Controlling dynamical thermal transport of biased bilayer graphene by impurity atoms Hamed Rezania, and Mohsen Yarmohammadi Citation AIP Advances 6, 075121 (2016); doi 10 1063/1 4960378 View online ht[.]
Controlling dynamical thermal transport of biased bilayer graphene by impurity atoms , Hamed Rezania and Mohsen Yarmohammadi Citation: AIP Advances 6, 075121 (2016); doi: 10.1063/1.4960378 View online: http://dx.doi.org/10.1063/1.4960378 View Table of Contents: http://aip.scitation.org/toc/adv/6/7 Published by the American Institute of Physics AIP ADVANCES 6, 075121 (2016) Controlling dynamical thermal transport of biased bilayer graphene by impurity atoms Hamed Rezania1,a and Mohsen Yarmohammadi2 Department of Physics, Razi University, Kermanshah, Iran Young Researchers and Elit Club, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran (Received 14 April 2016; accepted 21 July 2016; published online 29 July 2016) We address the dynamical thermal conductivity of biased bilayer graphene doped with acceptor impurity atoms for AA-stacking in the context of tight binding model Hamiltonian The effect of scattering by dilute charged impurities is discussed in terms of the self-consistent Born approximation Green’s function approach has been exploited to find the behavior of thermal conductivity of bilayer graphene within the linear response theory We have found the frequency dependence of thermal conductivity for different values of concentration and scattering strength of dopant impurity Also the dependence of thermal conductivity on the impurity concentration and bias voltage has been investigated in details C 2016 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/) [http://dx.doi.org/10.1063/1.4960378] I INTRODUCTION Graphene has received much attention in the last few years due to its unique properties In addition to the significant interests in fundamental physics, stemming in part from the relativistic-like behavior of the massless charge particles around the Dirac cone, this material is very attractive in many applications, particularly in high speed devices.1–3 However, the gapless electron spectrum of monolayer graphene makes it difficult to turn off the electrical current due to tunneling On the other hand, bilayer graphene (BLG) can provide a finite band gap up to hundreds of meV, when the inversion symmetry between top and bottom layers is broken by an applied perpendicular electric field.4,5 A current on/off ratio of about 100 was observed at room temperature, offering a much needed control for nonlinear functionality.6 With the experimental realization of graphene,2 a considerable literature has now accumulated which has uncovered a variety of exotic effects, such as an unusual quantum Hall effect7 giant farady rotations,8 plasmarons, and so on, some of which has been summarized in reviews.3 Bilayer graphene, which are made out of two graphene planes, have also been produced by the mechanical isolation and motivated a lot of researches on their transport properties.9–11 In contrast to the case of single-layer graphene (SLG) low energy excitations of the bilayer graphene have parabolic spectrum, although, the chiral form of the effective 2-band Hamiltonian persists because the sublattice pseudospin is still a relevant degree of freedom The low energy approximation in bilayer graphene is valid only for small doping n < 1012cm−2, while experimentally doping can obtain 10 times larger densities For such a large doping, the 4-band model12 should be used instead of the low energy effective 2-band model Furthermore, an electronic bandgap can be introduced in a dual gate bilayer graphene,11,12 and it makes BLG very appealing from the point of view of applications It was shown theoretically12,4 and demonstrated experimentally13,11 that a bilayer graphene is the only material with semiconducting properties that can be controlled by electric field effect.5 Nevertheless, just as single layer graphene,14 bilayer graphene is also sensitive a Corresponding author Tel./fax: +98 831 427 4569., Tel: +98 831 427 4569 E-mail: rezania.hamed@gmail.com 2158-3226/2016/6(7)/075121/12 6, 075121-1 © Author(s) 2016 075121-2 H Rezania and M Yarmohammadi AIP Advances 6, 075121 (2016) to the unavoidable disorder generated by the environment of the SiO2 substrate: adatoms, ionized impurities, etc Disorder generates a scattering rate τ and √ hence a characteristic energy scale ~/τ which is the order of the Fermi energy EF = ~vF k F (k F ∝ n is the Fermi momentum and n is the planar density of electrons) when the chemical potential is close to the Dirac point (n → 0) Thus, one expects disorder to have a strong effect in the physical properties of graphene Indeed, theoretical studies of the effect of disorder in unbiased15 and biased16 bilayer graphene show that disorder leads to the strong modifications of its transport and spectroscopic properties The understanding of the effects of disorder in this new class of materials is fundamental for any future technological applications It is established that charged impurity scattering is primarily responsible for the transport behavior observed in monolayer graphene.17,18 A comprehensive and unabridged study of the electronic properties of the graphene in the presence of defects as a function of temperature, external frequency, gate voltage, and magnetic field has been presented by Peres and coworkers.14 Thermopower of clean and disordered biased bilayer graphene has been calculated for Bernal AB-stacking within Born approximation.19 This work shows band gap through the application of an external electric field leads to greatly enhance the thermopower of bilayer graphene, which is more than four times that of the monolayer graphene and gapless bilayer graphene at room temperature The thermal transport properties of graphene are of considerable importance for technological applications, all variants of graphene are also of potential interest and should be examined The dynamical conductivity of graphene has been extensively studied theoretically20–25 and experiments have largely verified the expected behavior.26 Some preliminary work on the absorption coefficient of undoped AA-stacked bilayer graphene in zero magnetic field has been reported.27 However most materials naturally occur with charge doping where the Fermi level is away from charge neutrality There have also been theoretical studies28,29 of the conductivity, including discussions of optical sum rules30 which continue to provide useful information on the electron dynamics In this paper, we study the effects of site dilution or unitary scattering and bias voltage on the dynamical thermal conductivity of AA- stacked bilayer graphene within the well-known self-consistent Born approximation.14,31,32 Thermal conductivity in the presence of time dependent temperature gradient is an important and interesting topic in AC Joule heating in modern processor chips performing at high frequencies AA- stacked bilayer graphene has the full symmetry in the view point of stacking of atoms and is similar to AB stacked which is synthesized in experiment Therefore AA-stacked bilayer graphene is permissible in order to perform theoretical calculation However the value of interlayer energy of AA- stacked is weaker than that of AB-stacked, this hopping amplitude of simple bilayer graphene can affect its transport properties The Born approximation allows for analytical results of electronic self-energies, allowing us to compute physical quantities such as spectral functions measured by angle resolved photoemission (ARPES),33,9 and density of states scanning tunneling microscopy STM,34,35 besides standard transport properties such as the DC and AC conductivities To ensure the applicability of self consistent Born approximation (SCBA), we restrict our calculations to relatively clean systems with low impurity concentrations Dynamical thermal conductivity of AA- stacked bilayer graphene as a function of impurity concentrations is calculated for different bias voltage and scattering potential strengths We also study effects of impurity concentration and scattering potential strengths on the frequency dependence of thermal conductivity II THEORETICAL METHOD We consider a bilayer graphene composed of two graphene single layers arranged in the simple stacking.36 A bilayer graphene composed of two graphene single layers arranged in the simple stacking36 has been considered The thermal properties of AA-stacked bilayer graphene has been calculated using the band structure and the electronic Green’s function For the case of AA-stacking, an A (B) atom in the upper layer is stacked directly above A(B) atom in the lower layer An on-site potential energy difference between the two layers is included to model the effect of an external voltage In the presence of impurity, the Hamiltonian consists of two parts: H = H0 + Himp Up to nearest neighbor hopping, the single spin tight binding model hamiltonian for 075121-3 H Rezania and M Yarmohammadi AIP Advances 6, 075121 (2016) AA-stacking reads the following form20,21 H= φ†k H0(k)φk, (1) k † † † in which the vector of fermion creation operators is defined as φ†k = (a1,k , b†2,k, a2,k , b†1,k) al,k ,b†l,k create l layer states with wave vector k on the A and B sublattices,respectively (See Fig 1) The nearest neighbor approximation gives us the following matrix form for H0(k) as V/2 t⊥ f (k) * + ∗ −V/2 f (k) t ⊥ // / H0(k) = (2) f (k) −V/2 // t⊥ ∗ t⊥ V/2 , f (k) ( ) f (k) = −t ∥ + exp(ik.a1) + exp(ik.a2) describes the intralayer nearest neighbor hopping with strength t ∥ Furthermore √ the primitive vectors of the triangular sublattice presented in Fig have property |a1| = |a2| = 3acc that acc = |a01| = |a02| = |a03| is the nearest carbon-carbon distance The hopping parameter between an A (B) site in one layer and the nearest A(B) site in the other layer is given by t ⊥ and is reported to be about 0.2 eV.27,37 V is the potential energy difference between the first and second layers induced by a bias voltage Since for every attainable carrier density, it is possible to find a bias voltage to make the potential difference between the two layers as V , we would not consider the Coulomb interaction between imbalanced electron densities of the two layers and also neglect the dependence of V on the carrier density n in this work Impurity scattering effects are included in the tight-binding description by the addition of a local energy term Himp = vi (aq† aq + b†qbq), (3) q FIG (a) Schematic of graphene sheet The A and B sublattice sites separeted by a distance acc The blue dashed lines denote the Bravais lattice unit cell Each cell includes two nonequivalent sites, which are indicated by A and B a and a are the primitive vectors of unit cell a 01 , a 02 and a 03 are three vectors connecting nearest neighbor sites (b) AA-stacked bilayer graphene with the intra and inter layer hopping t ∥ and t ⊥, respectively 075121-4 H Rezania and M Yarmohammadi AIP Advances 6, 075121 (2016) where vi is the electron-impurity potential at site Ri This term breaks the translational symmetry of crystal so that it introduces the scattering of electrons from impurities situated at randomly distributed but fixed positions Under half filling constraint corresponding to one electron per each lattice site chemical potential (µ) gets zero value Since unit cell of bilayer graphene includes four atoms, the Green’s function can be written as the × matrix Fourier transformation of the Green’s function matrix of the clean system (G0) can be readily obtained by the following equation G0(k,iω n ) = , (iω n + µ)1 − H0(k) (4) where ω n = (2n + 1)π/ β is the Fermionic Matsubara’s frequency.38 After substituting Eq (2) into Eq (4), the explicit form of Green’s function matrix of clean bilayer system has been found The explicit expression for each matrix element of Green’s function is quite lengthy and has not been presented here According to Born approximation in the scattering theory,39 Using T matrix,39 the electronic self-energy matrix of disordered system in the presence of finite but small density of boron impurity atoms, ni = Ni /N, could be obtained as Σ(iω n ) = Ni Timp(iω n ) = n i vi − vi G0(iω n ) , (5) where N is the number of unit cells and vi denotes the electronic on-site energy which shows the strength of scattering potential The local propagator of clean system is given by G0(iω n ) = G0(k,iω n ) (6) N k In order to include some contributions from multiple site scattering, we replace the local bare Green’s functionG0(iω n ) by local full one (G(iω n )) in the expression of the self-energy matrix in Eq (5), leading to full self-consistent Born approximation Under neglecting intersite correlations, the self-consistent problem requires the solution of the equation Σαα (E) = n i vi n i vi = , − vi Gαα (E) − vi G A A(E + i0+ − Σαα (E)) (7) where a simple analytical continuation as iω n −→ E + i0+ has been performed to obtain retarded self-energy The electronic self-energy should be found from a self-consistent solution of Eq (7) The perturbative expansion for the Green’s function of disordered system is obtained via the Dyson equation38 given by G(k,iω n ) = [(G0(k,iω n ))−1 − Σ(iω n )]−1 (8) E The thermal conductivity is obtained as the response of the energy current (J ) to a temperature E gradient Imposing the continuity equation for the energy density, ∂H ∂t + ∇.J = 0, the explicit form of the energy current can be calculated After some calculations, the component of the energy current operator along x direction (see Fig 1) for AA type is given in terms of Fourier transformation of fermionic operators20,21 † JxE = it 2∥ i.Ri al,k al,keik.Ri k,i,l − it ∥t ⊥ ) ( ′ ′ † ′ ′ † b1,k − e−ik.∆ b†1,k a2,k , ∆′.i eik.∆ a1,k b2,k − e−ik.∆ b†2,k a1,k + eik.∆ a2,k (9) k,∆′ that Ri are the four vectors connecting the nearest neighbor unit cells and is given by Ri=1, ,5 = ±a1, ±a2, and ∆′ = 0, a1, a2 In analogy to energy current, there is an equation of charge conservation so that electrical current(Je ) satisfies it as ∇.Je + ∂ρ ∂t = 0, where ρ is the density operator of electrons Let us to define the polarization operator (P) as38 P= Rcm (al,† m al, m + b†l, m bl, m ), (10) l, m 075121-5 H Rezania and M Yarmohammadi AIP Advances 6, 075121 (2016) where Rcm denotes the position vector of m th unit cell in honeycomb lattice Also l = 1, denotes the index of layer in the lattice The electrical current is readily obtained via Je = dP dt = i[H, P] In terms of Fourier transformations of creation and annihilation operators, the final expression for the electrical current operator for type AA along the x direction is given by ′ ′ † (11) bl,k + e−ik.δ b†l,kal,k), i.δ ′(−eik.δ al,k Jxe = it δ ′,k,l where δ ′ = 0, a1, a2 are the vectors connecting nearest neighbor unit cells The heat current (JQ) is related to the energy current and electrical one by JQ = J E − µJe where µ is the chemical potential The linear response theory is implemented to obtain the thermal conductivity under the assumption of a low temperature gradient (as a perturbing field) Within linear response theory, the charge and thermal current are related to the gradients ∇V and ∇T of the electric potential and the temperature, respectively, by *J1(ω)+ = * L 11(ω) ,J2(ω)- , L 21(ω) L 12(ω)+ * E(ω) + L 22(ω)- ,−∇T(ω)- (12) J1(2) = Je (JQ) implies electrical (heat) current The Kubo formula38 gives us the transport coefficient L ab (ω) in terms of a correlation function of energy current operators +∞ β i iωt x x L Ret (ω) = dte θ(t)⟨[J (t), J (0)]⟩ = lim dτeiω nτ ⟨Tτ (Jax (τ)Jbx (0))⟩, a b ab βω −∞ βω iω n −→ ω+i0+ (13) where a = 1, 2; b = 1, Moreover, β is the inverse of temperature and ω n = 2nπ β is the bosonic Matsubara frequency Here, we can start the derivation of transport matrix Using Eq (13), the matrix element L 11(ω) is given as β ( dτeiω nτ ⟨Tτ Jxe (τ)Jxe (0)⟩) (14) L 11(iω n ) = ωβ We can calculate the function in Eq (14) within an approximation by implementing Wick’s theorem The correlation functions between current operators can be interpreted as multiplication of two disordered Green’s function According to Lehman representation40 the Matsubara Green’s function could be related to spectral function as ∞ dE Aα β (k, E) Gα β (k,iω n ) = , (15) −∞ 2π iω n − E where Aα β (k, E) = −2ℑGα β (k,iω n −→ E + i0+) is the spectral function of electronic system Using Eq (15) and performing Matsubara frequency summation, the final expression for dynamical spectral function L 11(ω) ≡ ℑL 11(iω n −→ ω + i0+) of AA stacking gets the following form 3t 2∥ +∞ L 11(ω) = − dE(n F (E − ω) − n F (E)) cos2(k y /2) πω β −∞ k ( √ × cos( 3k x ) 2A B1 A1(k, E)A B1 A1(k, E − ω) + 2A A1 B1(k, E)A A1 B1(k, E − ω) + A B1 A2(k, E)A B2 A1(k, E − ω) + A B2 A1(k, E)A B1 A2(k, E − ω) ) + A A1 B2(k, E)A A2 B1(k, E − ω) + A A2 B1(k, E)A A1 B2(k, E − ω) + 4A A1 A2(k, E)A A1 A2(k, E − ω) + A A1 A1(k, E)A A2 A2(k, E − ω) + A B2 B2(k, E)A B1 B1(k, E − ω) + A B1 B1(k, E)A B2 B2(k, E − ω) ) + A B1 B1(k, E)A A1 A1(k, E − ω) − 4A A1 A2(k, E)A A1 A2(k, E − ω) − A A1 A1(k, E)A A2 A2(k, E − ω) − A B2 B2(k, E)A B1 B1(k, E − ω) − A B1 B1(k, E)A B2 B2(k, E − ω) − A B1 B1(k, E)A A1 A1(k, E − ω) , (16) 075121-6 H Rezania and M Yarmohammadi AIP Advances 6, 075121 (2016) where n F (x) = 1/(e x/k BT + 1) is the Fermi Dirac distribution function In a similar way, one can also calculate the other elements of transport matrix introduced in Eq (13) In Appendix we present the final results for dynamical transport coefficients L 12(ω) and L 22(ω) of AA-stacked bilayer graphene In the presence of a dynamical temperature gradient (∇T(ω)) and in open circuit situation, i.e.Je = 0, heat current is related to temperature gradient via JQ(ω) = κ(ω)∇T(ω) where κ(ω) is the dynamical thermal conductivity and is obtained using transport coefficients as38 κ(ω) = L 212(ω) ) (L (ω) − 22 L 11(ω) T2 (17) The study of behavior of κ in bilayer graphene constitutes the main aim in this work III NUMERICAL RESULTS We have obtained the dynamical thermal conductivity of the impurity doped AA- stacked bilayer graphene along the x direction as shown in Fig We have implemented a tight binding model Hamiltonian including local energy term so that this term describes the scattering of electrons from impurity atoms We have obtained the electronic spectrum of the disordered tight binding model by means of Green’s function approach which gives the thermal conductivity by calculating the energy current correlation function The electronic self-energy of the disordered system is calculated within a self-consistent solution of Eq (7) The process is started with an initial guess for Σ A A(E) and is repeated until convergence is reached The final results for self-energy matrix elements have been employed to obtain electronic Green’s function of disordered bilayer graphene Afterwards static transport coefficients have been calculated using Eqs (16), (A1) The transport coefficients are obtained based on the linear response approximation which relates the response with perturbative potential via a linear equation This approximation preserves its validity as long as the intensity of perturbing field gets low values In the present problem we have obtained the numerical results of dynamical thermal conductivity under condition of low amounts of gradient of temperature as perturbing potential In obtaining numerical results, the intralayer nearest neighbor hopping parameter (t ∥ ) is set to Therefore the other parameters in the model Hamiltonian is expressed as V/t ∥ , vi /t ∥ , µ/t ∥ The impurity concentration dependence of thermal conductivity of AA- stacked bilayer graphene (κ A A−S BG (ω)) for different values of chemical potential µ/t ∥ is plotted in Fig This figure indicates that thermal conductivity increases monotonically with impurity concentration, however it decreases quite slowly for all chemical potentials µ/t ∥ Also we see the value of thermal conductivity reduces with chemical potential at fixed impurity concentration as shown in Fig It can be justified from the fact that the increase of µ/t ∥ raises the scattering rate between electrons Furthermore we see a drastic reduction in thermal conductivity when µ/t ∥ changes from 0.3 to 0.6 In Fig 3, we have plotted thermal conductivity of biased bilayer graphene as a function of impurity concentration for the various electron-impurity scattering strength, namely vi /t ∥ = 0.1, 0.2, 0.4, 0.6 for fixed parameters k BT/t ∥ = 0.06, µ/t ∥ = 1.0, ω/t ∥ = 2.0, V/t ∥ = 1.25 As a result, the thermal conductivity is found to be monotonically increasing with impurity concentration ni for higher values of vi /t ∥ = 0.4, 0.6, however it presents uniform behavior with ni in low values, i.e vi /t ∥ = 0, 0.2 In addition, at fixed values of impurity concentrations, the increase of vi /t ∥ leads to enhance thermal conductivity It can be understood from the fact that the increase of higher vi /t ∥ causes more electronic transition rate between energy bands and consequently higher values in thermal conductivity In fact impurities acts as scattering centers which can causes to increase the transition rate for electrons from valence band to conduction one The overlap of π electrons of carbon atoms leads to appear thin layer above and below graphene sheet This thin layer for electrons of each graphene sheet causes to overlap of electron cloud of both carbon layers However the stacking type of bilayer graphene affects the overlap intensity Such the electronic overlap of above and below layers turns out the inter layer hopping amplitude The combination of electronic cloud of impurity atoms with mentioned interlayer overlap changes the propagation of electron 075121-7 H Rezania and M Yarmohammadi AIP Advances 6, 075121 (2016) FIG Dynamical thermal conductivity of biased AA stacked bilayer graphene (κ A A−S BG (ω)) as a function of n i for various amounts of normalized chemical potential µ/t ∥ for fixed temperature k BT /t ∥ = 0.06 at fixed frequency ω/t ∥ = 2.0 wave between two layers In the current problem, we deal with a time dependent temperature difference changes the electronic wave The overlap between electron cloud of impurity atoms and that of electronic layer enhances due to increase of scattering potential strength This leads to change the position of electron wave Therefore the time dependent thermal conductivity increases We have also studied the effect of impurity concentration on temperature behavior of thermal conductivity of AA stacked bilayer graphene In Fig we plot κ A A−S BG (ω) versus normalized temperature for different values of impurity concentration, namely ni = 0.0, 0.03, 0.05, 0.09, 0.1 Two features are pronounced in this figure The increase of temperature reduces thermal conductivity for any value of impurity concentration ni Higher temperature causes more scattering of FIG Dynamical thermal conductivity of biased AA stacked bilayer graphene (κ A A−S BG (ω)) as a function of impurity concentration n i for various amounts of electron-impurity scattering strength v i /t ∥ for fixed temperature k BT /t ∥ = 0.06 at fixed frequency ω/t ∥ = 2.0 075121-8 H Rezania and M Yarmohammadi AIP Advances 6, 075121 (2016) FIG Dynamical thermal conductivity of biased AA stacked bilayer graphene (κ A A−S BG (ω)) as a function of normalized temperature k BT /t ∥ for various amounts of impurity concentration n i for fixed chemical potential µ/t ∥ = 1.0 at fixed frequency ω/t ∥ = 2.0 electrons which reduces the thermal conductivity Also the thermal transport is unaffected by the increase of ni where all plots fall on each other on the whole range of temperature as shown in Fig This situation is similar for scattering strength vi /t ∥ The temperature dependence of dynamical thermal conductivity of biased bilayer graphene is studied for various scattering strengths at fixed impurity concentration ni = 0.07 and main features are depicted in Fig A monotonically decreasing behavior for temperature dependence of thermal conductivity is clearly observed for each plot in FIG Dynamical thermal conductivity of biased AA stacked bilayer graphene (κ A A−S BG (ω)) as a function of normalized temperature k BT /t ∥ for various amounts of scattering strength v i /t ∥ for fixed chemical potential µ/t ∥ = 1.0 at fixed frequency ω/t ∥ = 2.0 075121-9 H Rezania and M Yarmohammadi AIP Advances 6, 075121 (2016) FIG Dynamical thermal conductivity of biased AA stacked bilayer graphene (κ A A−S BG (ω)) as a function of normalized scattering strength v i /t ∥ for various normalized chemical potentials µ/t ∥ for fixed normalized temperature k BT /t ∥ = 0.06 at fixed frequency ω/t ∥ = 2.0 Fig Moreover the variation of electron-impurity scattering strength has no remarkable effect on temperature behavior of κ A A−S BG (ω) In Fig 6, we present in-plane thermal conductivity of the biased undoped simple bilayer graphene versus normalized scattering strength (vi /t ∥ ) for different chemical potential amounts, namely µ/t ∥ = 0.0, 0.3, 0.6, 1.0, 2.0 for fixed temperature k BT/t ∥ = 0.06 and V/t ∥ = 1.25 By increasing scattering strength vi /t ∥ , thermal conductivity takes constant value for each chemical potential value At a fixed value of vi /t ∥ , thermal conductivity reduces with normalized chemical FIG Dynamical thermal conductivity of biased AA stacked bilayer graphene (κ A A−S BG (ω)) as a function of normalized frequency ω/t ∥ for various impurity concentrations n i for fixed normalized temperature k BT /t ∥ = 0.06 at fixed bias voltage V /t ∥ = 1.25 075121-10 H Rezania and M Yarmohammadi AIP Advances 6, 075121 (2016) potential as shown in Fig In addition, a rapid reduction in thermal conductivity has been clearly observed when chemical potential gets the value 0.6 We have also studied the frequency dependence of dynamical thermal conductivity of biased simple bilayer graphene Fig indicates κ A A−S BG (ω) versus normalized frequency ω/t ∥ for different values of impurity concentration, namely ni = 0, 0.03, 0.06, 0.09 for vi /t ∥ = 0.3, V/t ∥ = 1.25 This plot implies a weak dependence of thermal conductivity on ni except for frequencies around 0.2 and 0.5 At sufficiently low temperatures, the scattering rate of electrons from impurities has a little amount because of low value of temperature gradient Therefore the thermal conductivity shows divergent behavior and the variation of impurity concentration has no remarkable effect on the thermal conductivity behavior In fact this divergency at low frequencies for thermal conductivity arises from Drude weight contribution of conductivity The increase of frequency causes to decrease of thermal conductivity since Drude weight contribution has non zero value very low frequency region According to Fig the variation of impurity concentration has not any effect on frequency behavior conductivity up to frequency around 0.2 In this frequency a major interband transition takes place where the impurity concentration gives rise the increases of transition rate This interband transition takes place between upper valnce band and lower conduction band around Dirac points in the first Brillouin zone Dirac points are six vertexes of hexagonal Brillouin zone of honeycomb structure Thus we observe thermal conductivity rises with ni around normalized frequency ω/t ∥ = 0.2 In addition, this dependence of thermal conductivity on impurity concentration ni is clearly observed at frequencies around 0.5 Similar to the case of ω/t ∥ = 0.2, an electronic interband transition takes place at frequency 0.5 and higher values of ni leads to higher transition rate and consequently thermal conductivity increases with ni In other words, frequency dependence of thermal conductivity is almost independent of impurity concentration on the whole range of frequency However for frequency regions ω/t ∥ ≈ 0.2 and ω/t ∥ ≈ 0.5, thermal conductivity shows sensitive dependence on impurity concentrations IV SUMMARY In conclusion, we have presented the dependence of the dynamical conductivity of disordered biased bilayer grapehene in the simple stacking case on various physical parameters Using a tight binding model Hamiltonian including random on-site energy term and a Green’s function approach the thermal conductivity has been studied Thermal conductivity has been found within linear response theory based on the correlation function between energy current operators Particularly, the effects of impurity concentration on frequency dependence of thermal conductivity of biased bilayer graphene The results shows thermal conductivity gets the considerable effects due to impurity concentration around normalized frequencies 0.2 and 0.5 Also the behavior of dynamical conductivity versus impurity concentration shows the conductivity at fixed frequency increases with scattering strength Also our results show the impurity concentration and scattering strength has no remarkable effect on the temperature dependence of thermal conductivity APPENDIX: THE EXPLICIT EXPRESSION OF TRANSPORT COEFFICIENTS In this Appendix, we present the expressions of dynamical transport coefficients L 12(ω), L 22(ω) for simple type of bilayer graphene L 22(ω) = t 2∥ +∞ dE(n F (E − ω) − n F (E)) βπω −∞ ( ) √ √ × δ1x 4cos( 3k x )cos(k y ) + cos( 3k x ) − − cos(k y )) k ( × A A1 A1(k, E)A A1 A1(k, E − ω) + 2A A1 A2(k, E)A A2 A1(k, E − ω) ) + A A2 A2(k, E)A A2 A2(k, E − ω) ( + t ⊥ A A1 A1(k, E)A B2 A1(k, E − ω) + A A1 A1(k, E)A A1 B2(k, E − ω) 075121-11 H Rezania and M Yarmohammadi AIP Advances 6, 075121 (2016) + A A1 A2(k, E)A B1 A1(k, E − ω) + A A1 B1(k, E)A A2 A1(k, E − ω) + A A2 A1(k, E)A B2 A2(k, E − ω) + A A2 B2(k, E)A A1 A2(k, E − ω) + A A1 A1(k, E)A B1 A2(k, E − ω) ) + A A2 A2(k, E)A A2 B1(k, E − ω) )( ( √ √ + 2t ⊥2 cos( 3k x )cos(k y ) + cos( 3k x ) A B2 A1(k, E)A B2 A1(k, E − ω) + A A1 B2(k, E)A A1 B2(k, E − ω) + A B1 A2(k, E)A B1 A2(k, E − ω) + A A2 B1(k, E)A A2 B1(k, E − ω) + A A2 B1(k, E)A A2 B1(k, E − ω) ) + 2A B2 A2(k, E)A B1 A1(k, E − ω) + 2A A1 B1(k, E)A A2 B2(k, E − ω) ( )( − 4t ⊥2 + 2cos(k y ) A B2 B2(k, E)A A1 A1(k, E − ω) + A B2 B1(k, E)A A2 A1(k, E − ω) ) + A B1 B2(k, E)A A1 A2(k, E − ω) + A B1 B1(k, E)A A2 A2(k, E − ω) , t 2∥ +∞ dE(n F (E − ω) − n F (E)) L 12(ω) = βπω −∞ ) ( √ √ × δ1x − 2t 3∥ cos( 3k x )cos(k y ) + cos( 3k x ) − cos(k y ) k ( × A B1 A1(k, E)A A1 A1(k, E − ω) + A B1 A2(k, E)A A2 A1(k, E − ω) + A B2 A1(k, E)A A1 A2(k, E − ω) ) + A B2 A2(k, E)A A2 A2(k, E − ω) ) ( √ √ + 2t 3∥ + cos(k y ) − cos( 3k x )cos(k y ) − cos( 3k x ) ( × A A1 A1(k, E)A A1 B1(k, E − ω) ) + A A1 A2(k, E)A A1 B1(k, E − ω) + A A2 A1(k, E)A A1 B2(k, E − ω) (A1) K S Novoselov, A K Geim, S V Morosov, D Jiang, Y Zhang, S V Dubonos, I V Grigorieva, and A A Firsov, “Electric field in atomically thin carbon films,” Science 306, 666 (2004) K S Novoselov, A K Geim, S V Morosov, D Jiang, M I Katsnelson, I V Grigorieva, S V Dubonos, and A A Firsov, “Two-dimenensional gas of massless Dirac fermions graphene,” Nature 438, 197 (2005) A H Castro Neto, F Guinea, n M R Peres, K S Novoselov, and A K Geim, “The electronic properties of graphene,” Rev Mod Phys 81, 109 (2009) E McCann, “Asymmetry gap in the electronic band structure of bilayer graphene,” Phys Rev B 74, 161403 (2006) H Min, B Sahu, S K Banerjee, and A H MacDonald, “Ab initio theory of gate induced gaps in graphene bilayers,” Phys Rev B 75, 155115 (2007) K M Borysenko, J T Mullen, X Li, Y G Semenov, J M Zavada, M B Buongiorno Nardelli, and K W Kim, “Electron-phonon interactions in bilayer graphene,” Phys Rev B 83, 161402 (R) (2011) Y Zhang, Y -W Tan, H L Stormer, and P Kim, “Experimental observation of the quantum Hall efect and Berry’s phase in graphene,” Nature 438, 201 (2005) I Crassee, J Levallois, A L Walter, M Ostler, A Bostwick, E Rotenberg, T Seyller, D van Der Marel, and A B Kuzmenko, “Giant Farady rotation in single-and multilayer graphene,” Nature Phys 7, 48 (2011) T Ohta, A Bostwick, T Seyller, K Horn, and E Rotenberg, “Controlling the electronic structure of Bilayer graphene,” Science 313, 951 (2006) 10 M Koshino and T Ando, “Transport in bilayer graphene: Calculations within a self-consistent Born approximation,” Phys Rev B 73, 245403 (2006) 11 E V Castro, K S Novoselov, S V Morosov, N M R Peres, J M B Lopes dos Santos, J Nilsson, F Guinea, A K Geim, and A H Castro Neto, “Biased Bilayer graphene: Semiconductor with a gap tunable by the Electric field effect,” Phys Rev Lett 99, 216802 (2007) 12 Ed McCann and V I Falko, “Landau level degeneracy and quantum Hall effect in a Graphite Bilayer,” Phys Rev Lett 96, 086805 (2006) 13 J B Oostinga, H B Heersche, X Liu, A F Morpurgo, and L M K Vandersypen, “Gate induced insulating state in bilayer graphene devices,” Nature Mat (2007), doi:10.1038/nmat2082 14 N M R Peres, F Guinea, and A H Castro Neto, “Electronic properties of disordered two-dimenensional carbon,” Phys Rev B 73, 125411 (2006) 15 J Nilsson, A H Castro Neto, F Guinea, and N M R Peres, “Electronic properties of graphene Multilayers,” Phys Rev Lett 97, 266801 (2006) 16 J Nilsson and A H Castro Neto, “Impurities in a biased graphene bilayer,” Phys Rev Lett 98, 126801 (2007) 17 K Nomura and A H MacDonald, “Quantum Transport of massless Dirac fermions,” Phys Rev Lett 98, 076602 (2007) 075121-12 18 H Rezania and M Yarmohammadi AIP Advances 6, 075121 (2016) S Adam, E H Hwang, V M Galitski, and S Das Saram, “A self-consistent theory for graphene transport,” Proc Natl Acad Sci U S A 104, 18392 (2007) 19 L Hao and T K Lee, “Thermopower of gapped bilayer graphene,” Phys Rev B 81, 165445 (2010) 20 H Rezania and A Abdi, “Thermal conductivity of disordered AA-stacked bilayer graphene in the presence of bias voltage,” European Physical Journal B 88, (2015) 21 H Rezania and M Yarmohammadi, “Dynamical thermal conductivity of bilayer graphene in the presence of bias voltage,” Physica E 75, 125 (2015) 22 H Rezania and M Yarmohammadi, “Dynamical thermoelectric properties of doped AA-stacked bilayer graphene,” Superlattice and Microstructures 89, 15 (2016) 23 H Rezania and M Yarmohammadi, “The effects of impurity doping on the optical properties of biased bilayer graphene,” Optical Materials 57, (2016) 24 H Rezania and M Yarmohammadi, “Optical conductivity of AA-stacked bilayer graphene in presence of bias voltage beyond Dirac approximation,” Indian J Phys 90, 811 (2016) 25 T Ando, Y Zhang, and H Suzuura, “Dynamical conductivity and Zero-Mode Anomaly in Honeycomb Lattices,” J Phys Soc Jpn 71, 1318 (2002) 26 R R Nair, B Blake, A N Grigeronke, K S Novoselov, T J Booth, T Stauber, N M R Peres, and A K Geim, “Fine Structure constant difines visual transparency of graphene,” Science 320, 1308 (2008) 27 Y Xu, X Li, and J Dong, “electronic instabilities of the AA-Honeycomb Bilayer,” Nanotechnology 21, 065711 (2010) 28 L A Falkovaky and A A Varlamov, “space time dispersion of graphene conductivity,” Euro Phys J B 56, 281 (2007) 29 V P Gusynin, S G Sharapov, and J P Carbotte, “Magneto optical conductivity in graphene,” J Phys: Condense Mtter 19, 026222 (2007) 30 V P Gusynin, S G Sharapov, and J P Carbotte, “Sum rules for the optical and Hall conductivity in graphene,” Phys Rev B 75, 165407 (2007) 31 Nguyen Hong Shon and Tsuneya Ando, “quantum transport in Two-dimenensional graphite system,” J Phys Soc Jpn 67, 2421 (1998) 32 X-Z Yan and C S Ting, “Hall coefficient of Dirac in graphene under charged impurity scatterings,” Phys Rev B 80, 155423 (2009) 33 S Y Zhou, G H Gweon, and A Lanzara, “Low energy excitations in graphite: the role of dimenensionality and lattice defects,” Annals of Physics 321, 1730 (2006) 34 P Mallet, F Varchon, C Naud, L Magaud, C Berger, and J.-Y Veuillen, “Electron states of mono-and bilayer graphene on SiC probed by scanning -tunneling microscopy,” Phys Rev B 76, 041403 (2007) 35 G Li and E V Andrei, “Observation of Landau levels of Dirac fermions in graphite,” Nature Physics 3, 623 (2007) 36 J Nilsson, A H Castro Neto, F Guinea, and N M R Peres, “Tranmission through a biased graphene bilayer barrier,” Phys Rev B 76, 165416 (2007) 37 I Lobato and B Partoens, “Muliple Dirac particles in AA-stacked graphite and multilayers of graphene,” Phys Rev B 83, 165429 (2011) 38 G D Mahan, Many Particle Physics (Plenumn Press, New York, 1993) 39 S Doniach and E H Sondheimer (World Scientific, Singapore, 1988) 40 A L Fetter and J D Walecka, Quantum Theory of Many Particle Systems (MacGraw-Hill, New York, 1971) ... ADVANCES 6, 075121 (2016) Controlling dynamical thermal transport of biased bilayer graphene by impurity atoms Hamed Rezania1,a and Mohsen Yarmohammadi2 Department of Physics, Razi University,... the dynamical thermal conductivity of biased bilayer graphene doped with acceptor impurity atoms for AA-stacking in the context of tight binding model Hamiltonian The effect of scattering by dilute... increase of temperature reduces thermal conductivity for any value of impurity concentration ni Higher temperature causes more scattering of FIG Dynamical thermal conductivity of biased AA stacked bilayer