Scientia Iranica F (2015) 22(6), 2752{2756

Sharif University of TechnologyScientia Iranica

Transactions F: Nanotechnologywww.scientiairanica.com

Electronic transmission of benzene metallic junctions:Reliability range of perturbation values

R. Mohebbi and J. Seyed-Yazdi�

Department of Physics, Faculty of Science, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran.

Received 6 January 2015; received in revised form 23 July 2015; accepted 5 October 2015

KEYWORDSElectronic transport;Benzene;Metallic junctions;Perturbation theory.

Abstract. In this paper, the electronic transmission of an electrode-benzene-electrodesystem using equilibrium Green's and �rst order perturbation theory is investigated. Thee�ect of di�erent values of perturbation on substitution of the electrode into the di�erentatomic sites is presented. The range of the perturbation values are from zero to the energygap between the highest occupied and the lowest unoccupied molecular orbitals. Ourcalculations indicate that increasing the perturbation value does not greatly a�ect the casewhere the connection is with maximum transmission (para), whereas, doing likewise forthe case of minimum transmission (ortho), we �nd a much more sensitive transmissionresponse.

© 2015 Sharif University of Technology. All rights reserved.

1. Introduction

The new era of miniaturized circuits has paved the waytowards molecular electronics. Aviram and Ratner,in 1974, were the �rst people to calculate electronictransmission through a molecular bridge [1]. Thisbridge, in a donor-spacer-acceptor structure, may be-have as a diode [2]. Electronic transmission througha variety of di�erent molecules has been studied, andthis has paved the way for investigating the electronicbehavior of molecules and organic materials, and theirapplications in the technology of microelectronics [3]and biosensors [4]. In between molecules, benzene withnon-localized � orbitals represents a noble conductivity.Interference e�ects in transport through aromatic sys-tems in general and through benzene in particular havebeen studied both experimentally [5,6] and theoreti-cally [7-18]. The mechanism of molecular transmissioncalculation in this paper is by �rst calculating theGreen's function of an isolated molecule. Here, we

*. Corresponding author. Tel.: +98 913 7661445;E-mail address: j.seyedyazdi@vru.ac.ir (J. Seyed-Yazdi)

are working with a model Hamiltonian at the H�uckellevel of electronic structure and then locating themolecule between two metal electrodes. The e�ect ofconnection to the electrodes is then added to the H�uckelHamiltonian as a perturbation [7]. The value of thisperturbation a�ects the transmission, and the range ofits variation is between chemical potential reactivity,Highest Occupied Molecular Orbital (HOMO), andLowest Unoccupied Molecular Orbital (LUMO). Here,the e�ect of di�erent values of perturbation on di�erentsubstitution via two pathways is studied.

2. Method

The H�uckel Hamiltonian governing � orbitals of ben-zene is [7]:

H =6Xi=1

� jiihij+6Xi=1

��jiihi+ 1j

+ ji+ 1ihij�; (1)

R. Mohebbi and J. Seyed-Yazdi/Scientia Iranica, Transactions F: Nanotechnology 22 (2015) 2752{2756 2753

where jii represents � orbital on atom i, � is the onsiteenergy, which is usually set to zero, and � is the energyof jumping to the nearest neighboring atomic site. TheGreen's function of an isolated molecule derived fromthe Hamiltonian [7] is:

G0 = (E �H0 + i�)�1 ; (2)

where � is an in�nitesimal loading of the retardedGreen's function, and E is the energy eigenvalue. Theconnection to two metal electrodes is represented by animaginary part added to each atomic energy site [7]:

�! �� i�; (3)

where � is a spectral density and the negative shiftis reminiscent of a retarded Green's function. In thisstudy, we have used a wide band approximation inwhich the e�ect of connection is imaginary. So, thee�ective Hamiltonian of a connected molecule using theLowdin partitioning technique is [7]:

Heff ="�� i�=2 +

PAaa +

PBaa

PAab +

PBabPA

ba +PBba �� i�=2 +

PAbb +

PBbb

#;(4)

wherePA;Bab is the self-energy from pathways A and

B; a is the atomic site, to which the �rst electrodeis connected, and the second electrode b connects toatomic site 2, 3 or 4. Di�erent substitutions resultfrom coupling the electrodes to di�erent atomic sites(Figure 1). The Green's function of a perturbedmolecule via the o�-diagonal element of the Green'sfunction is evaluated using Dyson's equation as shownin Box I [7]. The molecular transmission is evaluatedwith [7]:

T (E) = �2 jG(E)j2 : (6)

In this paper, the e�ect of di�erent �'s on moleculartransmission is investigated.

3. Results and discussions

The studied system is demonstrated in Figure 1. The�rst electrode is �xed at atomic site 1 and the secondconnection can be substituted on di�erent atomic sites,

Figure 1. Di�erent pathways (A and B) and electrodesubstitution patterns (1! 2, 1! 3 and 1! 4) in thebenzene molecule.

Figure 2. Transmission function T (E), for di�erentperturbation values on ortho substitution. The reddashed-dot is for � = 0:2, blue � = 0:4, black � = 0:6,green � = 0:8, and solid red � = 1.

2, 3 or 4, to form, respectively, ortho (1 ! 2), meta(1! 3) and para (1! 4) substitutions. Transmissionof ortho substitution is plotted in Figure 2. The graphsfor � = 0:2 in all �gures are calculated transmissionsand they are in agreement with [7]. Very small valuesof perturbation (0.01) result in the �-function, likepeaks in a transmission plot. If the perturbation valueincreases, the width of peaks in the transmission plotincreases. When the perturbation had a real section,the peaks shifted. In an unperturbed molecule, thereare no anti-resonances, which can be seen in all plotsof the transmission. The reason is the interference oftwo di�erent spatial pathways (A and B) [7]. Increas-ing the value of perturbation broadens the width ofpeaks.

We have varied the value of perturbation (�)between 0.2 and 1, where 0.2 is the calculated transmis-

Gba =PAba +

PBba�

E � �+i�2�XA

aa�XB

aa

��E��+i

�2�XA

bb�XB

bb

��XA

ba+XB

ba

��XA

abA+

XB

ab

� : (5)

Box I

2754 R. Mohebbi and J. Seyed-Yazdi/Scientia Iranica, Transactions F: Nanotechnology 22 (2015) 2752{2756

Figure 3. Transmission function, T (E), for di�erentperturbation values on meta substitution. The reddashed-dot line is for � = 0:2, blue � = 0:4, black � = 0:6,green � = 0:8, and solid red � = 1.

sion in [7] and 1 is the energy gap between HOMO andLUMO orbitals. As � increases, the peaks broaden,but, we see no shift in peak positions, as the per-turbation value is purely imaginary. The validity ofthe �rst order perturbation is in the range of �, beingmuch smaller than 1. In Figure 2, di�erent values ofT on ortho substitution are illustrated. This variationhas in uenced this substitution more than that of thetwo others because of electron richness and the spatialsymmetry of substituting the two electrodes into thenearest neighbor of each.

In Figure 3, meta substitution, meaning locationof the second electrode on atomic site 3, is presented.It re ects the fact that increasing perturbation has lesse�ect than in the ortho situation. In Figure 4, parasubstitution, meaning location of the second electrodeon atomic site 4, is shown. As it has the maxi-mum transmission between these three substitutions,increasing the value of perturbation does not changethe width of the peaks too much. As a result, movingfurther from the �rst electrode increases transmissionand decreases the e�ect of perturbation.

The behavior of the electron with a scatteringregion (here, a benzene molecule) is like a potentialbarrier. The smallest barrier width is when theelectrodes are connected in a symmetric position (parasubstitution). As the connection gets far from thesymmetry, the barrier width increases and, therefore,reduces the probability of electron tunneling. As aresult, full symmetric substitution results in maximumtransmission. From the molecular orbitals point ofview, when para substitution is used, the overlap ofmolecular orbitals is in better agreement with theatomic orbitals of electrodes.

The e�ect of electron-electron interaction. Inorder to investigate the e�ect of electron-electron

Figure 4. Transmission function, T (E), for di�erentperturbation values on para substitution. The reddashed-dot line is for � = 0:2, blue � = 0:4, black � = 0:6,green � = 0:8, and solid red line � = 1.

interaction, we have placed the molecule betweentwo Au electrodes. The �rst principle calculationof electronic transport, based on Density FunctionalTheory (DFT) [19], has been used to calculate thetransmission (or the conductance) of nano-junctions.In this approach, we are treating the junction as amany body problem with electron-electron interac-tions. Transport properties are calculated using Non-Equilibrium Green's Function (NEGF) [20]. Usingthis approach, which is implemented in the Smeagol(non-equilibrium electronic transport) package [21], weare not considering the approximations mentioned inthe analytical approach. The results are depictedin Figure 5. In this �gure, we have plotted theProjected Density Of States (PDOS) of the carbonatom, which is connected to the right (or left) lead,and the peaks of the PDOS are in agreement withthe peaks of transmission. We have also revealedboth broadening and shifting in the peaks of trans-mission, in comparison with Figure 4, which are theresults of eliminating the approximations. Moreover,it should be noted that in Figure 4, four peaks arethe results of two di�erent pathways. However, inFigure 5, there is only one pathway and, therefore,the comparison is only between the positive sides ofFigures 4 and 5. Adding electron-electron interactionchanges our junction to a scattering region with manybody interactions.

4. Conclusions

In this paper, we have studied the transmission of anelectrode-benzene-electrode system. Di�erent substi-tution of electrodes results in di�erent transmissions.The greatest transmission is when the electrode is onpara substitution, and, the interference of di�erentpathways (A;B) results in new anti-resonances in thetransmission plot. The e�ect of the connection to

R. Mohebbi and J. Seyed-Yazdi/Scientia Iranica, Transactions F: Nanotechnology 22 (2015) 2752{2756 2755

Figure 5. The e�ect of electron-electron interaction onthe para substitution. The black and blue lines (that areon top of each other) are PDOS of right and left carbonatom connected to the electrodes, respectively. The redline is related transmission function.

the electrodes is an imaginary perturbation on theHamiltonian of an isolated molecule. So, increasingthis perturbation value broadens the peaks in thetransmission plot until the time at which this per-turbation value reaches the gap between HOMO andLUMO orbitals. As the perturbation value touchesthis energy gap, �rst order perturbation is invalid.We have also endeavored to indicate that whetherelectrode substitution with maximum transmission isless a�ected by increasing perturbation. Increasingperturbation has more in uence on ortho substitution,which has less transmission.

Acknowledgment

The authors would like to thank Dr. Alan Evans(University of Kent, Canterbury, UK) for helping withthe manuscript.

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Biographies

Razie Mohebbi obtained her BSc degree in Physicsfrom Vali-e-Asr University of Rafsanjan, Iran, and

2756 R. Mohebbi and J. Seyed-Yazdi/Scientia Iranica, Transactions F: Nanotechnology 22 (2015) 2752{2756

her MSc degree from Shahid Bahonar University ofKerman, Iran. She is currently a PhD candidate atVali-e-Asr University of Rafsanjan, Iran.

Jamileh Seyed-Yazdi obtained her BSc degree inPhysics from Ferdowsi University of Mashhad, Iran,her MSc degree from Shiraz University, Iran, and herPhD degree in Condensed Matter Physics from Iran

University of Science and Technology, Tehran. Shehas undertaken neutron scattering experiments andMD simulation at the University of Kent, Canterbury,UK, La Sapienza University, Rome, Italy, and VirginiaCommonwealth University, USA. She is currently As-sistant Professor of Physics at Vali-e-Asr University ofRafsanjan, Iran. Her present research interests includegraphene and other 2D electronic systems.