controlling dynamical thermal transport of biased bilayer graphene by impurity atoms

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[.]

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

Controlling dynamical thermal transport of biased bilayer graphene by impurity atoms

Hamed Rezania1,aand Mohsen Yarmohammadi2

1Department of Physics, Razi University, Kermanshah, Iran

2Young 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 addi-tion 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 3However, 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.4,5A 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,2a considerable literature has now accumulated which has uncovered a variety of exotic effects, such as an unusual quantum Hall effect7 giant farady rotations,8plasmarons, and so on, some of which has been summarized in reviews.3Bilayer 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 experimen-tally doping can obtain 10 times larger densities For such a large doping, the 4-band model12should

be used instead of the low energy effective 2-band model Furthermore, an electronic bandgap can

be introduced in a dual gate bilayer graphene,11,12and 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.5Nevertheless, just as single layer graphene,14bilayer 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.

to the unavoidable disorder generated by the environment of the SiO2substrate: 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= ~vFkF(kF ∝√nis 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, theoret-ical studies of the effect of disorder in unbiased15and biased16bilayer 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.19This 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 theoretically20–25and experiments have largely verified the expected behavior.26Some preliminary work on the absorption coefficient

of undoped AA-stacked bilayer graphene in zero magnetic field has been reported.27However 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 rules30which 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.14,31,32Thermal 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),33,9and density of states scanning tunneling microscopy STM,34,35besides standard trans-port 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 impu-rity 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 ef-fect of an external voltage In the presence of impurity, the Hamiltonian consists of two parts:

H= H + H Up to nearest neighbor hopping, the single spin tight binding model hamiltonian for

AA-stacking reads the following form20,21

H= k

φ†

in which the vector of fermion creation operators is defined as φ†k= (a†

1,k, b† 2,k, a† 2,k, b† 1,k) a†l,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

H0(k) =

*

,

V/2 0 t⊥ f(k)

0 −V/2 f∗(k) t⊥

t⊥ f(k) −V/2 0

f∗(k) t⊥ 0 V/2

+ / /

f(k) = −t∥

(

1+ 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.1have 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=

q

vi(aq†aq+ b†

FIG 1 (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 1 and a 2

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.

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 4 × 4 matrix Fourier transformation of the Green’s function matrix of the clean system (G0) can

be readily obtained by the following equation

G0(k,iωn) = 1

where ωn= (2n + 1)π/β is the Fermionic Matsubara’s frequency.38After 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,39Using T matrix,39the 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(iωn) = NiTimp(iωn) = nivi

1 − viG0(iωn), (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(iωn) = N1 

k

In order to include some contributions from multiple site scattering, we replace the local bare Green’s functionG0(iωn) by local full one (G(iωn)) 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) = nivi

1 − viGαα(E) = nivi

1 − viGA A(E + i0+− Σαα(E)), (7) where a simple analytical continuation as iωn−→ 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 equation38given by

G(k,iωn) = [(G0(k,iωn))−1− Σ(iωn)]−1 (8) The thermal conductivity is obtained as the response of the energy current (JE) to a temperature gradient Imposing the continuity equation for the energy density, ∂H∂t + ∇.JE = 0, the explicit form

of the energy current can be calculated After some calculations, the component of the energy cur-rent operator along x direction (see Fig.1) for AA type is given in terms of Fourier transformation

of fermionic operators20,21

JxE= it2

∥



k,i,l i.Ria†l,kal,keik.Ri

− it∥t⊥



k,∆ ′

∆′.i(

eik.∆′a1, k† b2, k− e−ik.∆′b†2, ka1, k+ eik.∆′a2, k† b1, k− e−ik.∆′b†1, ka2, k) , (9) that Ri are the four vectors connecting the nearest neighbor unit cells and is given by Ri=1, ,5

= ±a1,±a2,0 and ∆′= 0,a1,a2 In analogy to energy current, there is an equation of charge conser-vation 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(a†l, mal, m+ b†

l, mbl, m), (10)

where Rc

mdenotes the position vector of m th unit cell in honeycomb lattice Also l= 1,2 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

Jxe= it 

δ ′ ,k,l i.δ′ (−eik.δ′a†l,kbl,k+ e−ik.δ ′

b†l,kal,k), (11)

where δ′= 0,a1,a2are the vectors connecting nearest neighbor unit cells The heat current (JQ) is related to the energy current and electrical one by JQ= JE−µJewhere 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)

J2(ω)+

-= * ,

L11(ω) L12(ω)

L21(ω) L22(ω)+

-*

,

E(ω)

−∇T(ω)+

J1 (2)= Je

(JQ) implies electrical (heat) current The Kubo formula38gives us the transport coefficient

Lab(ω) in terms of a correlation function of energy current operators

LRetab(ω) = βωi

 +∞

−∞

dteiωtθ(t)⟨[Jx

a(t), Jbx(0)]⟩ = βω1 iω n −→limω+i0 +

 β 0 dτeiωnτ

⟨Tτ(Jax(τ)Jbx(0))⟩,

(13) where a= 1,2; b = 1,2 Moreover, β is the inverse of temperature and ωn= 2nπ

β is the bosonic Matsubara frequency Here, we can start the derivation of transport matrix Using Eq (13), the matrix element L11(ω) is given as

L11(iωn) =ω β (1

 β 0

dτeiω n τ

⟨TτJxe(τ)Jxe(0)⟩) (14)

We can calculate the function in Eq (14) within an approximation by implementing Wick’s theo-rem The correlation functions between current operators can be interpreted as multiplication of two disordered Green’s function According to Lehman representation40the Matsubara Green’s function could be related to spectral function as

Gαβ(k,iωn) =

 ∞

−∞

dE 2π

Aαβ(k, E)

where Aαβ(k, E) = −2ℑGαβ(k,iωn−→ E+ i0+) is the spectral function of electronic system Us-ing Eq (15) and performing Matsubara frequency summation, the final expression for dynamical spectral function L11(ω) ≡ ℑL11(iωn−→ ω + i0+) of AA stacking gets the following form

L11(ω) = − 3t

2

∥

πω β

 +∞

−∞

dE(nF(E − ω) − nF(E))

k cos2(ky/2)

×cos(

√ 3kx)(2AB1A 1(k, E)A B1A 1(k, E − ω) + 2A A1B 1(k, E)A A1B 1(k, E − ω) + AB1A 2(k, E)A B2A 1(k, E − ω) + A B2A 1(k, E)A B1A 2(k, E − ω)

+ AA1B2(k, E)AA2B1(k, E − ω) + AA2B1(k, E)AA1B2(k, E − ω))

+ 4AA1A 2(k, E)A A1A 2(k, E − ω) + A A1A 1(k, E)A A2A 2(k, E − ω)

+ AB2B 2(k, E)A B1B 1(k, E − ω) + A B1B 1(k, E)A B2B 2(k, E − ω)

+ AB1B1(k, E)AA1A1(k, E − ω))

− 4AA1A 2(k, E)A A1A 2(k, E − ω) − A A1A 1(k, E)A A2A 2(k, E − ω)

− AB2B2(k, E)A B1B 1(k, E − ω) − A B1B 1(k, E)A B2B 2(k, E − ω)

where nF(x) = 1/(ex/kBT+ 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) InAppendixwe pres-ent the final results for dynamical transport coefficients L12(ω) and L22(ω) of AA-stacked bilayer graphene In the presence of a dynamical temperature gradient (∇T(ω)) and in open circuit situa-tion, 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) = 1

T2(L22(ω) − L

2

12(ω)

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.1 We have implemented a tight binding model Hamiltonian including local energy term so that this term describes the scattering of elec-trons 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 calculat-ing 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 ΣA 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 re-lates 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 1 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 (κA A−S BG(ω)) for different values of chemical potential µ/t∥is plotted in Fig.2 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.2 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 kBT/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

nifor higher values of vi/t∥= 0.4, 0.6, however it presents uniform behavior with niin 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

FIG 2 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 B T /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 di ffer-ence 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 4 we plot κA 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 conduc-tivity for any value of impurity concentration ni Higher temperature causes more scattering of

FIG 3 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 B T /t ∥ = 0.06 at fixed frequency ω/t = 2.0.

FIG 4 Dynamical thermal conductivity of biased AA stacked bilayer graphene (κ A A−S BG (ω)) as a function of normalized temperature k B T /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.4

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.5 A monotonically decreas-ing behavior for temperature dependence of thermal conductivity is clearly observed for each plot in

FIG 5 Dynamical thermal conductivity of biased AA stacked bilayer graphene (κ A A−S BG (ω)) as a function of normalized temperature k B T /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.

FIG 6 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 B T /t ∥ = 0.06 at fixed frequency ω/t ∥ = 2.0.

Fig.5 Moreover the variation of electron-impurity scattering strength has no remarkable effect on temperature behavior of κA 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 kBT/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 7 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 B T /t ∥ = 0.06 at fixed bias voltage

V /t = 1.25.