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Optical properties of graphene superlattices

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 405304 (http://iopscience.iop.org/0953-8984/26/40/405304) View the table of contents for this issue, or go to the journal homepage for more

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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 405304 (10pp)

doi:10.1088/0953-8984/26/40/405304

Optical properties of graphene superlattices H Anh Le1 , S Ta Ho2 , D Chien Nguyen3 and V Nam Do1,4 1 Advanced Institute for Science and Technology, Hanoi University of Science and Technology (HUST), No. 01 Dai Co Viet road, Hanoi 10000, Vietnam 2 National University of Civil Engineering, No. 55, Giai Phong road, Hanoi, Vietnam 3 School of Engineering Physics, HUST, Hanoi 1000, Vietnam 4 Institute for Computational Science and Engineering, HUST, Hanoi 1000, Vietnam

E-mail: [email protected] Received 13 May 2014, revised 17 July 2014 Accepted for publication 5 August 2014 Published 17 September 2014 Abstract

In this work, the optical responses of graphene superlattices, i.e. graphene subjected to a periodic scalar potential, are theoretically reported. The optical properties were studied by investigating the optical conductivity, which was calculated using the Kubo formalism. It was found that the optical conductivity becomes dependent on the photon polarization and is suppressed in the photon energy range of (0, Ub ), where Ub is the potential barrier height. In the higher photon energy range, i.e. Ω > Ub , the optical conductivity is, however, almost identical to that of pristine graphene. Such behaviors of the optical conductivity are explained microscopically through the analysis of the elements of optical matrices and effectively through a simple model, which is based on the Pauli blocking mechanism. Keywords: graphene, superlattice, optical properties (Some ﬁgures may appear in colour only in the online journal)

range, showing a constant isotropic optical conductivity of σ = σ0 = e2 /4¯h [18, 19]. Regarding the current tendency of engineering graphene for particular applications, our aim in this work is to investigate the changes in the fundamental electronic and optical properties of graphene when it is patterned in a so-called superlattice structure. Technically, making a superlattice has been well known for a long time as a way of engineering the electronic and optical properties of semiconductors [20]. The concept of ‘graphene superlattice’ (GSL) was ﬁrst introduced by Barbier et al with the implication of the graphene sheets subjected to periodic potentials [21]. It was pointed out that the electronic properties of pz electrons in graphene can be altered in a controllable way using a periodic scalar potential. It was shown that new Dirac cones could be generated as the satellites of the two main cones located at the K points of the GSL Brillouin zone (see ﬁgure 1(c)) [22, 23]. Such Dirac cones, however, are anisotropic and robust even in high external magnetic ﬁelds [24]. In other words, a periodic potential can make electrons behave as a new kind of relativistic Dirac fermion whose motion is characterized by two different phase velocities. In some GSL structures with relevant parameters, the motion of such fermions are even restricted along only the direction perpendicular to the direction of the potential, i.e. the collimation effect

1. Introduction

Graphene, since its discovery in 2004 [1], has been attracting intensive consideration in many ﬁelds of research work due to its fascinating properties [2]. So far, much of the understanding of this material has been achieved, including half-integer quantum Hall effect, [3, 4] ﬁnite minimal conductivity at zero charge carrier concentration [3, 5, 6] and strong suppression of weak localization [7, 8], etc. Such fascinating properties of graphene were pointed out to be fundamentally governed by the low energy excited states which behave as the massless relativistic Dirac fermions [9, 10]. When studying graphenebased structures such as p-n and p-n-p junctions [11, 12], Cheianov et al realized that the variation of potential proﬁle in the graphene sheets can modulate the motion of such particles and hence may lead to surprising transport phenomena such as Klein tunneling [13–15] and focus of electron ﬂow through a p-n junction [11, 12]. In the ﬁelds of photonics and optoelectronics, graphene is expected to have advanced applications in the near future due to its very high transparency (about 97%) but strong interaction with light [16, 17]. Interestingly, it was predicted and then experimentally conﬁrmed that due to the linear dispersion the optical characteristics of graphene are independent of the light frequency from the visible to the terahertz 0953-8984/14/405304+10$33.00

1

© 2014 IOP Publishing Ltd

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J. Phys.: Condens. Matter 26 (2014) 405304

H Anh Le et al

Y

X

Su

bst

rat

Graphene

e

(a)

Insulator Electrode

U(x) U1 x1,i

ky

x2,i x4,i x3,i x1,i+1

(c) x

U2 K

(b) K’

A-GSL Primitive-cell

kx

Z-GSL Primitive-cell Figure 1. (a) Schematic diagram of the graphene superlattice structure. (b) Proﬁle of electrostatic potential induced by an array of back gate electrodes and its zoomed-out proﬁle showing the atomic pattern in a primitive cell of the A-GSL and Z-GSL conﬁgurations. (c) The ﬁrst Brillouin zone of the A-GSL conﬁguration with two K points.

transversal polarization of photons are presented in section 3.2. Discussions and explanations for the optical behavior of graphene in the GSL structure compared to that of pristine graphene are devoted to section 3.3. Finally, the conclusion of the paper is given in section 4.

[23]. Recently, we have ﬁgured out some other typical behaviors of pz electrons in the GSL structures [25]. We showed that, in general, the GSLs’ electronic structure can be seen as the superposition of the two energy band pictures of pristine graphene where they are relatively shifted by an amount of the potential barrier Ub . Moreover, we pointed out the dependence of the GSLs’ electronic structure on the way the potential induces the symmetrical breaking for the graphene hexagonal lattice. In particular, we demonstrated that under the periodic scalar potentials, states of pz electrons are strongly modulated; they even become localized in either regions of the potential barriers or wells. Though it is hard to recognize the localized states in transport measurements, the typical GSLs’ electronic structure in the energy range of (−Ub /2, Ub /2) were argued to be reﬂected signiﬁcantly through many physical properties, such as density of states [26], and dc conductivity [23, 27, 28], which were comprehensively reviewed in [29]. In this work, with the aim of complementing fundamental knowledge of graphene superlattices, we focus on the optical coupling between the modulated states of pz electrons. We show that the potential-induced symmetry breaking of the honeycomb lattice causes the dependence of the optical properties of graphene sheets on the polarization of incident photons. Additionally, the spatial modulation of pz states leads to the reduction of the optical coupling between occupied and unoccupied states compared to that in pristine graphene. The remainder of the paper is organized as follows. In section 2, we brieﬂy review efforts in introducing 1D potentials into graphene sheets in practice. We then present a physical model to describe the dynamics of pz electrons and calculations for quantities which characterize the electronic and optical properties of GSLs. Next, in section 3, we analyse the changes in the electronic structure of graphene versus parameters of the potential function. The calculation results for the optical conductivity in both cases of the longitudinal and

2. Graphene superlattice: model and calculation

The concept of ‘graphene superlattice’ has been commonly known, especially involving the Moir´e patterns of graphene epitaxially grown on the surface of metals and/or high quality insulators due to the lattice mismatch. Such structures have been considered as the natural form of two-dimension (2D) graphene superlattices. Though many interesting properties of pz electrons in one-dimension (1D) GSLs have been discussed [29], fabricating such structures remains a challenge. Practically, techniques such as imprinting adsorbates on the graphene surface [30] and corrugating graphene sheets [31, 32] have been used. Very recently, a quite simple technique of growing graphene on well-prepared high-index metallic surfaces was demonstrated to successfully synthesize 1D GSLs [33]. Savel’ev et al even showed an alternative way of engineering the electronic structure of graphene into one of the 1D GSLs by using a laser ﬁeld [34]. In optoelectronic applications, to be able to control the electronic structure and optical properties of graphene, the approach of using regular electronic gates may be technologically more relevant. In ﬁgure 1(a), we show a GSL conﬁguration adapted from the superlattice structure already demonstrated for conventional 2D electron gas [35, 36]. Accordingly, the periodic scalar potential is created by an array of metallic electrodes which are regularly buried inside a substrate. The graphene sheet may not be directly covered on the electrodes but on the top surface of an insulator layer. The electrodes are connected to a voltage source to control the magnitude of the electrostatic potential in the graphene sheet. 2

J. Phys.: Condens. Matter 26 (2014) 405304

H Anh Le et al ν (ν = a, b, c, d), only the explicitly assigned to, e.g. Vi,m,j last one is necessary to denote the spatial variation of these quantities in a period of the potential. Practically, we use a continuous function V (x) to describe the proﬁle of these ν energies, i.e. Vi,m,j = Vjν = V (xjν ), where V (x) is modeled by the expression:

The electrostatic potential is thus periodic and is characterized by the period length L = Wg + Ls , where Wg is the gate width and Ls is the gate-to-gate spacing, and by the potential barrier height Ub , which is tunable by an externally applied voltage. In this work, we consider two GSL conﬁgurations, namely A-GSLs and Z-GSLs. Since the calculation procedure for both GSL types is the same, in the following we present in detail the calculation for the A-GSL conﬁguration only. In ﬁgure 1(a), we illustrate a unit cell of the A-GSL sample. It is a rectangle √ with the length of L = 3N acc and the width of W = 3acc where acc 1.42 Å is the carbon atomic spacing and N the number of carbon atoms in a super-cell. The carbon atoms on each node of the armchair chain are denoted by characters a, b, c, and d which are associated with three indices i, m, j labelling the cell, the armchair line, and the real position of each node in the unit cell, respectively. The coordinates of each carbon atom are determined as follows: acc a xi,m,j = xi0 + 3acc (j − 1) + 2 √ 1 a = 3acc m − yi,m,j 2

V (x) = U1

θ(x − x1, )θ (x3, − x)fβ x − x2,

θ(x − x3, )θ (x1,+1 − x)fβ x4, − x − U2 , (4) where fβ (x) = 1/[1 + exp(βx)] takes the form of the FermiDirac distribution function to account for the smooth nature of the potential. The potential barrier is thus determined by Ub = U1 − U2 . Now taking into account the periodicity of the two indices i and m, we can Fourier transform all the involved operators, for instance,

b xi,m,j = xi0 + 3acc (j − 1) + acc √ b yi,m,j = 3acc (m − 1) c xi,m,j = xi0 + 3acc (j − 1) + 2acc √ c = 3acc (m − 1) yi,m,j 5acc d xi,m,j = xi0 + 3acc (j − 1) + 2 √ 1 d = 3acc m − yi,m,j , 2

(5)

† ai,m,j

(6)

a where Ncell is the number of A-GSL super-cells; ri,m,j = a 0 (xi , yi,m,j ) are the vectors positioning a-carbon atoms; and k = (kx , ky ) is the wave vector, which takes discrete values for both components. Concretely,

(1)

π n(3acc N ) π ky = ± √ n( 3acc )

kx = ±

where = 3acc (i − 1)N . The total Hamiltonian Htotal for pz electrons in a GSL sample is formally written as a summation of two terms: the one describing the hopping of pz electrons between two nearest neighbour carbon atoms H , and the other for the gate-generated electrostatic potential U . These two terms are speciﬁed as follows: xi0

H = tcc

† ai,m,j (bi,m,j + bi,m+1,j + di,m,j −1 )

ˆ where X(k) = (ak,1 , bk,1 , ..., dk,N )T is a column vector, and Hˆ (k) is a 4N × 4N matrix, which takes the cyclic block form as follows: ⎛ ⎞ P1 (k) Q†2 (k) . . . QN (k) ⎜ Q2 (k) P2 (k) . . . ⎟ 0 ⎜ ⎟ Hˆ (k) = ⎜ ⎟ (10) .. .. . . .. .. ⎝ ⎠ . .

† (ai,m,j + ai,m−1,j + ci,m,j ) + bi,m,j

U =

(2)

N

† † a b ai,m,j ai,m,j + Vi,m,j bi,m,j bi,m,j Vi,m,j

i,m j =1 † c + Vi,m,j ci,m,j ci,m,j

+

† d Vi,m,j di,m,j di,m,j

Q†N (k)

,

(8)

k

i,m j =1

† + di,m,j (ai,m,j +1 + ci,m,j + ci,m+1,j )

(7)

wherein n = 1, 2, . . . , Ncell . The set of such k points makes up the so-called ﬁrst Brillouin zone of A-GSLs. Substituting expressions (5) and (6) into equations (2) and (3), we can rewrite Htotal in the form ˆ Htotal = (9) Xˆ † (k)Hˆ (k)X(k),

N

† + ci,m,j (bi,m,j + di,m−1,j + di,m,j )

1 a ak,j eikri,m,j 2Ncell k † 1 a =√ ak,j eikri,m,j , 2Ncell k

ai,m,j = √

(3)

0

...

PN (k)

In equation (10) the matrix blocks Pj (k) and Qj (k) are given by:

where tcc ≈ −2.67 eV is the hopping energy of electrons between two nearest-neighbor pz orbitals. One should note that for the summation over the index j in equation (2), when j = 1 one has to replace di,m,j −1 by di−1,m,N and when j = N one replaces ai,m,j +1 by ai+1,m,1 . For the on-site energies in equation (3), though three indices i, m and j are

⎛

Vja α ∗ (k) 0 b ∗ ⎜ α(k) β (k) V j Pj (k) = ⎜ ⎝ 0 β(k) Vjc 0 0 α(k) 3

⎞ 0 0 ⎟ ⎟ α ∗ (k) ⎠ Vjd

(11)

J. Phys.: Condens. Matter 26 (2014) 405304

⎛

0 ⎜ 0 Qj (k) = ⎜ ⎝ 0 0

0 0 0 0

0 0 0 0

H Anh Le et al

⎞

β(k) 0 ⎟ ⎟ 0 ⎠ 0

where ΔRij = Rj − Ri are the vectors connecting two lattice points Rj and Ri , the velocity operators are calculated using the rule δHtotal [A] vx/y = − (18) δAx/y A→0

(12)

where the expressions for α(k) and β(k) in equations (11) and (12) read √ 3 1 (13) ky acc α(k) = 2tcc exp i kx acc cos 2 2 β(k) = tcc exp (ikx acc ) ,

After some tedious algebra, we proved that the resulted velocity matrices vx/y (k) are identical to those obtained from the gradient-approximation, i.e.

and Vjν satisfy equation (4). In the limit case of N = 1, equation (10) becomes ⎞ ⎛ α ∗ (k) 0 β(k) Va ⎜ α(k) β ∗ (k) 0 ⎟ Vb ⎟ (15) Hˆ (k) = ⎜ c ∗ ⎝ 0 β(k) V α (k) ⎠ β ∗ (k) 0 α(k) Vd

⎛ 1⎜ ⎜ vˆy (k) = ⎜ h ¯ ⎝

Qx† (k)

0

...

0 ... Py (k) 0 Py (k) . . . .. .. .. . . . 0 0 ...

Px (k) ⎞ 0 0 ⎟ ⎟ ⎟ .. ⎠ . Py (k)

(21)

where ⎛ ⎜ Px (k) = iacc ⎜ ⎝ ⎛

(16)

where fβ (x) = 1/(1 + exp[βx]) is the Fermi-Dirac function with μ the chemical potential √ and2 β = 1/kB T the inverse of the area of the super-cell thermal energy; Scell = 3 3N acc of GSLs; Nk the number of k-points needed to sample the Brillouin zone (BZ), which is in principle equal to the number nm of super-cells Ncell ; v˜ˆ α (k) the matrix elements corresponding to the α-component of the velocity operator vector. The spin degeneracy is already taken into account in the above equations by factor 2, and η is a phenomenological parameter characterizing electron scattering processes. In our work, we simply use it as an inﬁnitesimal number to eliminate the singularity of the sum. To complete equation (16), we need to determine all of nm the elements v˜ˆ x/y (k) of the velocity matrices. To do so, we follow the standard scheme by introducing the vector potential A representing an electromagnetic ﬁeld into the Hamiltonian Htotal . Using the Peierls substitution tcc → ti,j : e 1 (17) dξ A(Ri + ξ ΔRij ).ΔRij , ti,j = tcc exp −i h ¯ 0

α ∗ (k) 2

0 α(k) 2

0 0

0 ⎜ 0 = iacc ⎜ ⎝ 0 0 ⎛ 0 ⎜ αy (k) Py (k) = ⎜ ⎝ 0 0

Q x (k)

nm mn Nk v˜ˆ α (k)v˜ˆ β (k) ¯ 1 2e2 h iScell Nk k∈BZ m,n En (k) − Em (k) + Ω + iη

fβ [En (k) − μ] − fβ [Em (k) − μ] × En (k) − Em (k)

(19)

where Hˆ (k) is given by equations (10)–(14). Concretely, ⎞ ⎛ Px (k) Qx† (k) . . . Q x (k) ⎟ 0 1⎜ ⎟ ⎜ Qx (k) Px (k) . . . vˆx (k) = ⎜ ⎟ , (20) .. .. . .. .. ⎠ h ¯ ⎝ . . .

This result is in fact the Hamiltonian matrix for the case of pristine graphene, which is written for the rectangular unit cell, but not for the rhombus primitive one as usual. In our work, this limit case was used to check the validation of the calculation for the optical conductivity. For each value of k, the Hamiltonian matrix Hˆ (k) in equation (10) is numerically diagonalized. A set of 4N eigen-values {En (k)} and another of 4N corresponding eigen-vectors {|Ψn (k)} are obtained. The latter are then used to represent operators which deﬁne physical quantities of interest. To investigate the optical properties of GSLs, we analyse the behavior of the optical conductivity σαβ (Ω ) as a function of photon energy Ω , which was calculated using the Kubo formula in the representation of the eigenfunctions of the total Hamiltonian: σαβ (Ω ) =

1 ∂ Hˆ (k) , h ¯ ∂kx/y

vˆx/y (k) =

(14)

0 0 0 0

0 0 β ∗ (k) β(k) 0 α(k) 0 2 ⎞ 0 β(k) 0 0 ⎟ ⎟ 0 0 ⎠ 0 0

αy∗ (k) 0 0 0

0 0

α ∗ (k) 2

⎞ ⎟ ⎟ ⎠

(22)

0 (23)

⎞ 0 0 ⎟ 0 0 ⎟ (24) ∗ 0 αy (k) ⎠ αy (k)

and αy (k) is given by: αy (k)

√ √ 3ky acc 1 = − 3tcc exp i kx acc sin (25) 2 2

Based on the eigen-vectors of the Hamiltonian Hˆ (k), the nm elements v˜ˆ x/y (k) are determined as those of the matrices v˜ˆ x/y (k) = U † (k)vˆx/y (k)U (k),

(26)

where the basis transforming matrix U (k) is constructed from the eigen-vectors {|Ψn (k)} of the Hamiltonian Hˆ (k) given in equation (10). With all of the above ingredients, equation (16) was numerically calculated. In the next section, we will present and discuss the electronic and optical properties of the graphene superlattices in detail. 4

J. Phys.: Condens. Matter 26 (2014) 405304

H Anh Le et al

3. Results and discussion

2 1.5

3.1. Electronic structure

1

Energy (eV)

In our work, the proﬁle of periodic scalar potentials is modeled by equation (4). The natural smoothness of the potentials is described using the Fermi-Dirac function fβ (x) with a parameter β. We, however, observed that the effect of the potential smoothness simply makes overshadowing some subtle results, for instance, the conditions for the emergence of novel Dirac points and the determination of their position, but not qualitatively change the observations presented below. We accordingly present only results obtained from the calculations for GSLs with the periodic length of about 12 nm, i.e. containing more than 30 carbon atoms, and with quite sharp potential barriers (not need to be piecewise-constant). In ﬁgure 2, we ﬁrst present the energy band structure of pz electrons in a representative A-GSL sample for several values of the potential barrier heights Ub . Figure 2(a) is plotted for the case of Ub = 0. It shows the electronic energy band structure of free-standing graphene in the representation of the rectangular Brillouin zone of the A-GSL conﬁguration which is different from that of the hexagonal one of pristine graphene. The Dirac valleys are located at two K √ points with the coordinates of (0, ±kyK ) where kyK = 2π/3 3acc . In particular, there is a two-fold degeneracy of all energy surfaces except for two surfaces E2N (k) and E2N+1 (k), conﬁrming the ﬁndings of Ho et al in [26] in which cosine scalar potentials were used. When increasing the potential barrier height Ub to the values of U0 , 2U0 and 3U0 where U0 = 2π h ¯ vF /L, the degeneracy of several energy surfaces close to E2N (k) and E2N+1 (k) becomes broken, showing the appearance of additional energy surfaces. In ﬁgures 2(a)–(d) we add four ± red-lines E1,2 (ky ) = ±¯hvF (ky − kyK ) ± Ub /2 to guide the eyes to the changes in the electronic structure of pz electrons. Accordingly, it suggests the band structure of GSLs as the relative shift of the Dirac cones of graphene by an amount of exactly Ub . In the energy range close to the Fermi energy plane EF = 0, the energy surfaces possess local extrema which seem to be pinned at the values of integer multiples of U0 [25]. Because of this pinning, the potential results in the curvature of the energy surfaces, forming saddle-points which are signiﬁcantly manifested in the picture of the density of states as shown later in ﬁgure 5. Similarly, in the representation using the Z-GSL Brillouin zone, the π and π ∗ energy surfaces of pristine graphene are also folded into sub-bands (ﬁgure 3). The Dirac cones are also mapped into the interior of the Z-GSL rectangular zone at the K points whose positions are not ﬁxed as in the case of A-GSL Brillouin zone, but vary depending on the number of carbon atoms in the unit cell. In this representation, there is no degeneracy of the energy surfaces. However, when increasing the potential barrier height Ub , some energy surfaces becomes so close that they are hard to be distinguished, i.e. they become degenerated. We cannot observe any invariant point of the energy surfaces when changing U (x). Interestingly, when the potential barrier height Ub is sufﬁciently high, we can observe a narrow energy bandgap which distinguishes the whole energy band into the valence and the conduction bands. Recently,

0.5 0

a)

b)

d)

c)

−0.5 −1 −1.5 −2

1

1.5

1

1.5

1

1.5

kyacc

1

1.5

Figure 2. Variation of the energy band structure of pz electrons in a A-GSL sample with N1 = N2 = 15 and different potential barrier heights: Ub = 0 (a), Ub = U0 (b), Ub = 2U0 (c) and Ub = 3U0 (d) where U0 = 2π¯hvF /L. The red lines are added to show the change of the GSL band structure as the shift of the Dirac cones. The black dashed lines illustrate a pinning of some points on the energy surfaces. 2 1.5

Energy (eV)

1 0.5 0

b)

a)

c)

d)

−0.5 −1 −1.5 −2 −0.5

0

0.5

0

0.5 kyacc

0

0.5

0

0.5

Figure 3. Variation of the energy band structure of pz electrons in a Z-GSL sample with N1 = N2 = 20 as a function of the potential barrier height: Ub = 0 (a), Ub = 2U0 (b), Ub = 4U0 (c) and Ub = 6U0 (d).

Masir et al [37] and Dvorak et al [38] also theoretically showed an ability to open a mini electronic bandgap in graphene, which is, however, patterned by a regular array of magnets or lattice defects. To gain insights into the optical couplings between occupied and unoccupied electronic states of pz electrons in GSLs, it is worth knowing the behavior of the corresponding wave functions. In ﬁgure 4, we display the quantity Pn (kxj ) = |Ψ2N +n,k (xj )|2 where n = 1, 2, 3, 4 and xj are given by equation (1) as the contour plots in the ky −x plane (kx is set to zero), and Ψ2N +n,k (xj ) are the eigen-vectors of the Hamiltonian Hˆ (k) for a particular A-GSL sample. Our calculations show that for the case of Ub = 0, the probability densities Pn (ky , xj ) exhibit regular oscillations along the ox direction, but seemingly depend on ky (see the four upper panels in which P1 (ky , xj ) is 5

J. Phys.: Condens. Matter 26 (2014) 405304

P (k ,x)

y

3

y

17

P4(ky,x)

18 K

b

0

U =U

K

b

ky

Ub = 3U0

ky

x 10

17

ρ (E+U /2)

3

14

ρG(E−Ub/2)

2

12

ρ

1

16

ky

Density of states

2

G

b

(E)

x 10

DOS

P (k ,x)

U =0

P1(ky,x)

H Anh Le et al

GSL

10

0 −1

0

1

Energy (eV)

8 6 4

kK y

2 0 −10

ox Figure 4. Contour plots of the probability of ﬁnding pz electrons in a period of the potential. The four upper, middle and lower panels are plotted for Ub = 0, Ub and 3Ub , respectively. kx = 0 is chosen to display the probability densities. The vanishing of P1 (ky , x), P2 (ky , x), P3 (ky , x) and P4 (ky , x) is encoded in dark-blue.

−5

0

Energy (eV)

5

10

Figure 5. Density of states of pz electrons in GSLs. The inset is the zoomed-out of DOS, showing that ρGSL (E) > ρG (E) with multiple peaks of ρGSL (E) as the reﬂection of topological features of the energy surfaces in the energy range of potential variation.

This connection does not cause mixing of electronic states in different Dirac valleys. However, this is not the case for Z-GSLs which are seen as the combination of A-GNRs. Because of the properties of electronic states of pz electrons in A-GNRs [39, 40], the parity-based selection rule does not play any role to the connection of states in the potential and well regions at the armchair-edge of two corresponding ribbons. The unique condition here is the energy matching of the two states. The state combination in this case requires mixing of states in two Dirac valleys. The Dirac model for the spinors of two components is therefore not relevant for the description of the electronic states of pz electrons in the Z-GSL conﬁguration, though it is for the A-GSL case. To obtain an overview of the electronic structure of pz electrons in GSLs, we calculated the electronic density of states (DOS) ρGSL (E). The obtained results for the A-GSL and Z-GSL conﬁgurations were compared to each other. It was found that with the same potential proﬁle, the density of states of pz electrons in an A-GSL sample is almost identical to that in a Z-GSL sample. In ﬁgure 5, we display the DOS data for a representative A-GSL sample with N1 = N2 = 15 and Ub = 3U0 . The density of states of pz electrons in GSLs (the red curve) can be identiﬁed as the average value of two DOS pictures of graphene (the blue and green-moss ones) which are relatively shifted by an amount of ΔE = Ub , i.e. ρGSL (E) = [ρG (E − Ub /2) + ρG (E + Ub /2)]/2. Accordingly, it suggests that the behavior of pz electrons in the potential and well regions of GSLs can be roughly described using the electronic picture of graphene. In the energy range of (−Ub , Ub ), the GSL density of states, however, shows two typical features: (i) the appearance of a structure with many sub-peaks, and (ii) the non-vanishing of DOS at the zero energy (see the inset in ﬁgure 5). These behaviors are consistent with those founded by Ho et al for graphene sheets under the cosine scalar potentials though our calculations were not performed for GSLs with larger values of the potential period length [26]. The structure of DOS apparently reﬂects the electronic properties of pz electrons, particularly the topological features of the

simply a plane). However, when increasing Ub we realize not only the modulation of Pn (ky , xj ) in the real space, but also the decay versus ky . In the four panels for Ub = U0 and the others for Ub = 3U0 , we observe the vanishing of Pn (ky , xj ) the dark-blue color). For in the ky − x plane (encoded in √ K ky in the vicinity of ky = 2π/3 3acc , the wave functions Ψ2N +n,k (xj ) are not partially vanished, but their amplitudes vary along the ox direction. Particularly, the eigen-energy values E2N+n (k) of such states are determined in the range of (−Ub /2, Ub /2) (see ﬁgures 2 and 3). For ky far from kyK , the corresponding E2N+n (k), however, comes to be outside the range of (−Ub /2, Ub /2) and the wave functions Ψ2N +n,k (xj ) vanish in the potential barrier region. In general, we observed that for k far from the K points some eigen-states of Hˆ (k) can exhibit the localization either in the potential barrier or in the well regions. The existence of the localized states in A-GSLs can be easily understood in the framework of the continuum Dirac model. In the case of piecewise-constant potentials, the spatial dependence of the two-component spinors generally takes the exponential form, ψ(r) ∝ exp(ikx x) exp(iky y) with

kx = (E ± Ub /2)2 /(¯hvF )2 − ky2 . Accordingly, the spinor magnitude will exponentially decay when kx is a pure imagi¯ vF ky [29]. nary number, i.e. |E ∓ Ub /2| < h To explain the difference of the electronic structure of pz electrons in A-GSLs and Z-GSLs, we consider the graphene sheet as a connection of ribbons with certain edge shapes, namely the zigzag or armchair ones [25]. This consideration is guided from breaking the intrinsic symmetries of graphene induced by the applied scalar periodic potential U (x). The states of pz electrons in GSLs are thus the result of the connection of states deﬁned in the constituent ribbons. For A-GSLs, which are seen as the combination of zigzag-edge nano-ribbons (Z-GNRs), the conditions for the connection of states in two adjacent ribbons are governed not only by the energy matching, but also by the selection rule which is based on the parity of electronic states in Z-GNRs [25, 41, 42].

6

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(a)

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σG 2

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6

5 3 2

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H Anh Le et al

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8

4

10

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(b)

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2 1

U

0

b

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EF = 0.4 eV EF = 0.6 eV

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1

1 0 0

2

4

6

8

0 0

10

Photon energy (eV)

2

4

6

8

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Photon energy (eV)

Figure 6. (a, b) Optical conductivity of graphene superlattices and

Figure 7. Suppression of the optical conductivity of ‘doped’ graphene in the photon energy range of (0, 2EF ) where EF is the Fermi energy. Schema illustrates the mechanism blocking the interband transitions, namely the Pauli blocking.

of intrinsic graphene (red curve). Blue and moss-green curves are plotted for the cases of the longitudinal and transverse polarization of photons, respectively.

energy surfaces such as local extrema and saddle points. Barbier, Wang, Burset and their co-workers also pointed out that typical DOS structures are reﬂected signiﬁcantly in the transport properties of pz electrons through GSLs’ potential barriers [23, 26–29]. In the next section, we present the obtained results for the optical conductivity of GSLs. We show that, though the periodic potential results in the increase of DOS in the energy range of (−Ub , Ub ), i.e. making a transition of graphene from the semiconducting to semimetallic behaviors [26], it causes a reduction of the optical coupling between the modulated electronic states. As a consequence, the GSLs’ optical conductivity becomes suppressed in the photon energy range of (0, Ub ).

van Hove singularity states. Beyond this energy range, the optical conductivity is exponentially suppressed. Our calculation also conﬁrms that the optical response of graphene is isotropic, i.e. does not depend on the photon polarization. From the results presented in ﬁgure 6, we notice three of the following features: (i) the near-coincidence of the GSLs’ optical conductivities to that of intrinsic graphene in the high photon energy range, Ω > Ub , (ii) the suppression of the former ones, compared to the latter, in the photon energy range of (0, Ub ), and (iii) the dependence of GSLs’ optical conductivity on the incident photon polarization, i.e. σxx (Ω ) = σyy (Ω ) slightly. Feature (i) means that the potential does not affect the optical properties of graphene in a high photon energy range. We can actually realize the appearance of the GSLs’ optical conductivities with sub-peaks, which are especially signiﬁcant when the GSL period length is not too long. Obviously, this behavior reﬂects the essence of the GSLs’ electronic structure. When L is short, the electronic structure is composed of separate energy surfaces which exhibit distinctive local extrema and saddle points as shown in ﬁgure 3. The dependence of the GSL optical conductivity on the polarization of incident photons is explained as a consequence of the anisotropy of electronic energy surfaces. The appearance of the GSLs’ optical conductivities is similar to that observed for doped graphene sheets (see ﬁgure 7) [43–45]. However, one should remember that the suppression of the optical conductivity of doped graphene sheets is essentially governed by the Pauli blocking (see ﬁgure 7 as an illustration and/or read [44] for a detail explanation). For GSLs, the detailed behavior of the optical conductivity in the photon energy range of (0, Ub ) was found to depend on the geometrical factors. By changing the length of the potential period, we realized that the longer the length L, the more signiﬁcant the conductivity suppression. In the next section, we will show that the reduction of the optical coupling between modulated electronic states is the microscopic mechanism governing the suppression of the GSLs’ optical conductivity.

3.2. Optical conductivity

We calculated the optical conductivity for a number of GSL samples using equation (16) with all ingredients described in section 2. In ﬁgures 6(a) and (b), we display the obtained results for two representative GSLs, the A-GSL sample with N1 = N2 = 15 and Ub = 3U0 and the Z-GSL sample with N1 = N2 = 30 with Ub = 3U0 . The optical conductivities σxx (Ω ) and σyy (Ω ) as functions of photon energy Ω for both polarizations of incident photons along the ox and oy directions are shown as blue and green-moss curves, respectively. To have an insight into the change of the optical response of pz electrons in graphene under the scalar periodic potentials, we ﬁrstly calculated the intrinsic optical conductivity of graphene. The calculation was realized by simply setting Ub = 0 and with the unit cell chosen as that of GSLs. Our calculations correctly reproduce the well-known results, shown as the red solid curves in ﬁgures 6(a) and (b), which were theoretically predicted and experimentally conﬁrmed [16–19]. Speciﬁcally, the optical conductivity of graphene is independent of photon energy in the range of about (0, 1) eV and gets the value of σ0 = e2 /4¯h (guided by the black dashed line σ (Ω )/σ0 = 1). In the higher energy range, the conductivity gets a sharp peak exactly at the photon energy of 2|tcc | due to the dominance of transition of pz electrons between two 7

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Mxx

mn

2

x 10

1

1.5

c)

1

d)

m=n=3

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m=n=1

1

0

b)

a)

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1

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m=n=4

1

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Myy

mn

15

x 10

e)

f) m=n=2

m=n=1

10

g)

h)

m=n=3 m=n=4

5 0

1

1.5

1

1.5

1

1.5

1

1.5

kyacc Figure 8. Comparison of the elements of optical matrices of pristine graphene (the red curves) and of GSLs.

Despite that, it will be pointed out that this phenomenon can be qualitatively elucidated using the Pauli blocking picture.

functions of k are rapidly degraded when k goes far from the K point. As shown in ﬁgure 4, when k is far from the K point, the localization of eigen-wave functions becomes much more signiﬁcant. It means that the modulation of the electronic states reduces the optical coupling between such states. In other words, the suppression of the optical conductivity in the photon energy range of (0, Ub ) is a consequence of the reduction of optical coupling between the relevant electronic states in the energy range of (−Ub /2, Ub /2). Because of the large number of carbon atoms in the GSL unit cell, and difﬁculties in working out particular symmetrical properties of the wave functions, the analysis of all elements of the optical matrices is a formidable task. Instead of that, we divided the valence energy band into sub-energy ranges, namely B1v = (−EW , −Ub ), B2v = (−Ub , −Ub /2) and B3v = (−Ub /2, 0), where EW is the GSL energy bandwidth. We then calculated contributions σBxxv →B c (Ω ), σBxxv →B c (Ω ), σBxxv →B c (Ω ) 1 2 3 to the total optical conductivity due to the electron transitions from the sub-energy ranges up to the conduction band B c = (0, EW ). Our calculation results are presented in ﬁgure 9 for a representative A-GSL sample. We clearly see that the partial conductivities give different contributions to the total value (plotted as the red solid curve) in the different photon energy ranges. Particularly, for the curve σB3v →B c (Ω ) (the black solid curve), the presented result in ﬁgure 9 shows that pz electrons can only jump from the energy region of (−Ub /2, 0) up to the range of (0, Ub /2) and of (Ub , 3Ub /2). In ﬁgure 10, we illustrate dominant transitions of pz electrons from different energy ranges in the valence band up to the conduction band. As pointed out in section 3.2, in the energy range of (−Ub /2, Ub /2) the electronic states are extended ones which are typical for GSLs, i.e. different from those in pristine graphene because they are strongly affected by the variation of scalar periodic potential U (x). The transition of electrons from the energy range of (−Ub /2, 0)-(0, Ub /2) (indicated by a short black arrow in ﬁgure 10(a)) results in

3.3. Optical matrix elements

As shown in ﬁgure 5, the scalar periodic potential U (x) causes an increase of the density of states of pz electrons in the energy range around the Dirac points. However, it also causes a suppression of the optical conductivity in the photon energy range of (0, Ub ) (ﬁgure 6). These contradictory behaviors can be only explained on the basis of a microscopic analysis of the optical coupling between occupied and unoccupied states of pz electrons in the energy range of (−Ub /2, Ub /2). First of all, it is reasonable to calculate elements of the xx/yy optical matrix Mmn (k) = | Ψ 2N −m,k |vˆx/y |Ψ2N +n+1,k |2 where m, n = 0, 1, ...2N denote the eigen-modes and vˆx/y (k) are the velocity matrices calculated from the Hamiltonian matrix Hˆ (k) according to equations (10) and (19). We have performed calculations for the case of intrinsic graphene by setting Ub = 0. In both cases of using the graphene unit cells of the A-GSL and Z-GSL conﬁgurations illustrated in ﬁgures 1(a) and (b), we obtained only the optical coupling between modes |Ψ2N−m,k and |Ψ2N +m+1,k . This conclusion is not a surprise because on the model of graphene with only pz electrons as charge carriers, there is only the coupling between modes π and π ∗ , which in the GSL representation behaves as the folding of the energy surfaces to form the subbands. Checking this point is the basis allowing us to proceed with calculations for the cases of GSLs. For GSLs we realized that, in general, all of the elements xx/yy of the optical matrices Mmn (k) are non-null. However, their values are much smaller than those of graphene as shown in ﬁgure 8. The optical coupling between two modes |Ψ2N,k and |Ψ2N+1,k is the strongest but its maximum at the K points is just equal to the corresponding value in the case of xx/yy graphene. Particularly, the matrix elements Mmn (k) as the 8

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7 v

Conductivity (σ0)

6 5

B1 → B

c

v B2 v B 3 v

→B

c

→B

c

ok

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EF

4

Potential profile

no ok

3

Ub

2

no

2Ub 3Ub

1

Barrier region 0 0

2

4

6

8

10

Figure 11. Diagram illustrates an effective model explaining the behavior of the GSLs’s optical conductivity.

Photon energy (eV) Figure 9. Analysis of the contributions of optical transitions from different energy ranges in the valence band up to the conduction band to the total optical conductivity.

(3Ub /2, 2Ub ) via the states localized in the potential barrier region, see ﬁgure 10(b). The vanishing of σB2v →B c (Ω ) and σB1v →B c (Ω ) in the photon energy range of (0, Ub ) is due to the forbiddance of the transition from the energy ranges of B2v and B1v up to the range of (−Ub /2, 0) due to the Pauli blocking mechanism. The non-zero value of σBxxv →B c (Ω ) in the photon 1 energy range of (Ub , 3Ub ) is attributed to the contribution of the electron transitions between states localized in the potential well regions, i.e. from the range of (−2Ub , −Ub ) up to the range of (0, Ub ), see ﬁgure 10(c). The contribution of the electron transitions between states localized in the potential barrier and potential well from the energy range of (−EW , −Ub )-(2Ub , EW ) totally govern the value of the optical conductivity of GSLs in the photon energy range of (3U0 , 2EW ) as in the case of graphene. In summary, the GSLs’ optical conductivity is the result of the transitions of pz electrons from two kinds of states in GSLs: the extended states and the localized states in the potential barrier and well regions. Though the potential microscopically can cause changes in the optical coupling between electronic states, an effective model sketch in ﬁgure 11 can be used to directly explain the behavior of the optical conductivity. Accordingly, the potential causes the graphene sheets doped locally due to the shift of the electrostatic potential proﬁle relative to the Fermi energy level. The optical transitions are thus forbidden when the photon energy is in the range of (0, 2| ± Ub /2| = Ub ) due to the Pauli blocking mechanism, and allowed when Ω > Ub . It thus clearly explains the suppression of the optical conductivity in the photon energy range of (0, Ub ). The non-vanishing of σxx/yy (Ω ) in this energy range, however, is due to the optical coupling between typical extended states in GSLs.

2 2U 1.5

b

1.5Ub

Energy (eV)

1

1U

0.5 0

b

0.5U

b

c)

b)

a)

EF

−0.5

−0.5Ub

−1

−1Ub −1.5Ub

−1.5 −2

−2Ub 1

1.5

1

k a

1.5

1

Well region

1.5

y cc

Figure 10. Illustration for the dominant transition of pz electrons from the valence band to the conduction one: (a) from B3v to B c , (b) from B2v to B c and (c) from B1v to B c .

a typical contribution to the optical conductivity of GSLs because it reﬂects the symmetry of the GSL energy band structure. The very high tail at Ω = 0 is determined due to the contribution of the transition from mode 2N to mode 2N + 1. Meanwhile, the transitions from the range of (−Ub /2, 0) up to the range of (Ub , 3Ub /2) (indicated by two long black arrows in ﬁgure 10(a)) are explained as due to the transition between electronic states localized in the potential region. As pointed out in section 3.1, when the width of the potential barrier/well is sufﬁciently large, such states reﬂect much of the essence of states deﬁned in the constituent ribbons. As shown in ﬁgure 10, the two energy ranges of (−Ub /2, 0) and (Ub , 3Ub /2) can be seen as belonging to the Dirac cone deﬁned by the generation lines E+± (k) = ±¯hvF k + Ub /2 and are symmetric through the energy plane of E = Ub /2. Similarly, the behavior of the curve σBxxv →B c (Ω ) also shows 2 that the pz electrons in the energy range of (−Ub , −Ub /2) can only be excited into the range of (Ub /2, Ub ), i.e. via the typical extended states in GSLs, and into the range of

4. Conclusion

A theoretical study of the electronic and optical properties of graphene sheets under the action of periodic electrostatic potentials is presented. The investigation of the electronic structure of GSLs is based on the tight-binding description 9

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for pz electrons. The optical properties are analysed on the basis of the optical conductivity, which is calculated using the Kubo formalism. The detail of the electronic structure of GSLs depends on the way the symmetries of the graphene hexagonal lattice are broken by the potential. The calculations show that the states of pz electrons in the energy range of (−Ub /2, Ub /2) are strongly inﬂuenced by the action of the potential and thus behave as extended states. The energy surfaces in this energy range show the extrema and saddle points which exhibit signiﬁcantly as clear peaks in the density of states. Compared to the case of pristine graphene, the optical coupling between the occupied and unoccupied states of pz electrons in GSLs are strongly reduced. It results in the suppression of the optical conductivity in the photon energy range of (0, Ub ). The electronic states whose energies are outside the range of (−Ub /2, Ub /2) are weakly affected by the spatial variation of the potential. Such states reﬂect much of the essential features of those of pz electrons in pristine graphene. The optical coupling between those states always plays the dominant role. It thus explains why the optical conductivity of GSLs is almost identical to that of graphene. The microscopic mechanism is clariﬁed through the analysis of the elements of optical matrix, and a simple effective explanation is introduced for the behavior of the GSLs’ optical conductivity. The obtained results on the changes in the optical properties of graphene against the characteristics of periodic scalar potentials may, therefore, suggest the possibility of engineering graphene for applications in photonics and optoelectronics.

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Acknowledgments

This work was supported by the National Foundation for Science and Technology Development (NAFOSTED): 103.1-2013.04 and partially by the National ApplicationOriented Basic Research Program: DTDL.05-2011-NCCB. References [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] Geim A K and Novoselov K S 2007 Nat. Mater. 6 183 [3] 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 [4] Zhang Y, Tan Y-W, Stormer H L and Kim P 2005 Nature 438 201 [5] Tworzydlo J, Trauzettel B, Titov M, Rycerz A and Beenakker C W J 2006 Phys. Rev. Lett. 96 246802 [6] Miao F, Wijeratne S, Zhang Y, Coskun U C, Bao W and Lau C N 2007 Science 317 1530 [7] Suzuura H and Ando T 2002 Phys. Rev. Lett. 89 266603 [8] Morozov S V, Novoselov K S, Katsnelson M I, Schedin F, Ponomarenko L A, Jiang D and Geim A K 2006 Phys. Rev. Lett. 96 016801 [9] Wallace P R 1947 Phys. Rev. 71 622

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