016). Under the above assumption, we obtain

[TABLE]

is satisfied with the above conditions. The new DPs will exist when $(\frac{V}{\hbar v_{f}})^{2}-(\frac{2m\pi}{f\Lambda})>0$. Therefore, when $f>1$, $k_{y,m}$ can obtain real solutions with smaller $\Lambda$ and can make the generation of new DPs easier. This conclusion is consistent with Fig. 5(d).

Eq. 17 can also be changed to

[TABLE]

with $V_{A}=-V_{B}=2\pi l\frac{\hbar v_{f}}{\Lambda}$. Here, $l$ represents potential since it is proportional to $V_{A}$. In flat GSLs with $f=1$, a new pair of new DPs are generated once $l$ is a positive integer number larger than one Barbier et al. (2010); Fan et al. (2016). In CGSLs, $f>1$, so new DPs can arise with smaller $l$.

Fig. 6 is plotted to verify the above conclusions. In Figs. 6 (a)-(c), band structures with different $a_{0}$ and fixed $l=1$ are plotted. Here, we choose $a_{0}=0$, 15nm and 55nm to get clearly visible 1, 3 and 5 DPs, and we still use symmetric potential and $T_{f}=\Lambda$. It is illustrated that the slope of the band first decreases such that it nearly vanishes, and then, a new pair of DPs arise with increasing $a_{0}$. These phenomena are totally different from flat ones with $l=1$, where the first pair of new DPs just appear. Then, we set some $l$ and fixed $a_{0}=15$ nm to calculate the band structures in Figs. 6 (d)-(f), showing the appearance of the first and second pair of new DPs. It is demonstrated that new DPs can generate in a smaller potential when graphene is curved. For example, there are only three DPs when $l=2$ in flat GSLs, but five in the CGSL with $a_{0}=15$ nm. In addition, the upper row of Fig. 6 suggests that the locations of new DPs can be tuned by $a_{0}$ even with fixed potential or $l$, which cannot be realized in flat GSLs. Eq. (18) also denotes this. By comparing the two rows of Fig. 6, one can also find that the coordinates of new DPs in upper rows are obviously smaller than those in the low row due to smaller $l$. Our discussions provide a possible method to adjust the locations of new DPs. Since the above effects originated from the change of the Fermi velocity, our results suggest that forming that periodic Fermi velocity by other ways Pellegrino et al. (2012); Raoux et al. (2010); Krstajić and Vasilopoulos (2011) may also be available.

Finally, we discuss the number of DPs under different $l$ and $a_{0}$. We compute the zero-energy electronic states with different $k_{y}$ and $l$ in Fig. 7. When new DPs appear, states away from $k_{y}=0$ will be present in these figures. Referring to the flat situation in Fig. 7(a), we can indicate $l$ at which new DPs arise in (b) and (c). It is shown that $l$ under this condition is smaller with increasing $a_{0}$. The distinctions between the flat and curved situations also increase with larger $l$. We can conclude that with space-dependent Fermi velocity induced by curvature, new DPs can arise with smaller potentials, which means this changed velocity works as an effective potential. In view of the fact that the Fermi velocity can be measured indirectlyLuican et al. (2011); Yan et al. (2013b), and there have been experiments that have observed the appearance of new Dirac points in quasi-periodic grapheneYan et al. (2013b), our conclusion may be verified experimentally. This finding is consistent with the analysis in the Results and discussions section and provides a possible simple way to acquire new DPs and then investigate properties near them.

We have discussed several situations in this part and here we made a list. (a). For the simplest situation with equal periods and symmetric potentials, the DPs will not move, but the slopes of bands decreased with increasing $a_{0}$. (b). For asymmetric potentials, there will be tunable shifted DPs which move to higher energy when $w_{A}>w_{b}$ and to lower energy when $w_{A}<w_{B}$. DPs shift more with increasing $a_{0}$. (c). For unequal periods, there will be extra gaps. So it is another way to construct aperiodic GSLs and acquire tunable band gaps by changing $T_{f}$. (d). By increasing the periods of potentials we can obtain new DPs and show that new DPs can exist with smaller $w$ in curvature. (e). We have also illustrated that new DPs can arise with smaller potentials by curvature.

IV conclusions

In summary, we proposed CGSLs by combining curved graphene and periodic potentials and then investigated their electronic properties. Since we focus on the effects of the space-dependent Fermi velocity, we use the Dirac equation in curved spacetime and the transfer matrix to obtain the band structures. For the simplest situation with equal periods and symmetric potentials, the DPs will not move, but the slopes of bands dramatically decrease in both the $k_{x}$- and $k_{y}$-direction. For asymmetric potentials and unequal periods, tunable shifted DPs and extra gaps can appear, respectively. We also discussed the condition for DPs’ appearance with symmetric potentials and equal periods. The condition for obtaining new DPs can be met more easily when graphene is curved. One can use this property to create new DPs experimentally to investigate physical phenomena near those points.

We would like to state that we reveal part of the space-dependent Fermi velocity’s impacts on electronic properties. Although curvature can bring other effects such as the pseudomagnetic field, they are not reflected by our model and need to be analyzed by other means, such as the tight-binding model with elastic theory. Thus, it is valuable to discuss these effects with a more perfect method in the future. We would also like to emphasize that the effects of ripples in GSLs should not be neglected in some cases experimentally since they are intrinsic. That may cause a slight shift of DPs or extra gaps. The controllability of the DPs also provides a new way to adjust electronic structures in experiments.

Acknowledgements.

T.M. thanks CAEP for partial financial support. This work was supported by NSFC (Nos. 11774033, 11974049, 11674284 and 11974309), the Beijing Natural Science Foundation (No. 1192011) and the Zhejiang Provincial Natural Science Foundation of China under Grant No. LD18A040001. H.Q. Lin acknowledges financial support from NSAF U1930402 and NSFC 11734002, as well as computational resources from the Beijing Computational Science Research Center.

Appendix A. Hamiltonian of curved graphene

We rewrite the Dirac equation into a covariant form to obtain it in curved spacetime and the Hamiltonian with the method proposed by previous researchers Vozmediano et al. (2010). According to Eq. (1) and the Dirac equation $i\hbar\frac{\partial}{\partial t}\psi=\hat{H}\psi$, we obtain its covariant form

[TABLE]

with $\mu=0,1,2$ representing time and $x$ and $y$ as the coordinates. We first use natural units $v_{f}=1$ during calculations. The short lines above $\bar{\gamma}^{\mu}$ illustrate the flat case and $\bar{\gamma}^{\mu}$ should satisfy the anticommutation relation $\{\bar{\gamma}^{\mu},\bar{\gamma}^{\nu}\}=2\eta^{\mu\nu}I$ with the Minkowski metric $\eta_{\mu\nu}=\text{diag}(1,-1,-1)$. $\bar{\gamma}^{\mu}=(\sigma_{3},-i\sigma_{2},i\sigma_{1})$ with $(\sigma_{1},\sigma_{2},\sigma_{3})=(\sigma_{x},\sigma_{y},\sigma_{z})$ as the Pauli matrices. These $\gamma$ matrices fit the above anticommutation relation.

Then, we rewrite Eq. (19) into the curved case Vozmediano et al. (2010), which reads

[TABLE]

with metric $g_{\mu\nu}$. There are two differences between the curved and flat equations Vozmediano et al. (2010) (1) Fielbein fields $e^{\mu}_{a}$ need to be introduced to indicate the change of $\gamma^{\mu}$, which is $\gamma^{\mu}=e^{\mu}_{a}\bar{\gamma}^{a}$. Here, the symbols without short lines mean the curved case. Fielbein fields should fit conditions such that $g_{\mu\nu}=\eta_{ab}e_{\mu}^{a}e_{\nu}^{b}$, $\gamma^{\mu}$ calculated from Fielbein fields should have $\{\gamma^{\mu},\gamma^{\nu}\}=2g^{\mu\nu}I$ and the determinants of the metrics are $[\text{det}(g_{\mu\nu})]^{1/2}=\text{det}(e_{\mu}^{a})$. (2) The differential operator should be $D_{\mu}=\partial_{\mu}+\Omega_{\mu}$ and the spin connection is

[TABLE]

with the Christoffel symbol $\Gamma_{\mu\nu}^{\lambda}=\frac{1}{2}g^{\sigma\lambda}(\frac{\partial g_{\nu\sigma}}{\partial x^{\mu}}+\frac{\partial g_{\mu\sigma}}{\partial x^{\nu}}-\frac{\partial g_{\mu\nu}}{\partial x^{\sigma}})$.

In this paper, we consider a one-dimensional periodic curved surface, which refers to ripples that are only dependent on one coordinate and are written as $z=h(x)$. The line elements read

[TABLE]

with $g(x)=\frac{dz}{dx}=h^{\prime}(x)$, so the metric is

[TABLE]

According to Eq. (23), we obtain that

[TABLE]

and $\Omega_{\mu}=0$. After substituting them into Eq. (20), multiplying $\sigma_{z}$ on both sides of the equation and adding $v_{f}$, we obtain Eq. (3).

Appendix B. Transfer matrix for CGSL

After acting Eq. (5) on $\Psi=(\tilde{\psi}_{A},\tilde{\psi}_{B})^{T}=(\psi_{A},\psi_{B})^{T}e^{ik_{y}y}$, we can get

[TABLE]

where $k=\frac{E-V(x)}{\hbar v_{f}}$ represents the wave vectors inside potentials. Due to the square barriers of potentials, $V(x)$ in the $m$th potential region maintains constant $V_{m}$. Next, we divide the potential region into $n$ parts, so $f(x)$ changes little inside each part; then, $f(x)$ in the $j$th part can be regarded as constant $f_{j}$. With this approximation, Eq. (25) is expressed as

[TABLE]

The following processes will be the same as in Ref. Wang and Zhu (2010), and we can obtain Eq. (6).