Electronic properties of impurity-infected few-layer graphene nanoribbons

Physica B 458 (2015) 107–113

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Electronic properties of impurity-infected few-layer graphene nanoribbons Hamze Mousavi a,b,n, Mehran Bagheri c a

Department of Physics, Razi University, Kermanshah, Iran Nano Science and Nano Technology Research Center, Razi University, Kermanshah, Iran c Laser and Plasma Research Institute, Shahid Beheshti University, G.C., Evin, Tehran 19835-63113, Iran b

art ic l e i nf o

a b s t r a c t

Article history: Received 11 March 2014 Received in revised form 1 November 2014 Accepted 13 November 2014 Available online 17 November 2014

Spurred by achievements in devising different multilayered graphene-based nano-systems, based on the random tight-binding Hamiltonian model and within the coherent potential approximation, the influence of varying the number of layers and the effect of doping by the boron and nitrogen impurities on the density of states of a mono- and few-layer armchair- and zigzag-edge graphene nanoribbons are theoretically investigated. When the nanoribbons are pristine, with increasing the number of layers the band gap of the armchair nanoribbons is decreased, yet the zigzag ribbons remain metallic and depending on the number of the layers few peaks are appeared around the zero-energy level. Moreover, in the presence of impurities, the band gap of the armchair nanoribbons is decreased for each number of layers. The VanHove singularities are steadily broadened and the density of states move to a higher (lower) value of the energy as a result of doping with boron (nitrogen) atoms. This study could provide with us to explore and devise new optoelectronic devices based on the impurity-infected graphene nanoribbons with tunable widths and edges. & 2014 Elsevier B.V. All rights reserved.

Keywords: Graphene Nanoribbons Tight-binding Green's functions Density of states

Graphene is an atom thick allotrope of carbon (C ) atoms in twodimensional (2D) hexagonal honeycomb lattice, with unusual electronic properties, owing mainly to its structure [1]. Nanoribbons with a finite width (in the y-direction) of graphene (Fig. 1), referred to as graphene nanoribbons (GNRs), have been extensively studied [2–7]. Recent experiments using the mechanical [1,8] and the epitaxial growth methods [9,10] have clarified that it is now possible to make GNRs with various widths. The C atoms on the edge of GNRs have two typical topological shapes, namely zigzag (Fig. 1a) and armchair (Fig. 1b). GNRs with zigzag (armchair) edges on both sides are classified by the number of zigzag (dimer) lines across the ribbon width. In the following, M-GNR refers to such a prototypical GNR, M being the number of zigzag (dimer) lines (Fig. 1). It is predicted that all zigzag GNRs (zGNRs) are metallic while armchair GNRs (aGNRs) are either metallic or semiconductor, depending on their widths [11,12]. In addition, the perfect aGNR gap's, i.e., Δ's, are separated into three categories as Δ3m ≥ Δ3m + 1 ≥ Δ3m + 2 = 0 (m is a positive integer number), while the dangling bonds on the edge sites of GNRs will be assumed to be terminated by hydrogen atoms, so the gap of the deformed n Corresponding author at: Department of Physics, Razi University, Kermanshah, Iran. Fax: þ 98 83 3427 4556. E-mail address: [email protected] (H. Mousavi).

http://dx.doi.org/10.1016/j.physb.2014.11.021 0921-4526/& 2014 Elsevier B.V. All rights reserved.

edge of aGNRs is separated into three groups given by Δ3m ≥ Δ3m + 1 ≥ Δ3m + 2 ≠ 0 [13]. It is also interesting to study the multilayer GNRs in their own right, as they could help us in understanding the evolution of the electronic structure from graphene to bulk graphite [14–16]. The electronic structure of the multilayer GNRs has been investigated by Sahu et al. [14] using the first-principle electronic-structure method. They have addressed the interplay of magnetism, a perpendicular external electric field, and the energy gap. They found that there are three classes of the energy gap in the multilayer aGNRs, and a strong dependence of the magnetic properties on the edge alignment and the number of layers in the multilayer zGNRs in a vertical external electric field. Using the tight-binding (TB) and Pariser–Parr–Pople model Hamiltonians and at the mean-field Hartree–Fock level, the electronic structure and optical response of multilayer aGNRs, both with and without a gate bias, have been studied by Gundra and Shukla [15]. They result in the optical response depends on the type of the edge alignment and the number of layers. Since their electronic properties are strongly connected to the delocalized electrons, the GNR systems could be deliberately tuned by proper choosing from different kinds of dopants like the functional groups, adsorbed species, and defect sites [17–21]. In fact, the knowledge of the consequences of doping on the electronic performance of the C-based low-dimensional devices is

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(a)

y

B1

A1

B1

A1

B1

A1

B1

A1

B1

A1

B1

A2

B2

A2

B2

A2

B2

A2

B2

A2

B2

A2 W=4

x

A3

B3

A3

B3

A3

B3

B3

A3

B3

A3

B3

B4

A4

B4

A4

b0 A4

B4

A4

A4

B4

A4

B4 b

A2

A1

B1

A1

(b)

A2

B2

B1 A2

B2

B2

A3 y

B3

b0 A4

A3

B3 A4

B4

A3

B3 A4

B4

W=7

B4

x A5

B5 A6

A5

B5 A6

B6 A7

A6

B6 A7

B7

A5

B5 B6

B7

b Fig. 1. Geometry of GNRs are shown by (a) panel (4-zGNR) and (b) panel (7-aGNR). The rectangular unit cells illustrate the Bravais lattice unit cells. Each cell includes Na = 2 M sites which are denoted by Ar and Bs where r , s = 1, …, M . Primitive vector is pointed by b and b0 is the interatomic distance.

now a key point for future developments of their novel functionalities. The substitution of C by atoms with a different number of the valence electrons will thus introduce additional states in the density of states (DOS). Because Boron (B ) and nitrogen (N ) atoms differ only by one in their number of the valence electrons compared to the C atoms, they are the natural choices as dopants applicable in GNR devices [18,21]. Using nonequilibrium Green's function technique combined with density functional theory (DFT), Padilha et al. [18] have investigated the transport properties of pure and the B- and N-doped GNRs. Also, using the ab initio DFT, the substitutional of one N atom per 154 C atoms has been performed by Yu et al. [21]. In this study, via implementing the coherent potential approximation (CPA) and based on Green's function formalism of the random TB (RTB) model, we explore the influences of B- and N-dopants on DOS of mono- and few-layer zGNRs and aGNRs [22,23]. We start with the following RTB Hamiltonian model [22,23] on a quasi-1D lattice (Fig. 1)

^ ^ ^, / = /0 + =

(1)

^ is the TB Hamiltonian model of the pristine ribbon, where / 0 without B- and N-atoms, and the random impurity is denoted by ^ . The second quantization form of Eq. (1) is given the potential =

by

^ / =−

Nc

Np

∑∑ ∑

0αβ ^ α† ^ β tipjq aip a jq +

Nc

Np

α†

∑ ∑ ∑ εipα a^ip

α , β i , j = 1 p, q = 1

α

i=1 p=1

α a^ip ,

(2)

where α and β refer to the Ar or Bs (r , s = 1, 2, 3, … , M ) sub-sites inside the Bravais lattice unit cell (Fig. 1), i and j denote the 0αβ repreposition of the Bravais lattice unit cell in the system, tipjq sents the amplitude for a π electron to hop from sub-site α in layer p of the Bravais lattice site i to the sub-site β in the layers q of the α α† nearest-neighbor site j, a^ (a^ ) indicates the creation (annihilaip

ip

tion) operator of an electron on sub-sites α in the layer p of the Bravais lattice site i and εipα shows random on-site energy of subsites α in the layer p of the Bravais lattice site i due to the presence of the impurities. This takes value of zero with probability 1 − c for the host sites (related to C sites) and ηB (ηN ) with probability c for the impurity sites (related to B (N ) sites) where c is the concentration of the impurities. Also, Nc implies either the number of the Bravais lattice unit cells or the number of the modes in the first Brillouin zone (FBZ), and Np displays the number of layers in the nanoribbon. In our calculations, we set the chemical potential equal to zero which corresponds to contribution of one electron per pz orbital in the system. We also take a rectangular real space Bravais lattice unit cell with the primitive vector b = be x (e x is the

H. Mousavi, M. Bagheri / Physica B 458 (2015) 107–113

unit vector in the x-direction), where b = 3 b0 (b = 3b0 ) for zGNRs (aGNRs) with b0 ∼ 1.42 Å shows the interatomic distance (Fig. 1). We assume = = 1. Now, we explore the Hamiltonian in Eq. (2) utilizing Green's function approach. One would obtain that the Bravais lattice unit cell for the nanoribbon with an arbitrary width W ¼M and the number of layers Np contains Na = 2MNp atoms. So, the corresponding Green's function can be depicted by a Na × Na matrix given by

⎛ . AA (i, j; τ) . AB (i, j; τ) ⎞ ⎟, G(i, j; τ) = ⎜⎜ ⎟ ⎝ .BA (i, j; τ) .BB (i, j; τ) ⎠

(3)

in which τ = ıt denotes the imaginary time. In Eq. (1), the typical MNp × MNp sub-matrix . AA (i, j; τ) is presented by . AA (i , j ; τ) ⎛ A1A1 ⎜ G11 ⎜ AA ⎜ G 211 1 ⎜ ⎜ G A1A1 ⎜ 31 ⎜ ⋮ ⎜ A A = ⎜ G Np1 11 ⎜ ⎜ G A2 A1 ⎜ 11 ⎜ A2 A1 ⎜ G 21 ⎜ ⎜ ⋮ ⎜⎜ A M A1 G ⎝ Np 1

A1A1 G12

A1A1 1A1 ⋯ G1ANp G13

A1A2 G11

A1A1 G 22

A1A1 1A1 ⋯ G 2ANp G 23

A1A2 G 21

A1A1 G 32

A1A1 1A1 ⋯ G 3ANp G 33

A1A2 G 31

⋮

⋮

⋱

⋮

⋮

A1A1 G Np 2

A1A1 A1A1 ⋯ G NpNp G Np 3

A1A2 G Np 1

A2 A1 G12

A2 A1 2 A1 ⋯ G1ANp G13

A2 A2 G11

A2 A1 G 22

A2 A1 2 A1 ⋯ G 2ANp G 23

A2 A2 G 21

⋮

⋮

⋮

⋮

⋮

A M A1 A M A1 A M A1 A M A2 ⋯ G NpNp G Np G Np G Np 2 3 1

A1A2 1A M ⎞ ⋯ G1ANp G12 ⎟ ⎟ A1A2 1A M ⋯ G 2ANp G 22 ⎟ ⎟ A1A2 1A M ⎟ ⋯ G 3ANp G 32 ⎟ ⎟ ⋮ ⋮ ⋮ ⎟ A1A2 A1A M ⎟ ⋯ G Np G 2 NpNp , ⎟ A2 A2 2AM ⎟ ⋯ G1ANp G12 ⎟ ⎟ A2 A2 2AM ⋯ G 2ANp G 22 ⎟ ⎟ ⋮ ⋱ ⋮ ⎟ AM AM ⎟ A M A2 ⎟ ⋯ G NpNp G Np 2 ⎠

S

)

(4)

(5)

where E = , + I and δ i S (δij) exhibit a Na × Na unit matrix and the Kronecker delta function. The random Green's function matrix G(i, j; E) given in Eq. (5) could not be calculated exactly. One would expand it in terms of the pure Green's function matrix G0 (i, j; E) and the random potential VSS [25] as follows: ′

G (i, j; E) = G0 (i, j; E) +

∑ G0(i, S; E) VSS ′G (S′, j; E), SS ′

(6)

where G0 (i, j; E) takes the form

1 Ω

FBZ k

(7)

total volume of the system, in of the unit cell and rij denotes the lattice unit cell to its nearest Fourier transformation of t 0ij . For

example, G0 (i, j; E) for a 4-zGNR with Np ¼2 is a 16  16 matrix as follows:

1 Ω

0

0

0

ϵ⁎k

0

0

0

0

0

0 ϵk 0

0

0

0

0 ϵk 0

0

0

0 ϵ⁎k 0

0

t0

0 t0

0

0

0

0 ϵ⁎k 0

0

0 t0

0

0

0 t0

0

0

0

0 ϵk

0 t0

0

0⎞ ⎟ 0⎟ ⎟ 0⎟ 0⎟ ⎟, 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ ϵk ⎠

(10)

D 0 (, ) = −

1 ℑ∑ πNa Nc Ωc α

N p FBZ 0αα (k; E). ∑ ∑ Gpp

(11)

p=1 k

In Eq. (6), the random potential V SS is defined by ′

⎛ η B (η N ) 0 ⎜ B (η N ) ⎜ 0 η 12V SS = ⎜ ′ ⋮ ⎜ ⋮ ⎜ 0 0 ⎝

0 ⋯ 0 ⋯ ⋱ ⋮ 0 0

0 ⎞⎟ 0 ⎟δ . ⎟ SS ⋮ ⎟ ′ ⎟ η B (η N ) ⎠

(12)

Therefore, the Dyson equation for the averaged Green's function corresponding to Eq. (6) is given by

G¯ (i, j; E) = G0 (i, j; E) +

∑ G0(i, S; E) Σ (S, S′; E) G¯ (S′, j; E), SS ′

(13)

where the self-energy Σ (S, S′; E) is defined by

∑ Σ (S, S′; E) G¯ (S′, j; E) = ∑ S′

S′

VSS G (S′, j; E) . ′

(14)

Here, 〈⋯〉 denotes the configurational average. One would write the Fourier transformation of Eq. (13) as follows:

∑ eık·rij (EI − ϵk )−1,

where Ω = Nc Ωc exhibits the which Ωc presents the volume vectors connecting a Bravais neighbors. Also, ϵk shows the

G0 (i, j; E) =

⎛ ϵ⁎ ⎜ k ⎜0 ⎜ ⎜ t0 ⎜0 † .0AB (k; E) = ⎡⎣.0BA (k; E) ⎤⎦ = ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎝0

(9)

hopping to a nearest neighbor site and t⊥0 shows the inter hopping overlap integral. It is straight to calculate DOS of a pristine nanoribbon using the following expression:

ı0+ ,

G0 (i, j; E) =

⎛ E t0 0 0 0 0 0 0 ⎞ ⊥ ⎜ ⎟ ⎜t 0 E 0 0 0 0 0 0 ⎟ ⊥ ⎜ ⎟ ⎜ 0 0 E t⊥0 0 0 0 0 ⎟ ⎜ ⎟ ⎜ 0 0 t⊥0 E 0 0 0 0 ⎟ 0 AA 0 BB . (k; E) = . (k; E) = ⎜ ⎟, ⎜ 0 0 0 0 E t⊥0 0 0 ⎟ ⎜ ⎟ 0 ⎜ 0 0 0 0 t⊥ E 0 0 ⎟ ⎜ 0⎟ ⎜ 0 0 0 0 0 0 E t⊥ ⎟ ⎜ ⎟ 0 ⎝ 0 0 0 0 0 0 t⊥ E ⎠

αβ Gpq

⎡ ⎤ ⎢ EI + εi δi S + ti0S ⎥ G (S, j; E) = Iδij, ⎢⎣ ⎥⎦

∑ (

whose 8  8 sub-matrices . 0αβ (k; E) are

in which ϵ k = t0 [1 + exp(ık·b)], with t∥0 ≡ t0 , refers to the intra

α β† where j ; τ) ≡ = − 〈;a^ip (τ) a^ jq (0) 〉, with ; served as the time ordering operator. The equation of motion of the electrons can thus be written as [24] αβ Gpq (i,

109

⎛ .0AA (k; E) .0AB (k; E) ⎞−1 ı k·r ij ⎜ ⎟ , e ∑ ⎜ 0BA ⎟ ⎝ . (k; E) .0BB (k; E) ⎠ k FBZ

(8)

1 G¯ (i, j; E) = Ω

FBZ

−1

∑ eık·rij { [G0(k; E)]−1 − Σ (k; E) } k

.

(15)

In general, there is no analytical solution for such random systems, since the impurities are randomly doped. Hence, Green's function exploited in the equation of motion is random and the local behavior of the system could be different from its total behavior. So, the equation of motion should be solved approximately and we have to calculate the configurational average properties. The CPA formalism has paved the way for taking the average over all possible impurity configurations. In this method, the inter-site correlations are neglected and each lattice site is replaced by an effective site except one, which is called the impurity site denoted by i. The self-energy is thus local and it takes the same value for all

110

H. Mousavi, M. Bagheri / Physica B 458 (2015) 107–113

5 4.5

4zGNR

4

D0(ε)

Np =1 3.5 Np =2 Np=4 3

2.5 2 1.5 1 0.5 0

3

2

1

0

ε /t 0

1

2

3

4

4aGNR Np=1 Np=2 Np=4

3.5 3

0

D (ε )

2.5 2

1.5 1 0.5 0

3

2

1

0

ε /t 0

1

2

3

Fig. 2. DOS of the pristine 4-zGNR (panel (a)) and 4-aGNRs (panel (b)) for Np = 1, 2, 4 . (For interpretation of the references to color in this figure, the reader is referred to the web version of this paper.)

sites, Σ (i, j; E) = Σ (E) δ ij . So, Eqs. (14) and (15) at the impurity site take the following forms:

Σ (E) G¯ (i, i; E) = Vii G imp (i, i; E) ,

¯ αβ (k; E) are given by where 8  8 sub-matrices .

(16)

and

1 G¯ (i, i; E) = Ω

FBZ

−1

∑ { [G0(k; E)]−1 − Σ (E) }

.

k

(17)

¯ i, i; E) for a 4-aGNR with Np ¼ 2 is a 16  16 matrix For instance, G( as follows:

1 G¯ (i, i; E) = Ω

FBZ

−1 ⎛. ¯ AA (k; E) . ¯ AB (k; E) ⎞ ⎟ , ⎟ ¯ BA (k; E) . ¯ BB (k; E) ⎠ ⎝.

∑ ⎜⎜ k

(18)

¯ AA (k; E) . ⎛ ξ A1A1 t 0 0 0 0 0 0 0 ⎞⎟ ⊥ ⎜ 11 ⎜ t0 A1A1 ξ22 0 0 0 0 0 0 ⎟ ⎜ ⊥ ⎟ A2 A2 ⎜ 0 ξ11 0 t⊥0 0 0 0 0 ⎟ ⎟ ⎜ A2 A2 ⎜ 0 0 ξ22 0 0 0 0 ⎟ t⊥0 ⎜ ⎟, = A3 A3 ⎜ 0 0 0 0 ξ11 t⊥0 0 0 ⎟ ⎜ ⎟ A3 A3 ⎜ 0 ξ22 0 0 0 t⊥0 0 0 ⎟ ⎜ ⎟ A4 A4 ⎜ 0 0 0 0 0 0 ξ11 t⊥0 ⎟ ⎜ ⎟ ⎜ 0 A4 A4 ⎟ 0 0 0 0 0 t⊥0 ξ22 ⎝ ⎠

(19)

H. Mousavi, M. Bagheri / Physica B 458 (2015) 107–113

4.5

3.5

D(ε)

3 2.5 2

4.5

4zGNR Np=1 Bdoped c=0.00 c=0.05 c=0.10 c=0.20

4 3.5 3

D(ε)

4

2 1.5

1

1

0.5

0.5 3

2

1

0

ε /t 0

1

2

0

3

4aGNR Np=1 B doped c=0.00 c=0.05 c=0.10 c=0.20

3

2

1

0

ε /t 0

1

2

3

2.5 2

1

2

3

4aGNR Np=1 Ndoped c=0.00 c=0.05 c=0.10 c=0.20

3 2.5 2

D(ε)

3

1.5

1.5 1

1

0.5

0.5

0

4zGNR Np=1 Ndoped c=0.00 c=0.05 c=0.10 c=0.20

3.5

3.5

D(ε)

2.5

1.5

0

111

3

2

1

0

ε /t 0

1

2

0

3

3

2

1

0

ε /t 0

Fig. 3. Panels (a) and (b) are DOS of B (N)− doped mono-layer (Np ¼1) 4-zGNR for different impurity concentrations c¼ 0 (red-solid line), c¼ 0.05 (blue-dotted line), c ¼ 0.1 (green-dashed line), and c ¼0.2 (purple-shorted dash line). Panels (c) and (d) are the same for 4-aGNR. Increasing the number of impurities broadens the VHS. The impurity potential of B (N) shifts right (left) the positions of singularities in DOS. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

and

¯ BB (k; E) . ⎛ ξ B1B1 t 0 0 0 0 0 0 0 ⎞⎟ ⊥ ⎜ 11 ⎜ t 0 ξ B1B1 0 0 0 0 0 0 ⎟ 22 ⎜ ⊥ ⎟ B2 B2 0 ⎜ 0 0 0 0 0 0 ⎟ t⊥ ξ11 ⎜ ⎟ B2 B2 ⎜ 0 0 0 0 0 0 ⎟ t⊥0 ξ22 ⎟, =⎜ B3 B3 ⎜ 0 0 0 0 0 0 ⎟ t⊥0 ξ11 ⎜ ⎟ B3 B3 ⎜ 0 0 0 0 0 0 ⎟ t⊥0 ξ22 ⎜ ⎟ B4 B4 ⎜ 0 0 0 0 0 0 t⊥0 ⎟ ξ11 ⎜ ⎟ ⎜ B4 B4 ⎟ 0 0 0 0 0 t⊥0 ξ22 ⎝ 0 ⎠

and

⎛t0 ⎜ ⎜0 ⎜ ⎜t0 ⎜0 † ¯ AB (k; E) = ⎡⎣. ¯ BA (k; E) ⎤⎦ = ⎜ . ⎜0 ⎜ ⎜0 ⎜0 ⎜ ⎝0

0 t0

0

0

0

0

0

t0

0

0

0

0 t0 0

0

0 ϵ k 0 t0

0

t0

0 ϵk t0

0 t0

0 t0 0 t0

0

0

t0

0

0

0 t0 0 ϵ k

0

0

0

0 t0 0 t0

0 0

0⎞ ⎟ 0⎟ ⎟ 0⎟ 0⎟ ⎟, 0⎟ ⎟ t0 ⎟ 0⎟ ⎟ ϵk ⎠

(21)

αα αα where ξpp = E − Σ pp (E) and ϵ k = t0 exp(ık·b). Using Eqs. (6) and

(13), the impurity Green's function Gimp (i, i; E) relates to the ¯ i, i; E) via average Green's function G(

G imp (i, i; E) = G¯ (i, i; E) + G¯ (i, i; E) ⎡⎣Vii − Σ (E) ⎤⎦ G imp (i, i; E).

(20)

(22)

¯ i, i; E) is obtained by Finally, the new average Green's function G( taking average over all possible configurations of the impurities

G¯ (i, i; E) = 〈G imp (i, i; E) 〉.

(23)

Eqs. (6)–(23) can be solved self-consistently to provide the average ¯ i, i; E) within the CPA approach. Thus, DOS of a Green's function G(

112

H. Mousavi, M. Bagheri / Physica B 458 (2015) 107–113

2.5

2.5

c=0.00 c=0.05 c=0.10 c=0.20

4zGNR Np=2 2 Bdoped

4zGNR Np=2 2 Ndoped

c=0.00 c=0.05 c=0.10 c=0.20

1.5

D(ε)

D(ε)

1.5

1

1

0.5

0.5

0

3

2

1

0

ε /t 0

1

2

0

3

2

1

0

ε /t 0

1

2

3

2.5

2.5

4aGNR Np=2 2 Bdoped

c=0.00 c=0.05 c=0.10 c=0.20

c=0.00 c=0.05 c=0.10 c=0.20

4aGNR 2 Np=2 Ndoped 1.5

D(ε )

D(ε )

1.5

1

1

0.5

0.5

0

3

3

2

1

0

ε /t 0

1

2

3

0

3

2

1

0

ε /t 0

1

2

3

Fig. 4. The same as Fig. 3 for a double-layer graphene nanoribbon (Np ¼2). It is shown that with increasing the number of nanoribbon layers, for the sake of the inter-layer overlap hopping, additional VHS are manifested. An increase in the number of layers causes only an increment in the number of VHS around the Fermi energy without changing the metallic behavior of the zGNRs, while for the semiconducting armchair ones the band gap becomes narrower. (For interpretation of the references to color in this figure, the reader is referred to the web version of this paper.)

doped GNR is calculated by

D¯ (,) = −

1 ℑ∑ πNa Nc Ωc α

N p FBZ

∑ ∑ G¯ ppαα (k; E). p=1 k

(24)

It should be emphasized that a TB calculation provides with us a qualitative result, but it could be failed to quantitatively reproduce the experimental findings. To carefully explore the structural and thermodynamic stability of the B- and N-doped graphene nanoribbons of different edges, one would perform a systematic many-body calculation like many-body perturbation theory or DFT to include correlation effects. The stability of the system is a direct consequence of different types of many-body effects as electron–electron and electron–hole interactions [17– 21]. However, our calculation here is based on the single-body Green's function and no correlation is contained yet. In the numeric, the parameters are chosen as the intra (inter) hopping integral to the first nearest neighbors is t0 ≃ 2.8 eV (t⊥0 = 0.2t0 ) [26–28]. Also, one would select the hopping integral to the first nearest neighbors of the deformed edge sites of GNRs to be t0′ = 1.25t0 , which is appropriate to the hydrogen atoms. It should be noted that we taking into account the hopping parameter t0′ only for the monolayer nanoribbons where we assume that their dangling bonds are terminated by hydrogen atoms. The on-site energies of the impurities (B and N strengths) are ηB =+ 0.83t0 and ηN = − 0.89t0 , compared to C atoms [26–28]. The

impurity concentrations are assumed to be c¼ 0.0 (the pristine case), c¼ 0.05, 0.10, and 0.20. For the pristine (undoped) 4-zGNR and 4-aGNR, depicted in Fig. 2a and b, respectively, DOS of the ribbons are plotted for various number of layers Np ¼1 (red-solid line), Np ¼2 (blue-dotted line), and Np ¼ 4 (green-dashed line). As such, Fig. 2a exhibits that the pristine few-layer 4-zGNR remains metallic and some peaks appear near the Fermi energy , F = 0. The number of peaks is equal to the number of involved layers Np. However, the few-layer pristine 4-aGNR keeps still semiconducting, as pictured in Fig. 2b. As a result, the distance between two mirror Van-Hove singularities (VHS) is diminished with increasing the number of layers. This could be associated with the weak overlap of the nonhybridized pz orbitals in multilayer GNRs. In Fig. 3a,c and b,d, for B- and N-doped mono-layer (Np ¼1) 4-zGNR/4-aGNR, respectively, DOS of the nanoribbons is plotted for various impurity concentrations c ¼0 (red-solid line), c ¼0.05 (blue-dotted line), c¼0.1 (green-dashed line), and c¼ 0.2 (purpleshorted dash line). Depending sensitively on the nature of the impurity potential, DOS of the mono-layer nanoribbons moves either right for B atoms or left for N atoms. In Fig. 4a,c and b,d, for B- and N-doped double-layer (Np ¼2) 4-zGNR/4-aGNR, respectively, DOS of the nanoribbons is schemed for various impurity concentrations c ¼0 (red-solid line), c ¼0.05 (blue-dotted line), c¼0.1 (green-dashed line), and c¼ 0.2 (purpleshorted dash line). As shown, for the zGNRs, owing to the interlayer overlap integral, an increase in the number of layers yields

H. Mousavi, M. Bagheri / Physica B 458 (2015) 107–113

extra VHS around the Fermi level and keeps their metallic treatment. However, for the semiconducting zGNRs, increasing the number of layers causes a narrower band gap originated from widening the VHS towards either right for B-doped ribbons or left for N-doped ribbons. Compared to the pristine case, some important consequences could be resulted from doping process. We find that DOS of the nanoribbons is sensitive to the B and N concentrations. As such, with increasing doping concentration the VHS are steadily broadened and the corresponding DOS moves towards higher (lower) values of the energy for B (N ) atoms. The band gap of the aGNRs decreases from valence (conduction) band as well, when the nanoribbons are doped by B (N ) atoms. In other words, doping by the B and N gives rise to additional states (impurity levels) between the conduction and valence bands around the initial position of the Fermi energy. This is because the impurity potential created by the B (N ) atom is positive (negative), which yields a barrier (well) for electrons moving the surface of the nanoribbons [29]. For the B (N ) case the barrier (trap) causes a upward (downward) shift in the Fermi level, which yields a shift in DOS pattern towards higher (lower) energies. Then, the semiconducting gap is decreased with increasing further the B (N ) concentration of aGNRs. It is well known that the total number of available states, corresponding to the area under the profile of DOS, is proportional to the electron concentration of the system under study. In spite of a slight change in the overall behavior of DOS, the area under its curve remains constant. In fact, the VHS are broaden and some new levels in DOS are created. In conclusion, increasing the number of layers as well as doping by B and N atoms, changes the electronic properties of the zGNRs and aGNRs. The band gap of the aGNRs is dependent on the number of layers included in the ribbon. Also, altering the number of layers in zGNRs origins relocating the positions and rescaling the width as well as the hight of peaks near to zero energy. These are due to the weak overlap of the nonhybridized pz orbitals in multilayer GNRs. The potential created by the B (N ) atoms moves DOS towards higher (lower) energies by inducing additional states.

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Therefore, the number of layers and the rate of doping could be two significant factors to engineer the electronic properties of GNRs nanoribbons for application in devising novel optoelectronic devices.

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