The study of dynamical quasiparticle properties of undoped graphene nanoribbon

Accepted Manuscript The study of dynamical quasiparticle properties of undoped graphene nanoribbon Ameneh Abdi, Hamed Rezania PII:

S0038-1098(18)30134-0

DOI:

10.1016/j.ssc.2018.09.003

Reference:

SSC 13495

To appear in:

Solid State Communications

Received Date: 3 April 2018 Revised Date:

2 August 2018

Accepted Date: 6 September 2018

Please cite this article as: A. Abdi, H. Rezania, The study of dynamical quasiparticle properties of undoped graphene nanoribbon, Solid State Communications (2018), doi: https://doi.org/10.1016/ j.ssc.2018.09.003. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT The study of dynamical quasiparticle properties of undoped graphene nanoribbon

Hamed Rezania (Corresponding author) Faculty Member

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Affiliation: Razi University

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; Ameneh Abdi, Phd Student

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The study of dynamical quasiparticle

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properties of undoped graphene nanoribbon

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Ameneh Abdi and Hamed Rezania∗

Department of Physics, Razi University, Kermanshah, Iran

Abstract

We have calculated the quasiparticle particle self-energy and spectral function

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in a undoped graphene nanoribbon when a symmetry breaking of the sublattice is occurred. Hubbard model Hamiltonian has been applied to describe the electron dynamics in the structure. Using the many body G0 W approach and Green’s func-

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tion technique, we have found the dynamical quasiparticle properties of electrons

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on nanoribbon structure. Also random phase approximation has been exploited to calculate electronic self-energy of the system. The difference between on-site energies of atoms on two sublattices is named gap parameter. Specially, the effects of gap parameter and Hubbard parameter on spectral function, inelastic scattering lifetime and band gap renormalization have been studied. Also the wave vector dependence of inelastic scattering lifetime for different values of gap parameter has ∗

Corresponding author. Tel./fax: +98 831 427 4569. E-mail: [email protected]

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been investigated. In order to find the quasiparticle excitation spectrum of the interacting electronic system, we have studied the frequency dependence of spectral

Keywords: A. Armchair ;D. nanoribbon;D. Green’s function

Introduction

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Pacs numbers: 73.22-f

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function for different physical parameters.

Graphene is a single atomic layer of crystalline carbon on the honeycomb lattice consists of two interpenetrating triangular sublattices A and B, has opened up a new field for fundamental studies and applications [1, 2, 3, 4]. The single-particle energy spectrum in

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graphene contains two zero-energy at two points of the Brillouin zone which are called as valleys or Dirac points. The charge carriers in a pristine graphene show linear and isotropic energy dispersion relation and massless chiral behavior for the energy sclaes up

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to 1 eV. Graphene has revealed a variety of unusual transport phenomena characteristics

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of two-dimensional Dirac Fermions such as an anomalous integer quantum Hall effect at room temperature, am minimum quantum conductivity, Klein tunneling paradox, weak and anti-locallization, an absence of Wigner crystallization phase and Shubnikov-deHaas oscillations that exhibit a phase shift of π due to Berry’s phase[5, 6, 7, 8, 9]. Graphene, therefore, presents a new type of many body problem in which the noninteracting low energy quasiparticle dynamic is effectively described by a linear dispersion[10]. The presence of edges in graphene has strong implications for the low-energy spectrum of the π-electrons 2

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[11, 12, 13]. Recent experiments by using the mechanical method [5] and the epitaxial growth[14] method show it is possible to make graphene nanoribbons with various widths.

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Most electronic applications are based on the presence of a gap between the valence and conduction bands in the conventional semiconductors. The band gap is a measure of the threshold voltage and on-off ratio of the field effect transistors[15, 16]. Therefore for inte-

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grating graphene into semiconductor technology, it is crucial to induce a band gap in band structure of graphene like structures to control the transport of carriers. Several routes

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have being proposed and applied to induce and control a gap in graphene. One of them is using quantum confined geometries such as quantum dots and nanoribbons[17, 18]. Another way is spin-orbit coupling whose origin is due to both intrinsic spin-orbit interactions and Rashba interaction[19, 20]. Inversion symmetry breaking causes to induce a

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band gap in the electronic spectrum. This takes place when the number of electrons on A and B atoms are different[21] or Kekule[22] distortion, e.g. graphene on proper substrates [23] or adsorb of some molecules such as water, ammonia[24] on graphene sheets.

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It was shown that ribbons with zigzag edges possess localized edge states with energies close to the Fermi level[12, 25]. In contrast, edge states are absent for ribbons with

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armchair edges. Because zigzag edges can induce a strong magnetic response[25, 26, 27], considerable effort has been devoted to studying the effect of edges in graphitic nanomaterials.

The analytical wave function and energy dispersion of zigzag nanoribbon have been derived by several research groups[28, 29]. For armchair graphene nanoribbons, the analytical forms of wave functions within the low-energy range have been worked out based on

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the effective-mass approximation[30]. From the theoretical viewpoint, electronic transport through graphene nanoribbons exhibits a number of intriguing phenomena due to

current control and on/off switching in graphene devices[31, 32].

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their peculiar electronic properties. Graphene nanojunction structures can be used for

The strength of interaction effects in an ordinary two dimensional electron gas increases

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with decreasing the carrier density. At low densities, the effective velocity is suppressed, the charge compressibility changes sign from positive to negative, and the spin suscepti-

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bility is strongly enhanced. In other hand the abinitio estimates of the strength of the Hubbard U in graphene like structures suggests that on-site Coulomb repulsion is quite remarkable, 10 eV[33] Since this value is expected to be a local property of pz orbitals in carbon atoms, one does not expect the above value of U to be much different when

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the fermions of the underlying honeycomb lattice acquire a mass due to extrinsic effects, such as substrate, binding with ad-atoms such as hydrogen[34]. Sorella and Tossati [35] found that the Hubbard model in the half-filled honeycomb lattice would exhibit a Mott-

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Hubbard transition at finite U . Their Monte carlo results were confirmed by variational approaches and reproduced by other authors[36]. One important difference between con-

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ventional electron gas and graphene electrons is that the contribution of exchange and correlation to the chemical potential is an increasing rather than a decreasing function of carrier-density. This property implies that exchange and correlation increase the effectiveness of screening. This unusual property follows from the difference in sublattice pseudospin chirality between the Dirac model’s negative energy valence band states and its conduction band ones. Theoretical calculations of quasiparticle properties of electrons

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in conventional two-dimensional electron liquid are performed within the framework of Landau’s Fermi liquid theory[37], whose key ingredient is the quasiparticle concept and its

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interactions. As applied to the electron liquid model this entails the calculation of effective quasiparticle-quasiparticle interactions which enter the many-body formalism allowing the calculation of various physical properties. Our formalism is based on the Landau-Fermi

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liquid theory incorporating the G0 W- approximation for the self-energy. A number of calculations considered different variants of the G0 W -approximation for the self-energy

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in two dimensional electron gas [38, 39, 40, 41] from which density spin-polarization, and temperature dependence of quasiparticle properties are obtained. Coulomb interaction causes to create a mechanism for quasiparticle scattering lifetime. The carrier lifetime in an epitaxial graphene layers grown on SiC wafers has been recently measured[42]. The ex-

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perimental measurements are relevant for understanding carrier intraband and interband scattering mechanisms in graphene and their impact on electronic and optical devices [43]. In this paper we focus on the impacts of energy gap, quantum confinement and elec-

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tronic interaction upon some electronic properties of quasiparticles of armchair undoped graphene nanoribbons at finite temperature. In order to obtain these many-body prop-

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erties of quasiparticles, we use the random phase approximation and the G0 W approximation for making electronic self-energy. Our new results are based on the quasiparticle properties in the presence of a gap opening in the electronic spectrum. From self-energy we then obtain the quasiparticle energies, spectral function, inelastic scattering lifetime and band gap renormalization in undoped armchair graphene nanoribbon due to local electron-electron interaction. In the last section we discuss analyze our results to show

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how gap parameter and temperature affect the frequency dependence of spectral function which can be compared with ARPES spectra[44]. Also we analyze the behavior of

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inelastic scattering lifetime of quasiparticles for different gap parameters.

Theoretical formalism

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The dynamics of quasiparticles in a gapped armchair graphene nanoribbon in the presence

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of local electron-electron interaction are described by quasi two dimensional Hubbard model. In Fig.(1), we have shown the lattice structure of armchair graphene nanoribbons consisting two types of sublattices A and B. The unit cell contains n A-type atoms and B-type atoms. We consider the sublattice symmetry breaking mechanism in which the on-site energies for A and B sublattices are different. These different on-site energies are

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named ∆ and −∆ for A and B sublattices, respectively. The tight-binding part of the model Hamiltonian, HT.B , in terms of Fourier transformation of fermionic operators for

HT.B =

X kx ,p,σ

σ φ(kx , p)c†σ A,kx ,p cB,kx ,p + H.c. + ∆

X

σ c†σ A,kx ,p cA,kx ,p − ∆

kx ,p,σ

X

σ c†σ B,kx ,p cB,kx ,p

kx ,p,σ



X  †σ

σ cA,kx ,p cσA,kx ,p + c†σ B,kx ,p cB,kx ,p ,

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− µ

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π band electrons with gap parameter ∆ has given by

kx ,p,σ

φ(kx , p) = −t[2eikx a/2 cos(

pπ ) + e−ikx a ], p = 1, 2, ...w, w+1

(1)

where µ is chemical potential and w introduces the width of ribbon. Also c†α,kx ,p creates an electron at α = A, B sublattice with wave vector kx and quantum number p. Also the wave vector kx belongs to the first Brillouin zone of atomic chain with lattice constant π 3a ,i.e. − 3a < kx <

π . 3a

Also t is the nearest neighbor hopping integral for itinerant elec6

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trons on nanoribbon lattice. Using the following expansion for operator form (c†σ α,kx ,p )[45], Fermionic operators in band energy indexes kx , p are related to operators in real space as w 1 XX eikx xl φα (i)c†σ αi ,l , Nα l i=1

(2)

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c†σ α,kx ,p =

where xl introduces the position of l th unit cell along x direction as shown in Fig.(1). †σ Operator c†σ Ai ,l (cBi ,l ) creates an electron with spin σ in pz orbit of a carbon atom located

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at A(B) sublattices with position i along width of ribbon, i.e. y direction, (See Fig.(1)) Based on the translational invariance, the plane wave eikx xl basis along x direction has

√

φB (i) = sin(

3qy a i). 2

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been choosed. Employing hard-wall boundary condition[45], one can choose φA (i) = Then hard wall boundary conditions implies the discretized wave

vector qy gets the values

pπ √2 3a w+1

with p = 1, 2, ..., w.. a = 1.42A. is the bond length

between carbon atoms. Applying normalization condition, the normalized coefficients, N (w+1) [45] 2

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Nα , have been obtained as Nα =

q

so that N is the number of unit cells

along x direction. Based on Eq.(1), the final diagonalized tight binding part of model

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Hamiltonian gets the following form as HT B =

σ Eη (kx , p)c†σ η,kx ,p cη,kx ,p ,

X kx ,p,η=±,σ

q

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E± (kx , p) = ± ∆2 + |φ(kx , p)|2 − µ.

(3)

E± (kx , p) denotes the band energy of noninteracting electrons on armchair graphene like nanoribbon. The fermionic creation operators in diagonalized form of the Hamiltonian in Eq.(3), i.e. c†σ η,kx ,p can be expressed in terms of fermionic creation operators in Eq.(1), i.e. c†σ α,kx ,p as the following  |φ(kx , p)| †σ c†σ q η=±,kx ,p = r   cA,kx ,p 2 ∆2 + |φ(kx , p)|2 − η∆ ∆2 + |φ(kx , p)|2

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q

∆2 + |φ(kx , p)|2 − η∆

+ η

φ(kx , p)



c†σ B,kx ,p ,

(4)

Here, ± denotes the conduction and valence bands, respectively. Using Eq.(2), we can

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†σ readily expresse operators c†σ Ai ,kx and cBi ,kx , as i = 1, 2, ..., w, in terms of band operators

c†σ ±,kx ,p

w  X 1 pπ  †σ sin( i) fA,+ c†σ + f c A,− +,kx ,p −,kx ,p , (w + 1) p=1 n+1

s

w  X 1 pπ  †σ sin( i) fB,+ c†σ + f c B,− −,kx ,p , +,kx ,p (w + 1) p=1 n+1

= =

(5)

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c†σ Bi ,kx

s

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c†σ Ai ,kx

where coefficients fα,η are the complex functions and given by r

q   1 ∆ 1+ q ∆2 + |φ(kx , p)|2 − ∆ ∆2 + |φ(kx , p)|2 , fA,+ (kx , p) = |φ(kx , p)| ∆2 + |φ(kx , p)|2 r

q   1 ∆ ∆2 + |φ(kx , p)|2 + ∆ ∆2 + |φ(kx , p)|2 , fA,− (kx , p) = 1− q |φ(kx , p)| ∆2 + |φ(kx , p)|2

+ |φ(kx , p)|2 − ∆ ∆2 + |φ(kx , p)|2

v u u 2 ∆ −φ(kx , p) u t

+ |φ(kx , p)|2 + ∆ ∆2 + |φ(kx , p)|2

φ(kx , p) fB,+ (kx , p) = |φ(kx , p)|

|φ(kx , p)|

q

∆2 + |φ(kx , p)|2

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fB,− (kx , p) =

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v u u 2 u∆ t

,

q

∆2 + |φ(kx , p)|2

.

(6)

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Since each unit cell of armchair graphene nanoribbon includes 2w atoms with types A and B, the Green’s function can be written as the 2w × 2w matrix. At finite temperature[46], each matrix element of Matsubara’s Green’s function is defined by[46] Gσαi βj (kx , τ ) = −hT cσαi ,kx (τ )c†σ βj ,kx (0)i,

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where τ ≡ it is the real parameter and t implies the imaginary time according to Matsubara’ formalism. The Fourier transformation of each above matrix element of Green’s

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function is calculated by Gσαi βj (kx , iωn )

=

Z 1/(kB T ) 0

dτ eiωn τ Gσαi βj (kx , τ ) , α, β = A, B , i, j = 1, 2, ..., w.

(8)

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Here, ωn = 2(n + 1)πkB T denotes Matsubara fermionic frequency. n is integer number. Also kB is the Boltzmann constant and T introduces equilibrium temperature of the system. Based on the operator transformation in Eq.(5) the Fourier transformations of

w X f ∗ (kx , p)fβ,η (kx , p) pπ pπ 1 X i)sin( j) α,η . (9) sin( w + 1 p=1 η=+,− w+1 w+1 iωn − Eη (kx , p)

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Gσαi βj (kx , iωn ) =

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noninteracting Green’s function matrix elements are finally given by

The spin dependence of each component Green’s function in Eq.(9) originates from chemical potential µσ . The chemical potential, µ, in Eq.(3) is determined by the concentration of electrons with both spin direction

−∞

dED(E) , D(E) ≡

  1 X 1 q −2Im , (10) 2wN kx ,p,η E − η ∆2 + |φ(kx , p)|2 + i0+

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ne = 2

Z µ

that D(E) indicates the total density of states of noninteracting gapped graphene nanorib-

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bon. To determine µ, we use the definition of total occupation of electrons. Based on the values of electronic concentration ne in nonmagnetic system, the chemical potential

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for electrons, µ, can be obtained by means Eq.(10). We consider the intrasite Coulomb electron-electron interactions. Accordingly, the Hubbard model Hamiltonian including intrasite electron electron interaction is expressed as H = HT B + U

X

n↑αi ,l n↓αi ,l ,

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i,l,α=A,B

where l denotes the index of unit cell and i, α refers to atomic basis into each unit cell. nσαi ,l with σ =↑, ↓ introduces the electronic density operator with spin σ on atom located 9

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at unit cell l and sublattice α with position i along ribbon width. By exploiting the definition of transverse components of spin operators in terms of fermionic creation and

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annihilation operators, the local interacting part of Hamiltonian can be rewritten by the following HU ≡ U

X ↑

nαi ,l n↓αi ,l =

i,l,α

 U ne U X  + − − Sαi ,l Sαi ,l + Sα−i ,l Sα+i ,l , 2 2 i,l,α

(12)

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↓ †↓ ↑ − so that Sα+i ,l = c†↑ αi ,l cαi ,l and Sαi ,l = cαi ,l cαi ,l are the ladder components of spin operators on

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unit cell l at sublattice α with position i along width of ribbon. ne refers to the electron concentration. In terms of Fourier transformation of spin operators, local ineraction term is given by HU =

 Un U X  + − Sαi (qx )Sα−i (−qx ) + Sα−i (qx )Sα+i (−qx ) 2 2 i,qx ,α

(13)

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In order to obtain the expectation value of HU in Eq.(13) transverse static spin susceptibilities is needed to be calculated. Linear response theory gives us the noninteracting spin response functions based on the correlation function between transverse components (0)

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of spin operators. We introduce χ+− (qx , iΩn ) as transverse spin susceptibility matrix of

by

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noninteracting system with dimensions (2w ×2w) and each matrix element of that is given

(0)+−

χαi βj (qx , iΩn ) = −i

Z +∞ −∞

dτ eiΩn τ hT (Sα+i (qx , τ )Sβ−j (−qx , 0))i,

(14)

so that Ωn = 2nπkB T is the bosonic Matsubara’ frequency. The Fourier transformations of transverse components of spin density operators, (S +(−) ), are given by Sα+i (qx ) =

X †↑

ckx +qx ,αi c↓kx ,αi , Sα−i (qx ) =

kx

X †↓

ckx +qx ,αi c↑kx ,αi .

kx

10

(15)

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Substituting the operator form of S + and S − into definition of transverse spin susceptibility in Eq.(14), we arrive the following expression for each component of transverse (0)

(0)+− χαi βj (qx , iΩn )

= −

Z +∞

dτ eiΩn τ

−∞

1 X ↓ hT (c†↑ αi ,kx +qx (τ )cαi ,kx (τ )) 2 N kx ,k0 x

↓ (c†↑ βj ,kx0 −qx (0)cβj ,kx0 (0))i.

(16)

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×

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dynamical spin susceptibility (χ+− (qx , iΩn ))

After applying Wick’s theorem and taking Fourier transformation, we can write each

dependent Green’s function as (0)+−

χαi βj (qx , iΩn ) =

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element of transverse susceptibility matrix can be written in terms of one particle spin

kB T X ↑ G (kx , iωm )G↓βj αi (kx + qx , iΩn + iωm ). N kx ,m αi βj

(17)

However in this present study, the electronic system is nonmagnetic and consequently

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one particle electronic Green’s function elements are independent of spin direction. By replacing the noninteracting Green’s function matrix elements into Eq.(17) and summing

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over internal Matsubara frequency ωm , we obtain dynamical transverse spin susceptibility of electrons on the armchair graphene nanoribbon lattice. Here we present the final result (0)+−

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for matrix element χαi βj (qx , iΩn ) as follows χαi βj (qx , iΩn ) =

w X X X 1 pπ pπ p0 π p0 π sin( i)sin( j)sin( i)sin( j) (w + 1)2 kx p,p0 =1 η,η0 =± w+1 w+1 w+1 w+1

×

nF (Eη0 (kx + qx , p0 )) − nF (Eη (kx , p)) |fα,η (kx , p)|2 |fβ,η0 (kx + qx , p0 )|2 , (18) iΩn + Eη0 (kx + qx , p0 )) − Eη (kx , p)

(0)+−

where nF (x) =

1 ex/kB T +1

implies well known Fermi Dirac distribution function.

The random phase approximation[46] has been applied to express transverse spin susceptibility matrix of the interacting system in the presence of Hubbard interaction in terms 11

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of that of noninteracting one. The relation for transverse dynamical spin susceptibility of electronic system in the presence local interacting term is given by[47] (0)

(0)

1 − U χ+− (qx , iΩn )

.

(19)

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χ+− (qx , iΩn ) =

χ+− (qx , iΩn )

In this relation χ+− denotes transverse spin susceptibility matrix (2w × 2w) of interacting (0)

its elements have been presented in Eq.(18).

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system. Furthermore χ+− is spin susceptibility matrix of noninteracting system so that

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Using Dyson’s series for corrected interacting Green’s function, the first and second order perturbation theory gives the following the self-energy matrix elements of gapped graphene nanoribbon in terms of transverse spin susceptibility elements as kB T X (0)↓ G (kx − qx , iωn − iΩm )χ+− αi βj (qx , iΩm ) N qx ,m αi βj kB T X (0)↓ + U G (qx , iωn )δαβ δij , N qx ,n αi βj

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Σσαi βj (kx , iωn ) = U 2

(20)

where αi and βj are sublattice indices. In fact α, β = A, B and i, j = 1, 2, ...w. This

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result is similar to G0 W approximation for finding electronic self-energy of degenerate electron gas[49]. In order to perform summation over bosonic Matsubara frequencies Ωm

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in Eq.(20), we use the Lehmann representation as [46] χ+− αi βj (qx , iΩm )

=

Z +∞ −∞

+− 0 dω 0 −2Imχαi βj (qx , ω ) 2π iΩm − ω 0

(21)

Substituting Eq.(21) into Eq.(20), the summation over bosonic Matsubara frequency is calculated and we arrive the following relation for matrix elements of Σαi βj Σσαi βj (kx , iωn ) =

Z +∞ w X U 2 kB T 1 X X pπ pπ 0 sin( i)sin( j) dω 0 Imχ+− αi βj (qx , ω ) πN w + 1 qx p=1 η=± w+1 w+1 −∞

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∗ × fα,η (kx − qx , p)fβ,η (kx − qx , p)



1 + nB (ω 0 ) − nF Eη (kx − qx , p)

iωn − ω 0 − Eη (kx − qx , p) w X U X pπ pπ + sin( i)sin( j)f ∗ (kx , p)fβ,η (kx , p)nF (Eη (kx , p))δαβ δij , w + 1 p=1 η=+,− w+1 w + 1 α,η 1 ex/kB T −1

is Bose-Einstein distribution function.

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where nB (x) =

(22)

This result for self-energy has been expressed in atomic basis Hilbert space. It can be

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suitable to rewrite the self-energy matrix in band energy Hilbert space. Using Eq.(5), the unitary transformation matrix that relates two different mentioned Hilbert spaces to each

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other can be readily found. Afterwards diagonalized self-energy matrix elements in band indices Hilbert space,i.e. Ση,p , is expressed in terms of matrix elements Σαi βj as following Σσ+,p (kx , iωn ) =

h 1 X pπ pπ ∗ sin( i)sin( j) fA,+ (kx , p)fA,+ (kx , p)ΣσAi ,Aj (kx , iωn ) w + 1 i,j w+1 w+1

∗ ∗ + fA,+ (kx , p)fB,+ (kx , p)ΣσAi ,Bj (kx , iωn ) + fB,+ (kx , p)fA,+ (kx , p)ΣσBi ,Aj (kx , iωn )

Σσ−,p (kx , iωn ) =

i

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∗ + fB,+ (kx , p)fB,+ (kx , p)ΣσBi ,Bj (kx , iωn )

h pπ 1 X pπ ∗ i)sin( j) fA,− (kx , p)fA,− (kx , p)ΣσAi ,Aj (kx , iωn ) sin( w + 1 i,j w+1 w+1

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∗ ∗ (kx , p)fA,− (kx , p)ΣσBi ,Aj (kx , iωn ) − fA,− (kx , p)fB,− (kx , p)ΣσAi ,Bj (kx , iωn ) − fB,−

i

∗ + fB,− (kx , p)fB,− (kx , p)ΣσBi ,Bj (kx , iωn ) .

(23)

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Using electronic self-energy of the system, we can compute the inelastic scattering lifetime of quasiparticles due to carriers-carriers interactions at finite temperature for gapped armchair graphene nanoribbon. This is calculated through the imaginary part of the self-energy[48] when the frequency evaluated at the on-shell energy, 2 −1 τin (kx ) = − ImΣη,p (kx , Eη (kx , p) − µ), h ¯

(24)

−1 where τin (kx ) can be interpreted as quantum level broadening of the momentum with

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−1 eigenstate |η, p, kx i. It is worthwhile to note that the expression of τin is identical with

a result obtained by Fermi’s golden rule summing the scattering rate of electron and

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hole contributions at quantum numbers η, p, kx [49]. ARPES[44] plays as a central role to investigate quasiparticle properties such as group velocity and lifetime of carriers on the Fermi surface. ARPES measures the constant energy surface for all partially occupied

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states and the fully occupied band structure. The information of band dispersion can be elicited from date measured in ARPES experiments. The single particle excitation

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spectrum in the interacting electronic system can be derived by spectral function[46]. By using retarded self-energy, the spectral function for each eigenstate |η, p, kx i is obtained as

|=mΣη,p (kx , iωn −→ ω + i0+ )| .(25) [ω − Eη (kx , p) −
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Aη,p (kx , ω) =

The width of single particle peak in spectral function is proportional to |=mΣη,p |. For the noninteracting system we get A(0) η,p (kx , ω) = δ(ω − Eη (kx , p)). The Fermi liquid theory

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applies only when the interacting spectral function at the Fermi quantum numbers kF , pF Aη,pF (kx = kF , ω) behaves as a delta function, and has a broadened peak indicating

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damped quasiparticle at k 6= kF with width |=mΣη,p (kx , ω)| Finally we can calculate a band gap renormalization (BGR)[50, 51, 52]. The BGR for conductance band is given by the quasiparticle self-energy at the band edge, namely 



BGR =
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3

Numerical results and discussion

We turn to a presentation of our main numerical results. Some illustrative results for

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the quasiparticle dynamic properties such inelastic scattering lifetime, band gap renormalization and spectral function have been presented. These mentioned properties are calculated for undoped armchair graphene nanoribbon at finite temperature in the pres-

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ence of local electronic coulomb interaction U/t and gap parameter ∆/t. In the following the width of the nanoribbon for all numerical calculations is assumed to be w = 5. All

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physical properties have been studied for spin unpolarized case. The dynamical spin structure factor is associated with the imaginary part of dynamical correlation function between spin density operators introduced in Eqs.(18,19). Based on transverse dynamical spin susceptibility, the self-energy of gapped graphene nanoribbon is readily found using

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Eqs.(22,23).

Since we have intrested to study the behavior of physical parameters qualitatively, all units of axis labels in the following figures have been considered as arbitrary units. Energy

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quantities such as temperature, Hubbard interaction and frequency have been scaled by

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hopping amplitude as dimensionless parameters. This argument has been applied in Sec. 3 of revised manuscript.

Spectral function and inelastic scattering lifetime of quasiparticles can be found by Eqs.(24,25), respectively. In the following, the numerical calculation of dynamical quasiparticle properties have been performed for the first low energy conduction band, i.e. p = 4, η = + In Fig.(2), we have plotted the frequency distribution curves of spectral function of 15

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gapped graphene nanoribbon with width w = 5 for different Hubbard parameters, namely U/t = 0.2, 0.3, 0.4 and for undoped case at fixed temperature kB T /t = 0.05 for ∆/t =

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0.3. This spectral function has been measured at Fermi surface in half filling case, i.e. kx = π/(3a). The frequency positions of sharp peaks in spectral distribution function are related to the single particle excitation. For each curve we readily observe a quasiparticle

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peak at high frequencies. These high frequency excitation modes is an evidence of non Fermi liquid[53] behavior of the present electronic system. The position of peak tends to

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higher frequencies with electron-electron interaction strength U/t. it can be understood from this fact the energy gap in the excitation spectrum increases with U/t. In other hand, the peak in spectral function for U/t = 0.4 is more sharp in comparison with the other Hubbard parameters. Also the ARPES intensity or the height of peak in spectral

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function enhances with increase of U/t according to Fig.(2). The intensity of spectral function for noninteracting case is higher than that for U/t = 0.2, 0.3. The frequency position of excitation spectrum at U = 0 appears at lower frequency compared to the

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other values of U . This quasiparticle particle excitation spectrum has been obtained for gapped graphene like structure due to long range interaction under Dirac approximation

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in a theoretical work [54].

We have shown the spectral distribution function as a function of frequency for different wave vectors at non zero gap parameter ∆/t = 0.4 for U/t = 0.2 in Fig.(3). Although a quasiparticle excitation has been observed for fermi wave vector kx = π/3 at ω/t ≈ 1.8, the spectral function of graphene nanoribbon has no sharp peak in frequency region 0.0 < ω/t < 5.0.

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We have also studied the behavior of inelastic scattering lifetime of quasiparticles of gapped graphene nanoribbon in the presence of local electron-electron interaction in

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undoped case. For finding inelastic lifetime the absolute value of ImΣη,p (kx , ω) evaluated at on-shell energy ω = Eη (kx , p) − µ

−1 The behavior for wavevector dependence of inverse lifetime of quasiparticles (τin ) of

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gapped graphene nanoribbon for different gap parameter ∆/t has been shown in Fig.(4). The Hubbard has been fixed at U/t = 0.2. Except for kx ≈ 0.15, the inelastic scattering

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lifetime increases with gap parameter ∆/t. At kx ≈ 0.15, inelastic scattering lifetime for ∆/t = 0.2 is higher than that for ∆/t = 0.4. In addition, at kx ≈ 0.15, a considerable peak in inverse inelastic lifetime is observed. Also inelastic scattering lifetime tends to infinite for ∆/t = 0.4 in wave vector region 0.2 < kx < 0.6 as shown in Fig.(4).

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In Fig.(5), inelastic scattering lifetime of quasiparticles of graphene nanoribbon has been plotted as a function of wave vector for different values of Hubbard parameter U/t for fixed gap parameter ∆/t = 0.4. Based on this figure, the value of quasiparticle

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lifetime decreases with Hubbard parameter on the whole range of wave vector. This can be justified from this fact that turning on coulomb interaction between electrons

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leads to finite lifetime for quantum energy levels of electronic system. In other words the stability of quantum levels preserves in the absence of interaction between particles and thus lifetime of quasiparticles goes to infinity. However the interacting Hamiltonian causes a finite lifetime for quantum energy levels. A noticeable peak in inverse scattering lifetime has been observed at kx ≈ 0.62 for U/t = 0.4. A peak in the inverse lifetime has been observed at kx = π/3 for all values of U/t. The height of this peak increases with

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U/t as shown in Fig.(5). In Fig.(6) we have presented the temperature dependence of inverse inelastic scattering

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lifetime of gapless graphene nanoribbon for different Hubbard parameter, namely U/t = −1 0.2, 0.3, 0.4. Inverse lifetime τin increases by increasing of the temperature for each value

of U/t and it is less temperature dependent for kB T /t > 0.01. Also Fig.(6) shows that

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lifetime increases with reduction of U/t at fixed temperature. Such result is expectable since the increase of electron-electron interaction can lead to to stability of quantum

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energy levels.

The inverse scattering lifetime of quasiparticles in graphene nanoribbon has been plotted as a function of normalized temperature kB T /t for different gap parameters ∆/t at fixed Hubbard parameter U/t = 0.2 in Fig.(7). For normalized temperatures below 0.02,

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the lifetime of quasiparticles becomes infinite for all values of gapless parameter. Upon increasing kB T /t above 0.02 the scattering lifetime of quasiparticles for ∆/t = 0.2 decreases while the lifetime for the other gap parameters is also infinite. Two peaks with

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considerable height are clearly observed at normalized temperatures around 0.065 and 0.082. Another novel feature in Fig.(7) is weak dependence of lifetime on temperature for

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∆/t = 0.4, 0.6 in contrast to case ∆/t = 0.2. We have also studied the behavior of band gap renormalization, i.e. BGR, of gapped armchair graphene nanoribbon in the presence of local intrasite coulomb interaction between electrons. The BGR has been plotted in terms of gap parameter for different Hubbard parameters U/t for kB T /t = 0.03 in Fig.(8). This figure shows that there is a peak in BGR at ∆/t ≈ 0.05 for all values of Hubbard parameter. Upon more increas-

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ing ∆/t above peak position, BGR decreases with gap parameter as shown in Fig.(8). In addition, at fixed gap parameter, BGR increases with Hubbard parameter. This can

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be understood from this point that Hubbard parameter causes to induce band gap in excitation spectrum of the system and a Mott insulator develops.

Fig.(9) presents the temperature dependence of BGR of armchair graphene nanoribbon

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for different values of U/t for ∆/t = 0.4. There is no remarkable variation for temperature dependence of BGR for all values of Hubbard parameter. However a partial increase is

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observed for temperatures above 0.09. Moreover BGR increases with increase of Hubbard parameter at fixed temperature according to Fig.(9).

Finally, we have plotted BGR as a function of U/t for different gap parameters in Fig.(10). Based on this figure a monotonic increasing behavior for dependence of BGR on

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U/t has been readily observed for all ∆/t. A novel feature that is pronounced in Fig.(10) is linear dependence of BGR on U/t for all values of gap parameters. Also BGR reduces

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with increase of gap parameter at fixed Hubbard parameter U/t.

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Figure 1: Structure of an armchair graphene nanoribbon, consisting of sublattices, A and B. w is the width of the ribbon. Every unit cell contains w number of A and B sublattices. Two additional hard walls (j = 0, w + 1)are imposed on both edges.

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70

0.0

U/t

60

0.2 0.3

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0.4

40

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A(-π/3 ,ω )

50

30

20

0 0

1

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10

2

3

4

5

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ω/t

Figure 2: The quasiparticle spectral function of gapped undoped armchair graphene

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nanoribbon for conductance band p = 4, η = + as a function of frequency ω/t for different values of electron-electron interaction strength U/t and ∆/t = 0.3

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60

-π/3

kx

50

-π/4 -π/6

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40

A (ω)

30

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20

10

0

0

1

2

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-10

3

4

5

ω/t

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Figure 3: The quasiparticle spectral function of gapped undoped armchair graphene nanoribbon for conductance band p = 4, η = + as a function of frequency ω/t for differ-

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ent values of k points, namely kx = −π/6, −π/4, −π/3. The normalized local electronelectron interaction strength is assumed at U/t = 0.2. The gap parameter has been fixed at ∆/t = 0.4.

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0.003

0.0 ∆/t

0.2 0.4

0.0025

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0.0015

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τin-1

0.002

0.001

0.0005

0 0.1

0.2

0.3

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0

0.4

0.5

0.6

0.7

0.8

kx

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Figure 4: The inverse inelastic scattering lifetime of quasiparticles of gapped undoped

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armchair graphene nanoribbon for conductance band p = 4, η = + as a function of wavevector kx for different values of gap parameter, namely ∆/t = 0.0, 0.2, 0.4. The normalized local electron-electron interaction strength is assumed at U/t = 0.2.

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0.0014

0.2 U/t 0.0012

0.3 0.4

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0.001

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τin-1

0.0008

0.0006

0.0004

0.0002

0 0.1

0.2

0.3

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0

0.4

0.5

0.6

0.7

0.8

kx

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Figure 5: The inverse inelastic scattering lifetime of quasiparticles of undoped armchair

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graphene nanoribbon for conductance band p = 4, η = + as a function of wavevector kx for different values of Hubbard parameter U/t. The normalized gap parameter has been fixed at ∆/t = 0.4

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0.016

0.2

U/t

0.014

0.3

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0.4

0.012

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τin-1

0.01

0.008

0.006

0.002 0

0.01

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0.004

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

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kB T/t

Figure 6: The inverse inelastic scattering lifetime of quasiparticles of gapless undoped

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armchair graphene nanoribbon for conductance band p = 4, η = + as a function of temperature for different values of Hubbard parameter U/t.

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3.5

0.2

∆/t

3

0.4 0.6

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2.5

1.5

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τin-1

2

1

0.5

0

-0.5 0.01

0.02

0.03

0.04

0.05

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0

0.06

0.07

0.08

0.09

0.1

kB T/t

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Figure 7: The inverse inelastic scattering lifetime of quasiparticles of undoped armchair

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graphene nanoribbon for conductance band p = 4, η = + as a function of temperature for different gap parameters ∆/t. The local intrasite electronic interaction strength has been fixed at U/t = 0.2.

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26

0.0 U/t

24

0.2 0.4

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22 20

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BGR

18 16 14 12 10

6 0

0.1

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8

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

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∆/t

Figure 8: The band gap renormalization of armchair graphene nanoribbon as a function of

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gap parameter ∆/t for various Hubbard parameter U/t at fixed normalized temperature kB T /t = 0.03.

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20

0.2 U/t

0.3 0.4

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18

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BGR

16

14

12

8 0

0.01

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10

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

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kBT/t

Figure 9: The band gap renormalization of armchair graphene nanoribbon as a function of

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normalized temperature kB T /t for various Hubbard parameter U/t at fixed gap parameter ∆/t = 0.4

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50

0.0 45

∆/t

0.2 0.4

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40 35

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BGR

30 25 20 15 10

0 0

0.1

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5

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

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U/t

Figure 10: The band gap renormalization of armchair graphene nanoribbon as a function

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of normalized Hubbard parameter U/t for various gap parameter ∆/t at fixed normalized temperature kB T /t = 0.03

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References

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[1] A. K. Geim, a. H. MacDonald, Phys. Today 60, 35 (2009) [2] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubons, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004)

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[3] T. Ohta, A. Bostwick, T. Seyller, K. Horn, and E. Rorenberg, Science 313, 951 (2006)

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[4] M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, Nat. Phys 2, 620 (2006) [5] Y. Zhang, T-W Tan, H. L. Stormer, and Philip Kim, Nature (London) 438, 201 (2005)

[6] K. S. Novoselov, E. McCann, S. V. Morozov, V. I. Falko, M. I. Katsnelson, U. Zeitler,

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D. Jiang, F. Schedin and A. K. Geim, Nat. Phys 2, 177 (2006) [7] T. Ando and T. Nakanishi and R. Saito J. Phys. Soc. Japan 67, 2857 (1998)

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[8] M. I. Katsnelson, K. S. Novoselov and A. K. Geim, Nat. Phys 2, 620 (2006)

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[9] I. A. Luk’yanchuk and Y. Kopelevich Phys. Rev. Lett 93, 166402 (2004) [10] A. K. Geim and K. S. Novoselov, Nature Mater 6, 183 (2007) [11] M. Fujita, K. Wakabayashi, K. Nakada, and K. Kusakabe, J. Phys. Soc. Jpn 65, 1920 (1996)

[12] K. Nakada, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus, Phys. Rev. B 54, 17954 (1996)

30

ACCEPTED MANUSCRIPT

[13] M. Ezawa, Phys. Rev. B 73, 045432 (2006) [14] C. Berger, J. Phys. Chem. B 108, 19912 (2004)

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[15] Y. Lin, K. A. Jenkins, A. Valdes-Garcia, J. P. Small, D. B. Farmer, and P. Avouris, Nano Lett 9, 422 (2009)

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[16] J. Kedzierski, P. Hsu, P. Healey, P. W. Wyatt, C. L. Keast, M. Sprinkle, C. Berger, and W. A. de Heer IEEE Trans. Electron Devices 55, 2078 (2008)

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[17] M. Y. Han, B. Ozyilmaz, Y. Zhang and P. Kim. Phys. Rev. Lett 98, 206805 (2007) [18] L. Yang, C-H. Park, Y-W. Son, M. L. Cohen, and S. G. Louie, Phys. Rev. Lett 99, 186801 (2007)

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[19] X-F. Wang and T. Chakraborty, Phys. Rev. B 75, 033408 (2007) [20] Y. Yao, F. Ye, X. L. Qi, S. C. Zhang, and Z. Fang, Phys. Rev. B 75, 041401 (R)

EP

(2007)

[21] G. W. Semenoff, V. Semenoff, Fei Zhou, Phys. Rev. Lett 101, 087204 (2008)

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[22] C.-Y. Hou, C. Chamon, and C. Murdy, Phys. Rev. Lett 98, 186809 (2007) [23] S. Y. Zhou, D. A. Siegel, A. V. Fedorov, and A. Lanzara Physica E 40, 2642 (2008) [24] R. M. Riberio, N. M. R. Peres, J. Coutinho and P. R. Briddon, Phys. Rev. B 78, 075442 (2008) [25] K. Wakabayashi, M. Sigrist and M. Fujita, J. Phys. Soc. Japan 67, 2089 (1998)

31

ACCEPTED MANUSCRIPT

[26] Y.-W. Son, M. L. Cohen, S. G. Louie, Nature (London) 444, 347 (2006) [27] Y.-W. Son, M. L. Cohen, S. G. Louie, Phys. Rev. Lett 97, 216803 (2006)

RI PT

[28] K. Sasaki, S. Murakami, and R. Saito, J. Phys. Soc. Jpn 75,074713 (2006)

[29] K. Sasaki, S. Murakami, and R. Saito, Appl. Phys. Lett 88, 113110 (2006)

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[30] L. Brey and H. A. Fertig, Phys. Rev. B 73, 235411 (2006)

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[31] K. Wakabayashi and M. Sigrist, Phys. Rev. Lett 84, 3390 (2000) [32] K. Wakabayashi, Phys. Rev. B 64, 125428 (2001)

[33] T. O. Wehling, E. Sasioglu, C. Freidrich, A. I. Lichtenstein, M. I. Katsnelson, S.

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Blugel, Phys. Rev. Lett 106, 236805 (2011)

[34] A. Garg, H. R. Krishnamurthy, M. Randeria, Phys. Rev. Lett 97, 046403 (2006) [35] S. Sorella and E. Tosatti, Europhys. Lett 19, 699 (1992)

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[36] L. M. Martelo, M. Dzierzawa, L. Siffert, and D. Baeriswyl, Z. Phys. B: Condense.

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Matter 103, 335 (1997)

[37] L. D. Landau, Sov. Phys. JETP, 3, 920 (1957) [38] I. K. Mamorkos and S. Das Sarma, Phys. Rev. B 44, R3451 (1991); J. D. Lee and B. I. Min, Phys. Rev .B 53, 10988 (1996) [39] S. Yarlagadda and G. F. Giuliani, Phys. Rev . B 49, 7887 (1994); C.S. Ting, T. K. Lee, and J. J. Quinn, Phys. Rev .Lett 34, 870 (1975) 32

ACCEPTED MANUSCRIPT

[40] Y. Zahang , V. M. Yakovenko, and S. Das Sarma, Phys. Rev. B 71, 115105 (2005) [41] H. M. Bohm and K. Schorkhulber, J. Phys.:Condens. Matter 12, 2007 (2000)

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[42] J. M. Dawlaty, S. Shivaraman, M. Chandrasekhar, F. Rana, M. G. Spencer, Appl. Phys. Lett 92, 042116 (2008)

Kane, Appl. Phys. Lett 90, 253507 (2007)

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[43] G. GU, S. Nie, R. M. Feenstra, R. P. Devaty, W. J. Choyke, W. K. Chan and M. G.

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[44] E. H. Hwang and S. Das Sarma, Phys. Rev. B 77, 081412 (R) (2008) and A. H. MacDonald, Phys. Rev. B 77, 081411 (R) (2008)

[45] H. Zheng, Z. F. Wang, T. Luo, Q. W. Shi, and J. Chen, Phys. Rev. B 75, 165414

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(2007)

[46] G D Mahan Many Particle Physics, Plenumn Press, New York, 1993

(2010)

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[47] J. Solyom, Fundamentals of the Physics of Solids, Volume III, Springer: New York

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[48] E. H. Hwang, B. Y-K. Hu, and S. Das Sarma, Phys. Rev. B 76, 115434 (2007) [49] G. F. Giuliani and G. Vignale, Quantum Theory of the Electron Liquid (Cambridge University Press, Cambridge, 2005) [50] Y. Zhang and S. Das Sarma, Phys. Rev. B 72, 125303 (2005) [51] S. Das Sarma, R. Jalabert, and S. R. Eric Yang, Phys. Rev. B 41, 8238 (1990)

33

ACCEPTED MANUSCRIPT

[52] K. F. Berggren and B. E. Sernelius, Phys. Rev. B 29, 5575 (1984) [53] D. Pines and P. Nozieres, The theory of quantum liquids, Oxford Press, London

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(1994)

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[54] A. Qaiumzadeh and R. Asgari, New Journal Of Physics 11, 095023 (2009)

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The investigation of quasiparticle property of graphene nanoribbons using self-energy within GW approximation.

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The study of the effect of gap parameter and temperature on life time of quasiparticles