Scientia Iranica F (2014) 21(3), 1179{1182

Sharif University of TechnologyScientia Iranica

Transactions F: Nanotechnologywww.scientiairanica.com

Energy loss spectra of doped graphene and armchaircarbon nanotubes at �nite temperature

H. Rezaniaa;� and F. Taherkhanib

a. Department of Physics, Razi University, Kermanshah, P.O. Box 0786534239456, Iran.b. Department of Chemistry, Razi University, Kermanshah, P.O. Box 075764684762, Iran.

Received 19 May 2014; accepted 18 August 2014

KEYWORDSEnergy loss;Armchair;Random phaseapproximation;Graphene;Nanotube.

Abstract. We have studied the dynamical polarizability associated to energy loss spectraof armchair carbon nanotubes in the context of tight binding model Hamiltonian includinglong range electronic interaction. Linear response theory in the context of random phaseapproximation has been implemented to obtain charge response function via calculatingcorrelation function of density operators. The e�ects of both temperature and electrondoping as well as the diameter on the frequency behaviour of charge response have beeninvestigated. The sharp peak in the energy loss spectra of the armchair carbon nanotubedisappears upon raising the temperature. This is not the case of non interacting case wheretemperature has no considerable e�ect on the dynamical susceptibility. Also, the similarcalculations have been performed for graphene sheet.c 2014 Sharif University of Technology. All rights reserved.

1. Introduction

Carbon nanotubes (CNTs) are newly realized two andone dimensional electronic systems that have attractedgreat deals in the scienti�c community because of newphysics they exhibit and because of their potentialas new materials for technology [1]. This compound,which is made of a single graphite layer rolled upinto a hollow cylinder, is a one dimensional allotropeof carbon. In order to study electronic propertiesof electron gas in the nano structures and graphene,the dynamical polarizability, whereby the screeninge�ects have been found, is required [2]. The dynam-ical polarizability renormalizing the phononic Green'sfunction can explain the phonon softening and Kohnanomaly phenomenon [1] at the � point. So far, theextensive studies have been performed on the chargeresponse function of monolayer Graphene in the Dirac

*. Corresponding author. Tel.: +98 8334274556;Fax:+98 8334274556E-mail address: rezania.hamed@gmail.com (H. Rezania);bf.taherkhani@gmail.com (F. Taherkhani)

cone approximation so that the energy dispersion ofthe hexagonal lattice is considered to be linear formin terms of electronic wave vector. In the case ofundoped graphene at zero temperature, the polariz-ability has been found by Gonzalaz [3]. Their resultsdemonstrated the uctuations of density in a bilayercase can present either single-component massive-chiralcharacter or standard two layer character, dependingon energy and doping. But, the e�ect of full banddispersion over all the Brillouin zone on the chargeresponse of nano structures, such as nanotubes, isexpected to be signi�cant. In the obtaining of highenergy corrections to the charge response, both interand intraband transitions contribute to the frequencybehavior of dynamical polarizability. Recently, a the-oretical work provides a study of both imaginary andreal part of non-interacting polarizability of graphenewithin an analytical approach [4]. The authors foundthat there is no noticeable angle dependence forimaginary part of polarizability around the van hovesingularity, i.e ~!

t = 2 where t implies nearest neighborhopping integral. An extensive study on the behavior ofdynamical polarizability in various wave vector regions

1180 H. Rezania and F. Taherkhani/Scientia Iranica, Transactions F: Nanotechnology 21 (2014) 1179{1182

for graphene sheet beyond Dirac approximation hasbeen performed by Peres et al. [4]. In the presentpaper, we will extend the Stauber's work [4] so thatthe results of polarizability or energy loss spectra forthe other nano structures, such as armchairs CNTs,are presented. Here, we apply a random phase ap-proximation to �nd the analytical form of polarizabilityfor CNTs at �nite temperature. In addition, to derivethe results of charge response for graphene monolayerperformed by Stuber and co-workers [4], we also obtainthe noninteracting charge response of CNTs for variousamounts of temperature, concentration and diameter.Interacting charge response function can be readilyderived by using RPA approximation in which quantumcorrelation e�ects on the charge response are neglected.Also we investigate the plasmon excitations of electronsof CNTs for both interacting and noninteracting casesat �nite temperature. The e�ects of diameters andelectronic concentrations on the frequency behaviorof polarizability as criterion of electron energy lossfunction and the collective electronic excitation havebeen also addressed.

2. Materials and method

The dynamics of electrons on the honeycomb latticeas graphene or CNT is described by the followingtight binding model including a long-range Coulombinteraction term:

H =Xk;�

�kc+�k;Ack;B + h:c:+Xq;�;�

vq��(q)��(�q); (1)

where vq = 2�e2=q and �q;� are the Fourier transfor-mation of Coulomb potential in two dimensional andelectronic density operator, respectively. N is the num-ber of unit cells in honeycomb lattice. Furthermore,we have �k = 1 + cos(kx=2)e�iky

p3=2 where kx and ky

belong to the �rst Brillouin zone of honeycomb lattice.The quantity of interest for studying many parti-

cle properties such as plasmon oscillations and phononsoftening is the dynamical charge response function.Also this quantity determines the e�ective electron-electron interaction and the Friedel oscillations. Linearresponse theory gives us the noninteracting chargeresponse based on the correlation function of density-density operators as:

�(0)(q; i!n) = � 1A

X�;�=a;b;�

d�ei!n�

hT (���(q; �)���(�q; �))i=2X�;�

Z �

0ei!n��(0)

��(q; �);(2)

where � denotes the spin degree of freedom of electron.Also electronic density operator ���(q) is de�ned as:

���(q) =Xk

c+k+q;�ck;�:

Since paramagnetic phase is considered for CNT andgraphene sheet, the contributions of two di�erent spincomponents to the charge response are the same.Therefore the summation over � is readily performedand coe�cient 2 comes from spin degeneracy. AlsoA is the area of the nanostructure sample. !n =2n�=� denotes the Bosonic Matubara frequancy inwhich � is the inverse of equilibrium temperature.Wick's theorem has been applied to write the chargeresponse in terms of matrix elements of non-interactingelectronic Green's function. Using the Fourier trans-formation in Matsubara's representation [2], chargeresponse function can be written in the following form:

�(0)�� =

1N�

Xk;m

G(0)��(k + q; i!m)G(0)

��(k; i!m + i!m);(3)

where !m = (2m + 1)�=� denotes the fermionic Mat-subara frequency. According to the model Hamiltonianintroduced in Eq. (1), the elements of electronic Green'sfunction in the Fourier space get the following forms [4]:

G(0)AA(k; i!m)=G(0)

BB(k; i!m)=X�=�

12((i!m)� ��(k))

;

G(0)AB(k; i!m) =

X�=�

��2�+

�2((i!m)� ��(k))

;

G(0)AB(k; i!m) =

X�=�

�2�+

�2((i!m)� ��(k))

; (4)

where the factors in the above equations are given by:

�� = t�j�(k)j;�(k) = 1 + cos(kx=2)e�iky

p3=2:

�� introduces each of electronic band energies of hon-eycomb lattice.

Substituting Eq. (4) into Eq. (3) and performingMatsubara frequency summation over fermionic Mat-subara's energies gives us the following expressions forone of the elements of dynamical susceptibilty tensor:

�(0)AB(q; i!n) =

14N

Xk

nF (��;k+q)� nF (��0;k)i!n + ��;k+q � ��0;k ; (5)

where nF (E) = 1=(e�E + 1) is the Fermi-Dirac dis-tribution function. The charge response of electronson the honeycomb lattice has been found after sum-mation over the charge response elements in Eq. (5).Random phase approximation [2] has been employedto correct charge response function of electronic system

H. Rezania and F. Taherkhani/Scientia Iranica, Transactions F: Nanotechnology 21 (2014) 1179{1182 1181

in the presence of electron-electron interaction part ofHamiltonian in Eq. (1), and the interacting dynamicalresponse function is given by:

�(q; i!n) =�(0)(q; i!n)

1� vq�(0)(q; i!n): (6)

Under a simple analytical continuition as:

i!n ! ! + i0+;

retarded charge response function has been obtainedfrom its Matsubara form. In the next section, wepresent the results of interacting and noninteractingcharge response function for armchair CNT with vari-ous diameters and graphene sheet.

3. Numerical results

In this section, we present the numerical results ofenergy loss spectra corresponding to the imaginarypart of dynamical polarizability introduced in Eqs. (5)and (6) for both armchair CNT and graphene sheet.We address the e�ects of electronic interaction on theeneregy loss spectra within RPA approximation. Thee�ects of electron doping associated to the increase ofchemical potential, �=t, on the noninteracting energyloss spectra at wave vector, q0, in low energy region,0 < !=t < 0:5, have been presented in Figure 1. Theposition of the point q0 with amplitude jq0j = 1=(10

p3)

and polar angle ' = �=6 into the Brillouin zone hasbeen shown in Figure 1. The behavior of noninteractingenergy loss spectra is free of chemical potential e�ectsat high normalized frequencies and we have not shownit here. As it is obvious in Figure 1, a collectivemode associated with resonances in the charge responsefunction is excited around normalized frequency !=t �0:16. As it is known, the imaginary part of charge

Figure 1. Noninteracting energy loss spectra of graphemesheet as a function of normalized energy !=t for di�erentvalues of chemical potential �

t at a particular wave vectorq0. The normalized temperature is assumed to bekTt = 0:05.

response function is proportional to the electronicenergy loss spectra [2]. Therefore, the position of peaksin the energy loss spectra is associated with the energyamounts that cross section of scattering of particlesfrom sample get its maximum value. A novel featureshown in Figure 1 is the vanishing of frequency gap inthe energy loss with electronic concentration.

The e�ect of electron doping on the behaviorof energy loss spectra of armchair CNT (9,9) in thenoninteracting regime for wave vector q1 has beenpresented in Figure 2. The wave vector q1 is one ofthe points in the Brillouin zone with high symmetry.By increasing the electronic concentration, the widthof frequency gap in Im�(0)(q1; !) decreases. Energyloss spectra for half �lling case gets a zero value in theenergy region 0:0 < !

t < 1:0 while the energy gaprange is in region 0:0 < !

t < 0:5 for electron dopedcase �

t = 1:7. Figure 2 also implies plasmon modes ofarmchair CNT (9,9) shift to the lower frequency uponincrease of chemical potential. We have also studiedthe e�ect of diameter on the noninteracting energy lossof armchair CNT for half �lling case at kT

t = 0:01.In Figure 3, we plotted the frquency behavior

Figure 2. Energy loss spectra of CNT (9,9) for variouschemical potentials.

Figure 3. Interacting energy loss spectra of armchairCNT with di�erent diameters.

1182 H. Rezania and F. Taherkhani/Scientia Iranica, Transactions F: Nanotechnology 21 (2014) 1179{1182

of interacting energy loss spectra for three di�er-ent diamters of armchair CNT at half �lling caseand kT

t = 0:05. The frequency gap in the en-ergy loss spectra is approximately independent ofthe tube's diameter. Moreover, the intensity of en-ergy loss spectra decreases with the radios of nan-otube, and the peak position moves toward higherfrequency.

4. Conclusions

In conclusion, we have presented the frequency de-pendence of the energy loss spectra of armchair CNTand monolayer grapheme. Using random phase ap-proximation and a Green's function approach, theenergy loss spectra has been studied. In particular,the e�ect of electron doping and temperature havebeen investigated. We have found that the electronicconcentration has a major e�ect on the energy lossspectra. Also the results show that the increase ofdiameter leads to the intensity reduction of energy lossspectra of armchair CNT.

References

1. Ijiima, S. \Helical micronanotuble of graphitic carbon",Nature, 354(2), pp. 56-58 (1991).

2. Mahan, G.D. Many Particle Physics, 3rd Ed., pp. 295-331, Plenumn Press, New York, USA (1993).

3. Gonzalez, J. \Marjinal fermi-liquid behavior from twodimensional Coulomb interaction", Phys. Rev. B, 59(3),pp. R2474-2477 (1999).

4. Peres, N.M., Araujo, M.A.N. and Bozi, D. \Phase dia-gram and magnetic collective excitation of the Hubbardmodel in graphene sheet and layers", Phys. Rev. B,70(2), pp. 195122-195133 (2004).

Biographies

Hamed Rezania is Assistant Professor in theoreticalcondensed matter physics department of Razi Uni-versity of Kermanshah in Iran. His research inter-ests include: The study of electronic properties ofnanostructure systems and transport properties of spinmodel Hamiltonian as strongly correlated systems areof my interests.

Farid Taherkhani is Assistant Professor in Theoreti-cal Physical Chemistry Department of Razi universityof Kermanshah in Iran. His research interests include:molecular dynamics method in transport properties ofnanostructure systems using density functional theortmethod.