Theoretical verification of experimentally obtainedconformation-dependent electronic conductancein a biphenyl molecule

Santanu K. Maiti n

Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 Barrackpore Trunk Road, Kolkata 700 108, India

H I G H L I G H T S

� Conformation-conductance correla-tion is studied in a biphenyl mole-cule.

� A tight-binding framework is givento describe the molecular system.

� Two-terminal conductance is calcu-lated using Green's function techni-que.

� Theoretical results are in goodagreement with experimentalobservations.

G R A P H I C A L A B S T R A C T

1

3

4

6 5

2

12 11

10

98

7

Source Drain

Molecule

a r t i c l e i n f o

Article history:Received 20 November 2013Received in revised form4 March 2014Accepted 27 March 2014Available online 5 April 2014

Keywords:Conformation-dependent molecularconductanceBiphenyl moleculeGreen's function formalism

a b s t r a c t

The experimentally obtained (Venkataraman et al. [1]) cosine squared relation of electronic conductancein a biphenyl molecule is verified theoretically within a tight-binding framework. Using Green's functionformalism we numerically calculate two-terminal conductance as a function of relative twist angleamong the molecular rings and find that the results are in good agreement with the experimentalobservation.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

In a glorious experiment Venkataraman et al. [1] have estab-lished that electronic conductance of a molecular wire does notdepend only on the chemical properties of the molecule used, butalso on its conformation. It has been examined that for thebiphenyl molecule where two benzene rings are connected by asingle C–C bond, electronic conductance varies significantly withthe relative twist angle among these molecular rings. The con-ductance reaches a maximum for the planar conformation, while it

gets reduced with increasing the twist angle and eventually dropsto zero when the molecular rings are perpendicular to each other.The experimental results suggest a clear correlation betweenjunction conductance and molecular conformation which predictsthat the conductance of the biphenyl molecule decreases withincreasing twist angle obeying a cosine squared relation.

In this present communication we essentially verify theoreti-cally this conformation dependent molecular conductance andprove that our numerical results agree well with the experimentalrealization. A simple tight-binding (TB) Hamiltonian is given todescribe the model quantum system and we numerically computemolecular conductance using Green's function approach based onthe Landauer conductance formula [2]. Within a non-interactingelectron picture this framework is well applicable for analyzing

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/physe

Physica E

http://dx.doi.org/10.1016/j.physe.2014.03.0281386-9477/& 2014 Elsevier B.V. All rights reserved.

n Fax: þ91 33 2577 3026.E-mail address: santanu.maiti@isical.ac.in

Physica E 61 (2014) 125–128

electron transport through a molecular bridge system, as illu-strated by Aviram and Ratner [3] in their work where they havefirst described two-terminal electron transport through a mole-cule coupled to two metallic electrodes. Later several works [4–35]have been done to explore electron transfer through differentbridging molecular structures. A full quantum mechanicalapproach is needed [36] to study electron transport in suchmolecular bridge systems where transport properties are charac-terized by several key factors like, quantization of energy levels,quantum interference of electronic waves associated with thegeometry of bridging system adopts within the junctions andother several parameters of the Hamiltonian that are used todescribe a complete system.

Here we use a simple parametric approach [37–42] rather thanab initio methods to describe conformation-dependent electronconductance in a biphenyl molecule. The physical picture aboutconformation-conductance correlation that emerges from ourpresent study based on the single band TB model is exactly thesame as obtained in the experiment [1] and provides a very goodinsight to the problem.

The structure of the paper is as follows. In Section 2, wedescribe the molecular model and theoretical formulation for thecalculations. The essential results are analyzed in Section 3. Finally,in Section 4, we summarize our main results.

2. Molecular model and theoretical formulation

Let us refer to Fig. 1 where a biphenyl molecule is connected totwo semi-infinite one-dimensional (1D) non-interacting electro-des, commonly known as source and drain. The single particleHamiltonian for the entire system which describes the moleculeand side-attached electrodes becomes

H¼HMþHeleþHtun: ð1ÞThe first term HM represents the Hamiltonian of the biphenylmolecule coupled to source and drain electrodes. Within anearest-neighbor hopping approximation, the TB Hamiltonian ofthe molecule containing 12 (N¼12) atomic sites gets the form,

HM ¼∑iϵc†i ciþ∑

iv½c†iþ1ciþc†i ciþ1�

þ∑jϵc†j cjþ∑

jv½c†jþ1cjþc†j cjþ1�

þv4;7½c†4c7þc†7c4� ð2Þ

where the index i is used for the left ring and for the right ring weuse the index j. ϵ represents the site energy of an electron at i-(j-)th site and v gives the nearest-neighbor coupling strengthbetween the molecular sites. c†i (c

†j ) and ci(cj) are the creation and

annihilation operators, respectively, of an electron at the i-(j-)thsite. The last term in the right-hand side of Eq. (2) illustrates thecoupling among the molecular rings and in terms of the relativetwist angle θ between these two rings, the coupling strength v4;7 iswritten as v4;7 ¼ v cos θ.

Similarly the second and third terms of Eq. (1) denote the TBHamiltonians for the two semi-infinite 1D electrodes and theircouplings to the molecule. They are expressed as follows.

Hele ¼HSþHD

¼ ∑α ¼ S;D

∑nϵ0d

†ndnþ∑

nt0½d†nþ1dnþh:c:�

� �; ð3Þ

and,

Htun ¼HS;molþHD;mol

¼ τS½c†pd0þh:c:�þτD½c†qdNþ1þh:c:�: ð4Þ

The parameters ϵ0 and t0 correspond to the site energy andnearest-neighbor hopping integral in the source and drain elec-trodes. d†n and dn are the creation and annihilation operators,respectively, of an electron at the site n of the electrodes. Thehopping integral between the source and the molecule is τS, whileit is τD between the molecule and the drain. The source and drainare attached to the biphenyl molecule via the sites p and q,respectively, those are variable.

To calculate two-terminal conductance (g) we use the Landauerconductance formula g¼ ð2e2=hÞT , where the transmission func-tion T ¼ Tr½ΓS G

rM ΓD Ga

M� [36]. Here, GrM and Ga

M are the retardedand advanced Green's functions, respectively, of the moleculeincluding the effects of the electrodes. GM ¼ ðE�HM�ΣS�ΣDÞ�1,where ΣS and ΣD are the self-energies due to coupling of the chainto the source and drain, respectively, while ΓS and ΓD are theirimaginary parts.

3. Results and discussion

Throughout the analysis we choose the site energies in themolecule and side-attached electrodes to zero, ϵ¼ ϵ0 ¼ 0. Thenearest-neighbor hopping integral in the electrodes (t0) is set at2 eV, while in the molecule (v) it is fixed at 1 eV. The hoppingintegrals of the molecule to the source and drain electrodes(τS and τD) are also set at 1 eV. Here, we consider that the entirevoltage drop takes place across the molecule–electrode interfacesand it is a very good approximation for smaller size molecules.We also restrict ourselves at absolute zero temperature and choosethe units where c¼ e¼ h¼ 1. The energy scale is measured in unitof v.

Fig. 2 describes the variation of electronic conductance of thebiphenyl molecule for a typical energy as a function of twist angleθ when the source and drain electrodes are attached to themolecular sites 1 and 10, respectively. The results are shown fortwo different energy values. In (a) we set E¼0.25 eV, while in (b) itis fixed at 1.65 eV. The red dotted curves in the spectra aregenerated from the numerical results and they are superimposedon the blue dotted curves those are plotted from the cosinesquared relation A cos 2ðθÞ, where A is the conductance amplitudefor the planar conformation of the molecule. We evaluate thisamplitude A numerically. Very interestingly we notice that forE¼0.25 eV the red dotted curve sharply coincides with the blueone, and even for the other case i.e., when E¼1.65 eV the resultsare surprisingly close to each other. We also carry out extensivenumerical work for other possible energies within the allowedenergy band and find that the molecular conductance determinedfrom the Landauer conductance formula agrees well with thecosine squared relationship. Thus we can emphasize that ournumerical results can well fit the experimental data and providea detailed information of the behavior of the molecular conduc-tance on its conformation. Now the reduction of electronic con-ductance with the molecular twist can be clearly understood fromthe following interpretation. The degree of π-conjugation betweenthe molecular rings decreases with the rise of twist angle θ which

1

3

4

6 5

2

12 11

10

98

7

niarDecruoS

Molecule

Fig. 1. A biphenyl molecule attached to source and drain electrodes. The relativetwist among the molecular rings is described by the green arrow. (For interpreta-tion of the references to color in this figure caption, the reader is referred to theweb version of this paper.)

S.K. Maiti / Physica E 61 (2014) 125–128126

results a reduction of the molecular conductance because thetransfer rate of electrons through the biphenyl molecule scales asthe square of the π-overlap [43]. At the typical case when θreaches to π=2, the π-conjugation between the molecular rings

vanishes completely, and therefore, the conductance drops to zero.Obviously, it becomes a maximum for the planar conformation(θ¼ 01) of the molecule. Thus, twisting one molecular ring withrespect to the other electronic transmission through the biphenylmolecule may be controlled and eventually one can reach to theinsulating phase. This phenomenon leads to a possibility of gettinga switching action using this molecule.

Similar observations are presented in Fig. 3 when the sourceand drain electrodes are coupled to the molecule at the sites 5(p¼5) and 8 (q¼8), respectively. All the other parameters are thesame as in Fig. 2. It is interesting to note that the quantuminterference does not destroy the cosine squared dependencebetween junction conductance and molecular conformation,which proves the robustness of the conformation-conductancecorrelation. Our numerical results corroborate the experimentalfindings [1].

Before we end, it should be pointed out that though the resultspresented in this communication are worked out for absolute zerotemperature, they should be valid even for finite temperatures(�300 K) as the broadening of the energy levels of the biphenylmolecule due to its coupling with the electrodes will be muchlarger than that of the thermal broadening [36–40]. Throughoutour work, we numerically compute electronic conductance of themolecule for a typical set of parameter values and in our modelcalculations we choose them only for the sake of simplicity.Though the results presented here change numerically with theseparameter values, but all the basic features remain exactly invar-iant which we confirm through our extensive numericalcalculations.

4. Conclusion

In the present communication we have theoretically verifiedthe experimentally obtained [1] cosine squared relation of elec-tronic conductance in a biphenyl molecule within a tight-bindingframework. Using Green's function formalism we have numeri-cally calculated two-terminal conductance as a function of relativetwist angle among the molecular rings and found that the resultsare in very good agreement with the experimental observation.We have carried out extensive numerical work for all possibleenergies within the allowed energy band and noticed the mole-cular conductance determined from the Landauer conductanceformula agrees well with the cosine squared relationship. We havealso verified our results by coupling the electrodes at differentpositions of the molecule. Interestingly we have seen that thequantum interference does not destroy the cosine squared depen-dence between junction conductance and molecular conforma-tion, which justifies the robustness of the conformation-conductance correlation. Our numerical results exactly match theexperimental findings [1].

In short, we feel that there is much need for a clean exposure tothe conformation-conductance correlation in the molecular sys-tems and the experimental methods and observations need to beexplained consistently from theory – which is precisely ourmotivation.

References

[1] L. Venkataraman, J.E. Klare, C. Nuckolls, M.S. Hybertsen, M.L. Steigerwald,Nature 442 (2006) 904.

[2] R. Landauer, IBM J. Res. Dev. 1 (1957) 223.[3] A. Aviram, M. Ratner, Chem. Phys. Lett. 29 (1974) 277.[4] A. Nitzan, M.A. Ratner, Science 300 (2003) 1384.[5] S. Woitellier, J.P. Launay, C. Joachim, Chem. Phys. 131 (1989) 481.[6] K. Tagami, L. Wang, M. Tsukada, Nano Lett. 4 (2004) 209.[7] P. Orellana, F. Claro, Phys. Rev. Lett. 90 (2003) 178302.

Fig. 2. Electronic conductance (in unit of g0 ¼ e2=h) for a specific energy as afunction of twist angle for the biphenyl molecule when the electrodes areconnected at the molecular sites 1 (p¼1) and 10 (q¼10), as shown in Fig. 1. Theresults are computed for two typical energy values where we choose E¼0.25 eV in(a) and in (b) we fix the energy E at 1.65 eV. The red dotted curve, drawn fromnumerical results, is superimposed on the blue dotted curve generated from thecosine squared relation: A cos 2ðθÞ, where A represents the conductance amplitudeat θ¼ 01. (For interpretation of the references to color in this figure caption, thereader is referred to the web version of this paper.)

Fig. 3. Electronic conductance (in unit of g0 ¼ e2=h) for the same characteristic ofFig. 2, when p¼5 and q¼8. (For interpretation of the references to color in thisfigure legend, the reader is referred to the web version of this article.)

S.K. Maiti / Physica E 61 (2014) 125–128 127

[8] J.H. Ojeda, R.P.A. Lima, F. Domínguez-Adame, P.A. Orellana, J. Phys.: Condens.Matter 21 (2009) 285105.

[9] J.H. Ojeda, R.R. Rey-González, D. Laroze, J. Appl. Phys. 114 (2013) 213702.[10] J.H. Ojeda, M. Pacheco, L. Rosales, P.A. Orellana, Org. Electron. 13 (2012) 1420.[11] M. Araidai, M. Tsukada, Phys. Rev. B 81 (2010) 235114.[12] K. Walczak, Cent. Eur. J. Chem. 2 (2004) 524.[13] P. Dutta, S.K. Maiti, S.N. Karmakar, Org. Electron. 11 (2010) 1120.[14] M. Dey, S.K. Maiti, S.N. Karmakar, Org. Electron. 12 (2011) 1017.[15] S.K. Maiti, Physica B 394 (2007) 33.[16] S.K. Maiti, Solid State Commun. 150 (2010) 1269.[17] S.K. Maiti, Eur. Phys. J. B 86 (2013) 296.[18] M. Magoga, C. Joachim, Phys. Rev. B 59 (1999) 16011.[19] J.-P. Launay, C.D. Coudret, in: A. Aviram, M.A. Ratner (Eds.), Molecular

Electronics, New York Academy of Sciences, New York, 1998.[20] S.K. Maiti, Solid State Commun. 145 (2008) 126.[21] S.K. Maiti, J. Nanosci. Nanotechnol. 8 (2008) 4096.[22] S.K. Maiti, J. Comput. Theor. Nanosci. 6 (2009) 710.[23] S.K. Maiti, Phys. Lett. A 366 (2007) 114.[24] R.H. Goldsmith, M.R. Wasielewski, M.A. Ratner, J. Phys. Chem. B 110 (2006)

20258.[25] D. Vonlanthen, A. Mishchenko, M. Elbing, M. Neuburger, T. Wandlowski,

M. Mayor, Angew. Chem., Int. Ed. 48 (2009) 8886.[26] A. Mishchenko, D. Vonlanthen, V. Meded, M. Bürkle, C. Li, I.V. Pobelov,

A. Bagrets, J.K. Viljas, F. Pauly, F. Evers, M. Mayor, T. Wandlowski, Nano Lett.10 (2010) 156.

[27] M. Bürkle, J.K. Viljas, D. Vonlanthen, A. Mishchenko, G. Schon̈, M. Mayor,T. Wandlowski, F. Pauly, Phys. Rev. B 85 (2012) 075417.

[28] H. Kondo, J. Nara, H. Kino, T. Ohno, Jpn. J. Appl. Phys. 47 (2008) 4792.[29] Y. Xue, M.A. Ratner, Phys. Rev. B 68 (2003) 115407.[30] M.A. Reed, C. Zhou, C.J. Muller, T.P. Burgin, J.M. Tour, Science 278 (1997) 252.[31] J. Chen, M.A. Reed, A.M. Rawlett, J.M. Tour, Science 286 (1999) 1550.[32] T. Dadosh, Y. Gordin, R. Krahne, I. Khivrich, D. Mahalu, V. Frydman, J. Sperling,

A. Yacoby, I. Bar-Joseph, Nature 436 (2005) 677.[33] C.M. Fischer, M. Burghard, S. Roth, K.V. Klitzing, Appl. Phys. Lett. 66 (1995)

3331.[34] X.D. Cui, A. Primak, X. Zarate, J. Tomfohr, O.F. Sankey, A.L. Moore, T.A. Moore,

D. Gust, G. Harris, S.M. Lindsay, Science 294 (2001) 571.[35] J.K. Gimzewski, C. Joachim, Science 283 (1999) 1683.[36] S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University

Press, Cambridge, 1997.[37] S.K. Maiti, Org. Electron. 8 (2007) 575.[38] S.K. Maiti, Solid State Commun. 149 (2009) 1684.[39] S.K. Maiti, Solid State Commun. 149 (2009) 2146.[40] S.K. Maiti, Solid State Commun. 149 (2009) 973.[41] S.K. Maiti, Physica E 36 (2007) 199.[42] S.K. Maiti, Solid State Commun. 149 (2009) 1623.[43] A. Nitzan, Annu. Rev. Phys. Chem. 52 (2001) 681.

S.K. Maiti / Physica E 61 (2014) 125–128128