PHYSICAL REVIEW B 83, 195415 (2011)

Efficiency of thermoelectric energy conversion in biphenyl-dithiol junctions:Effect of electron-phonon interactions

Nikolai Sergueev,1,2 Seungha Shin,2 Massoud Kaviany,2 and Barry Dunietz1,*

1Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109, USA2Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA(Received 3 November 2010; revised manuscript received 1 February 2011; published 9 May 2011)

The electron-phonon interaction is the dominant mechanism of inelastic scattering in molecular junctions.Here we report on its effect on the thermoelectric properties of single-molecule devices. Using density functionaltheory and the nonequilibrium Green’s function formalism we calculate the thermoelectric figure of merit fora biphenyl-dithiol molecule between two Al electrodes under an applied gate voltage. We find that the effectof electron-phonon coupling on the thermoelectric characteristics strongly varies with the molecular geometry.Two molecular configurations characterized by the torsion angles between the two phenyl rings of 30◦ and 90◦

exhibit significantly different responses to the inelastic scattering. We also use molecular dynamics calculationsto investigate the torsional stability of the biphenyl-dithiol molecule and the phonon thermal transport in thejunction.

DOI: 10.1103/PhysRevB.83.195415 PACS number(s): 81.07.Nb, 68.37.Ef, 72.10.−d, 73.63.−b

I. INTRODUCTION

Charge and energy transport at the nanoscale is of cur-rent interest with implications in electronic, structural, andtransport properties of electronic devices.1–12 These alsoinclude applications based on the thermoelectric (TE) effectwhere thermal energy is converted into electrical energy.However, the TE effect has been mainly observed in bulkmaterials. Recent experiments13–16 on atomic and molecularjunctions have demonstrated that nanoscale devices can alsoexhibit a TE behavior. These experiments have inspiredtheoretical interest in predicting the TE properties of nanoscalejunctions.17–23 One of the key TE characteristics is the Seebeckcoefficient, which measures the magnitude of the electric po-tential generated in response to a temperature gradient acrossthe junction. The Seebeck coefficient is not only the keyto the TE efficiency of the device. It also provides insightinto the conductance mechanism. The overall efficiency ofthermal-to-electric energy conversion is described by thedimensionless figure of merit (ZT) which depends on electricaland thermal conductance in addition to the Seebeck coefficient.

As in the bulk materials, the charge transport in molecularjunctions is affected by the phonons. An electron interactswith phonons (molecular vibrations) and tunnel through thejunction inelastically, i.e. loosing or gaining energy. Theelectron-phonon (e-p) interaction substantially alters the trans-mission probability and leads to various inelastic effects suchas heat dissipation. The inelastic scattering and the effect ofe-p interaction in molecular scale junctions have been studiedextensively, both theoretically5,24–28 and experimentally.29–33

Despite the remarkable progress in theoretical modeling ofTE junctions, most of the studies are based on the elasticdescription of the electron tunneling processes. Others usescattering theory approaches to treat phonons.34,35 RecentlyGalperin et al.17 reported on the strong effect of the e-pcoupling on the calculation of the Seebeck coefficient using aself-consistent approach to treat molecular vibrations with asimple phenomenological model.

The purpose of this paper is to show the effect of inelasticscattering on the TE characteristics of molecular junctions

(fully from first principles). This requires that in calculatingthe ZT figure of merit the Seebeck coefficient and electricalconductance are calculated with the e-p interaction included.We use an atomistic description of a molecular junctionwithin density functional theory (DFT) and the nonequilibriumGreena’s function (NEGF) formalism for both electron andphonon transport. Self-consistency is obtained between theelectronic Hamiltonian and charge density under constant e-pcoupling. Our junction model consists of a biphenyl-dithiolmolecule sandwiched between two Al electrodes, as shownschematically in Fig. 1. Below, we show that the TE propertiesare greatly affected by electron inelastic scattering. We showthat the qualitative effect of e-p interactions on transportdepends strongly on the junction configuration.

The rest of the paper is organized as follows. We beginwith a theoretical model, including the theory of inelastic scat-tering, thermoelectricity, and phonon propagation within theharmonic approximation. Each theory describes an importantingredient in evaluating a ZT factor. Section III contains thenumerical results for the biphenyl-dithiol molecular junctionand discusses the dependence of the TE properties on the e-pcoupling, electron and phonon thermal transport, and the roleof anharmonicity on the phonon conductivity. Section IV is asummary.

II. THEORETICAL MODEL

A. Electron transport within DFT-NEGF

We start by writing the total Hamiltonian of the moleculartunneling junction including the e-p interaction in the follow-ing form:

H = He + Hp + He−p, (1)

where He is the electronic contribution represented by theKohn-Sham (KS) Hamiltonian. Hp is the phonon Hamiltonianwhich is written within the second quantization as

Hp =∑

ν

hων

(b†ν bν + 1

2

), (2)

195415-11098-0121/2011/83(19)/195415(12) ©2011 American Physical Society

SERGUEEV, SHIN, KAVIANY, AND DUNIETZ PHYSICAL REVIEW B 83, 195415 (2011)

pe

T

T T+ Δ

VΔ e−p

Torsion angle)b()a(

90°

30°

(c)Biphenyl-dithiol

Hot electrode (Al)

Cold electrode (Al)

SCH

On configuration

Off configuration

Gate electrode

Gate electrode

FIG. 1. (Color online) (a) The biphenyl-dithiol molecule betweentwo Al electrodes and the electron and phonon energy carriers andtheir coupling. The molecular on (b) and off (c) states are characterizedby the torsion angles (angle between the planes of the phenyl rings)of 30◦ and 90◦, respectively.

where b†ν (bν) is the creation (annihilation) operator of themolecular vibrational mode ν. By “phonons” we refer toquantized molecular vibrations in the junction region. The lastpart of Eq. (1), He−p, represents the interaction of electronswith phonons in the junction region and is calculated as

He−p =∑

ν

∑ij

Mνij |φi〉〈φj |(b†ν + bν). (3)

Here the e-p coupling matrix Mν is expressed using localizedatomic orbital (LCAO) basis functions (|φi〉) as

Mνij =

∑α

(h

2mαων

)1/2

eνα〈φi | ∂He

∂Rα

|φj 〉, (4)

where eν and ων are the eigenvectors and eigenfrequencies ofthe dynamical matrix K,

K = {Kαβ} = (mαmβ)−1/2 ∂2Etot

∂Rα∂Rβ

. (5)

The quantum transport is well represented in the space ofthe eigenvectors projected onto the junction model region.The bridging region includes the six outermost surface layersof the functionalized surface with a molecular bridge asdescribed in detail below.

Once the total Hamiltonian is defined, we describe many-body effects using the Green’s functions. In the case where noe-p interactions are considered (i.e., the e-p coupling matrixMν is zero for all the vibrational modes) the electrons andphonons can be treated separately. In this limit, the bareretarded electron Green’s function becomes

GR0 (E) = [

ES − HKS − �RL(E) − �R

R(E)]−1

. (6)

Here �RL/R is the retarded self-energy that describes interaction

of the scattering region with the left/right (L/R) electrode.Under nonequilibrium effects resulting from an applied bias,the lesser nonequilibrium electron Green’s functions areevaluated using

G<0 (E) = iGR

0 (E)[�<L (E) + �<

R (E)]GA0 (E). (7)

Here GA0 = (GR

0 )† is the bare advanced electron Green’sfunction. The bias-driven electron injection into the junctionregion is represented by the �<

L/R as

�<L/R(E,μL/R) = ifL/R(E,μL/R)�L/R(E)

= −fL/R(E,μL/R)[�R

L/R(E) − �AL/R(E)

].

(8)

In the last expression we denote by fL/R the Fermi distributionfunction of the left/right electrode shifted by the chemical po-tential μL/R and by �L/R the electrode broadening functions.The lesser Green’s function then also depends on the chemicalpotentials. In what follows we may omit the notations for thechemical potentials and for the electron energy for simplicitywhen needed.

Having defined the electronic Green’s functions, we nextconsider the bare phonon ones. In what follows we assume aninfinite lifetime of phonons, i.e., the characteristic time scaleof electronic processes is much smaller than that of phononprocesses. Electrons repeatedly scatter off the phonons beforeundergoing any energy changes. Within this approximationthe bare retarded and nonequilibrium Green’s functions of thephonon with the frequency ων are defined as

DR0,ν(�) = 1

� − hων + iη− 1

� + hων + iη(9)

and

D<0,ν(�) = −2πi[(Nν + 1)δ(� + hων)

+Nνδ(� − hων)], (10)

where Nν is the phonon occupation number, δ(� ± hων) isthe Dirac delta function, and η is an infinitesimally small realnumber. In this paper we also assume an equilibrium phonondistribution, i.e., the phonon occupation number is given bythe Bose-Einstein distribution function Nν = nB(ων).

We now consider the effect of e-p interaction on theelectronic system. In the presence of e-p interaction wesolve the many-body problem perturbatively, i.e., the electronGreen’s functions are corrected by self-energies representingthe e-p coupling. We use the lowest-order Feynman diagramsand the Langreth rules for analytical continuation to calculatethe e-p self-energies.36 In this approach, both the retardedand the nonequilibrium self-energies for the electron Green’sfunction corresponding to the Fock diagram are written as

�Re−p(E) = ih

∑ν

∫d�

2πMν

[G<

0 (E − �)DR0,ν(�)

+ GR0 (E−�)DR

0,ν(�) + GR0 (E−�)D<

0,ν(�)]Mν

(11)

�<e−p(E) = ih

∑ν

∫d�

2πMνG<

0 (E − �)D<0,ν(�)Mν . (12)

In this paper we neglect the self-energy contribution dueto the Hartree diagram as it is frequency independent andthus constant.36 In fact, it is often omitted, as, e.g., inapproaches based on the self-consistent Born approximation(SCBA).26,37–40 Such a contribution is expected to renormalizethe electron transmission function, leaving its derivative

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EFFICIENCY OF THERMOELECTRIC ENERGY . . . PHYSICAL REVIEW B 83, 195415 (2011)

intact.41 Given the e-p interaction self-energies, the electronGreen’s functions are written as

GR(E) = [ES − HKS − �R

L(E) − �RR(E) − �R

e−p(E)]−1

(13)

and

G<(E) = GR(E)[�<

L (E) + �<R (E) + �<

e−p(E)]GA(E).

(14)

As described above, the electrodes’ and e-p coupling effectson the electronic density are represented by the self-energy ex-pressions in the corresponding Green’s functions expressions.This requires that the perturbative treatment considers theself-consistency between the self-energies and the calculatedGreen’s functions. While the e-p self-energies are obtainedusing the electronic Green’s functions [see Eqs. (11) and (13)]we restrict our treatment to the linear term in the expansionover the e-p self-energies, namely, the e-p self-energies arekept constant. An update of the e-p self-energies is required toguarantee current conservation and is invoked, for example, inthe SCBA approaches.26,38,40 The self-consistent treatment ofthe self-energies is, however, rather involved and beyond thescope of this work. The remaining self-consistent aspect, andin our opinion also the more important, is the relationshipbetween the Green’s function and the perturbed electroniccharge density under the effect of bulk coupling. In theimplemented DFT-NEGF combined formalisms, the electronicHamiltonian is a functional of the electron density extractedfrom the nonequilibrium Green’s functions,36,42–44

ρ = − 1

2π

∫dE G<(E). (15)

Therefore, we compute the e-p self-energies [see Eqs. (11) and(12)] using the bare unperturbed electron Green’s functionsand enforce the self-consistency between the biased chargedensity and the Green’s functions.

In DFT-NEGF the density matrix and the electronicHamiltonian are self-consistently iterated under the effect ofcoupling to the voltage-biased electrodes. In this scheme theself-consistency accounts for the nonequilibrium statistics ofthe injected electrons. Once the iterative solution is obtained,the transport properties of the tunneling junction are thenobtained from the fully responded electronic density. Theelectron current flowing into the scattering region from, e.g.,the left electrode can be generally written in terms of theGreen’s functions as36,45

J = 2e

h

∫dE Tr[�<

L (E)G>(E) − �>L (E)G<(E)]. (16)

The electron current can also be written in terms of thetransmission function as

J = 2e

h

∫dE Te(E,Vb)[fL(E,μL) − fR(E,μR)], (17)

where Vb is the bias applied across the junction, defined asμL − μR . The transmission function Te can be written as asum of elastic and inelastic components,

Te = T ele + T inel

e . (18)

The elastic component is given by the following relation:46

T ele = Tr[�LGR�RGA], (19)

where �L/R is the broadening function due to the interaction ofthe scattering region with the left/right electrode. The inelasticcontribution to the transmission function is written similarlyas

T inele = Tr[�LGR�e−p,RGA]. (20)

The �e−p is the total broadening function due to the e-pinteraction and can be expressed as a sum of two contact terms�e−p = �e−p,L + �e−p,R .47

B. Calculation of the thermoelectric figure of merit

The efficiency of the TE energy conversion is usuallydescribed by the dimensionless quantity called the TE figureof merit (ZT), which is defined as

ZT = S2Ge

Gt,e + Gt,p

T , (21)

where S is the Seebeck coefficient describing magnitude ofthe TE voltage induced by the temperature difference acrossthe tunneling junction, Ge is the electrical conductance, andGt,e and Gt,p are the thermal conductances due to electronsand phonons, respectively. T is the temperature at the absolutescale.

In order to calculate the TE figure of merit let us first writethe electric J and thermal Q currents in the junction as

J = 2e

h

∫ ∞

−∞dE Te[fL − fR], (22)

Q = Qe + Qp = 2

h

∫ ∞

−∞dE (E − Ef ) Te[fL − fR]

+ h

2π

∫ ∞

0dω ω Tp[nL − nR], (23)

where Ef is the Fermi energy defined by the externalelectrodes, Te and Tp are the electron and phonon transmissionfunctions, respectively, and nL/R is the Bose-Einstein distri-bution function for the left/right electrode. In what followswe focus on the electron thermal current Qe which is the firstpart of Eq. (23), leaving the discussion of the phonon thermalcurrent for the next subsection. As the junction TE formalismis based on the linear expansion approach (as discussed later),the electrical thermal current in Eq. (23) is calculated in thelimit of vanishing temperature differences. Thus the Fermienergy here is defined as an average of the electrochemicalpotentials of the two electrodes.

The thermopower of the tunneling junction, given by theSeebeck coefficient, measures the induced TE voltage built upin response to the temperature difference applied across thejunction. If the temperature difference between the two endsof the junction is small, then the Seebeck coefficient is definedas

S = − V

T, (24)

where V is the TE voltage difference between the electrodesgenerated in response to the temperature difference T across

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SERGUEEV, SHIN, KAVIANY, AND DUNIETZ PHYSICAL REVIEW B 83, 195415 (2011)

the tunneling junction. The electric current is linearly expandedin terms of the voltage and temperature differences across thejunction,17

J = ∂J

∂( V ) V + ∂J

∂( T ) T = Ge V + L T. (25)

Therefore, the Seebeck coefficient at the limit of zero currentis rewritten as

S = L

Ge

. (26)

Here Ge is the electrical conductance while L can be referredas the electrical thermal coefficient.

In a similar fashion the electron thermal conductance canbe obtained. Expansion of the thermal current yields thefollowing:

Qe = ∂Qe

∂( V ) V + ∂Qe

∂( T ) T = R V + F T. (27)

Then the electron thermal conductance can be written as

Gt,e = Qe

T= R

V

T+ F = −R S + F. (28)

In the Landauer approximation the Seebeck coefficientand the electron thermal conductance can take the followingform:18

S = K1

eK0, (29)

where Kn is defined as

Kn = 2

h

∫ ∞

−∞dE Te(E)

(−∂fL

∂E

)(E − Ef )n. (30)

In the zero-voltage- and zero-temperature-bias limits, theelectron thermal conductance becomes

Gt,e =(

K2 − K21

K0

)1

T. (31)

We, therefore, calculate both the Seebeck coefficient and theelectron thermal conductance via the more general formulationof Eqs. (26) and (28), respectively, to address the e-pinteraction.

At this modeling level, the Seebeck coefficient is obtainedby first calculating the TE current J ( T ) induced by thetemperature difference T (at V = 0). We then apply thevoltage V (at T = 0) that generates the same tunnelingcurrent, i.e., J ( T ) = J ( V ). Division of the voltagedifference by the temperature difference yields the Seebeckcoefficient. In order to obtain the electron thermal conductancewe first calculate the coefficients R and F in Eq. (28) by takinga numerical derivative of the electron thermal current Qe withrespect to the voltage V and temperature T differences,respectively. Use of R, F , and the Seebeck coefficient inEq. (28) yields the electron thermal conductance.

C. Phonon transport within the NEGF formalism

We extend the NEGF formalism to treat phonon transportin molecular scale junctions.7,48,49 A similar scheme asimplemented for electron transport can be used here, whereBose-Einstein statistics is used for the phonons instead of the

Fermi-Dirac statistics relevant for the electrons.36 We startby calculating the dynamical matrix defined in the previoussubsections by Eq. (5). As in the electron transport problem,the junction is divided into the central region and the left andright electrodes. The dynamical matrix of the junction then hasthe form

K =

⎡⎢⎣

KLL KLC 0

KCL KCC KCR

0 KRC KRR

⎤⎥⎦ . (32)

Here KLL and KRR are the dynamical matrices of the semi-infinite left and right electrodes, respectively. Each electrode isdefined as a set of layers periodically repeated in the transportdirection with the interaction of only the nearest-neighboringlayers. Then, for example, the left semi-infinite electrodedynamical matrix has the following form:

KLL =

⎡⎢⎢⎣

. . .. . .

. . .

Kl,l−1 Kl,l Kl,l+1

0 Kl,l−1 Kl,l

⎤⎥⎥⎦ , (33)

where Kl,l is the dynamical matrix of the layer with index l

while Kl,l+1 or Kl,l−1 describe interaction of the layer l withneighboring layers l + 1 and l − 1. The dynamical matrix ofthe right semi-infinite electrode is written in a similar fashion.

The central region is further partitioned into the scatteringregion and the regions representing the bulk left and rightelectrodes. The models representing the bulk regions arechosen to be large enough to effectively screen any effectof the molecule. The dynamical matrix of the central region isthen written as

KCC =

⎡⎢⎣

Kl,l Kl,c 0

Kc,l Kc,c Kc,r

0 Kr,c Kr,r

⎤⎥⎦ , (34)

where Kl,l and Kr,r are the dynamical matrices of the finiteleft and right electrodes, Kc,c is the dynamical matrix of thescattering region, and Kl,c and Kr,c are the interaction matrixelements. Finally, the matrices KLC and KRC describe inter-action of the central region with the semi-infinite electrodes,where the only nonvanishing terms are due to nearest-neighborinteractions (KLC = Kl,l+1 and KRC = Kr,r+1).

The semi-infinite nature of the electrode, as in the electrontransport system, is represented by phonon self-energies(�L/R). The phonon Green’s function of the central regionconnected to the semi-infinite electrodes is then written as

DR(ω) = [ω2I − KCC − �R

L(ω) − �RR(ω)

], (35)

where �RL(ω) and �R

R(ω) are the phonon retarded self-energiesdue to the interaction of the central region with the left andright electrodes, respectively.

To obtain the phonon self-energies we employ the iterativemethod proposed by Sancho et al.,50 where the surface andbulk Green’s functions are calculated iteratively and sepa-rately. Within this approach the self-energies and the surfaceGreen’s functions are obtained via the transfer matrices. Werefer to Ref. 50 for more details.

Provided with the phonon retarded Green’s functions ofthe scattering region, the phonon transport properties of the

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junction can be calculated. The phonon thermal current is thengiven by

Qp = h

2π

∫ ∞

0dω ωTp(ω)[nL(ω,TL) − nR(ω,TR)], (36)

where TL/R is the temperature of the left/right electrode andthe phonon transmission function Tp(ω) is obtained as

Tp(ω) = T r[�L(ω)DR(ω)�R(ω)DA(ω)]. (37)

Here DA = (DR)† is the advanced phonon Green’s function,and �L/R is the broadening function due to the interaction ofthe central region with left/right electrode defined as

�L/R(ω) = i[�R

L/R(ω) − �AL/R(ω)

]. (38)

The phonon thermal conductance is then written as

Gt,p = Qp

T, (39)

where T is defined as the difference between the tempera-tures of the left, TL, and right, TR , electrodes. In this paper thephonon thermal conductance is computed directly by takinga numerical derivative of the phonon thermal current withrespect to the temperature difference.

III. RESULTS AND DISCUSSION

We investigate the effect of e-p coupling on the TEproperties of a molecular junction by calculating its Seebeckcoefficient, electrical and thermal conductances, and the TEfigure of merit in the presence of the e-p interaction. As amodel for a molecular bridge we couple a biphenyl-dithiolmolecule to two Al(111) electrodes and apply a temperatureand voltage gradient across the junction, as shown in Fig. 1(a).The scattering region of the junction includes the moleculeand some portion of Al(111) electrodes of finite cross section(about 12.2 A) at each side of the junction. Each such portionhas three (4 × 4) layers of Al atoms. We perform a periodiccalculation of the external electrodes, where k sampling isevoked only along the transport axis due to the finite crosssection of the electrode models. The molecule is bound to theAl surface at the hollow sites. The calculated binding energyof the molecule to the Al electrodes is 2.37 eV which is largerthan that for gold electrodes as reported.51

The dynamical matrices and vibrational spectrum are ob-tained at the DFT level using the SIESTA package.52 Here we usethe Troullier-Martins (TM) norm-conserving nonlocal pseu-dopotentials to treat atomic cores, Ceperley-Alder functionalsfor the exchange-correlation energy, and single-ζ localizedatomic orbitals (LCAO) basis functions. For the quantum-transport calculations we use our developed numerical packagebased on the DFT-NEGF approach, discussed in Sec. II. Itutilizes the LCAO basis set and the TM pseudopotentialswith the Perdew-Wang 1992 (PW92) exchange correlationfunctional.

A. Biphenyl-dithiol: Two distinct junction configurations

Recent studies of the transport and conformationalproperties of biphenyl-based molecular junctions, both

theoretical53,54 and experimental,54,55 suggest that the con-ductance greatly depends on the torsion angle between thetwo phenyl rings. The conductance is high when the twophenyl rings are in the same plane and decreases drasticallywhen the torsion angle increases. When the phenyl ringsare in perpendicular planes, the conductance is at minimum.The origin of such conductance variations lies in the overlapbetween the carbon π orbitals of different phenyl rings. Thegreater overlap corresponds to higher tunneling probability.The biphenyl-based junction can then be assigned withtwo transport regimes characterized by the high and lowconductance values. In what follows we address the effectof the e-p coupling on the junction in both regimes.

We start by first studying the conformational propertiesof the biphenyl-dithiol molecule in the junction. We followthe energy of the molecular junction, where one phenyl ringis rotated relative to the other. We find that the structuralminimum is a compromise between the tendency for parallelalignment of the two phenyl π systems and the intramolec-ular repulsion due to hydrogen atoms. Indeed, the stableconfiguration features about 30◦ rotation angle between thetwo phenyl rings as shown schematically in Fig. 1(b). Theconfiguration with the perpendicular orientation of the rings[see Fig. 1(c)] corresponds to the transition geometry along therelated torsion angle coordinate. The energy barrier betweenthe stable and transition configurations is about 0.3 eV. Below,we refer to the semiplanar configuration as the on state and tothe perpendicular as the off state, reflecting their conductanceproperties as indeed confirmed by our modeling.

The two configurations are used to demonstrate the richrelationship of structure and TE at the molecular level. Weemphasize that while the off configuration is energeticallyunstable as a transition state it serves as a model of possiblestable molecular orientations of related junctions. The perpen-dicular orientation can be stabilized by enhancing the repulsiveinteractions between the rings using steric substituents.56

For example, such substitutions were used experimentally todesign a study of the conductance dependence on the anglein biphenyl-derived junctions.56 In order to gauge the roleof steric substituents we added two methyl groups to oneof the molecule’s phenyl rings and performed total energyand quantum-transport calculations. Our results show thatindeed, the addition of methyl groups resulted in perpendicularalignment of the rings. Furthermore, we also confirmed thatthe transport and TE properties as evaluated for the off modelstructure are very similar to those of the dimethyl-substitutedjunction. In fact, the effect of the phonon-assisted tunneling onthe electron transmission function was found to be even morepronounced for the molecule with steric groups. This indicatesthat the off structure can successfully represent the junction inits low-conducting state.

Therefore, in what follows we remove the substituentsfrom the off model. This allows a focused analysis of theTE properties of two different related configurations. The twomodels also follow earlier studies on the effect of substituentson the conductance properties in similar junctions.53 Below weanalyze the effect of the e-p interaction on the TE properties,where the two configurations are used to sample the high-and low-conductance regimes of the junction. We demonstrate

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that indeed inelastic scattering effects on the TE propertiesexhibit a diverse behavior depending on the configuration andthe nature of the relevant vibrational mode.

B. Electronic structure and band alignment

We first describe the electronic structure of the biphenyl-dithiol junction in the absence of e-p interaction. We calculatethe electronic projected density of states (PDOS) on themolecular biphenyl-dithiol space using

De,m = − 1

πIm

∑i,j∈M

Sij GRij . (40)

Figure 2 shows the PDOS of the on and off molecularconfigurations. The PDOSs for both configurations exhibittwo peaks around the Fermi energy. The peaks are associatedwith the highest occupied (HOMO) and lowest unoccupied(LUMO) biphenyl-dithiol molecular states. We observe thatthe peaks are split due to the hybridization of the molecularand Al orbitals. Overall the electronic structure is similar: theDFT HOMO-LUMO gap is about 3.5–3.9 eV while the LUMOstate is closer to the Fermi energy of the junction. Thereforethe electron transport is mediated mainly by the LUMO state.The work function for Al is relatively small at 4.2 eV, where thegold value, for example, is 5.1 eV. In the case of Au electrodes,we expect the HOMO state to be closer to the Fermi energy.

-4 -2 40

3

6

0

3

6

(a) On

(b) Off

Transport

HOMOLUMO

LUMOHOMO

Gate electrode

Electric field(c)

(ar

b. u

nits

),em

D

20 (eV)fE E−

Idealized shift

1 5 eV ~ .

FIG. 2. (Color online) Electronic partial density of states of thebiphenyl-dithiol molecule versus electron energy for the junction on(a) and off (b) states. The Fermi energy (indicated with the verticaldashed line) is shifted to zero. (c) Schematic of the junction with theapplied gate electric field (shown with the red dashed arrow).

Gating the Al-contacted system moves the LUMO state andaligns it with the Fermi energy. This results in maximizingthe Seebeck coefficient and the e-p effect. Indeed, the Seebeckcoefficient is proportional to the derivative of the electronicdensity of states (DOS) with respect to energy at the Fermilevel.13 The same overall effect is found also when gatingis not applied. We note that gating can be effected either bychemical substitution57 or by adding a gating contact.58–60 Inthis paper, we maximize the modeled Seebeck coefficient byapplying a gating electric field, as shown in Fig. 2(c). We findthat the electric field of about 1–1.5 V/A effectively shifts theelectronic states of the molecule. In the rest of the paper theTE properties of the junction are obtained under the effect ofa fixed gating electric field of 1.2 eV/A.

C. Inelastic electron scattering

We next incorporate the e-p coupling effect. We start bycomputing the vibrational spectrum of the biphenyl-dithiolmolecule connected to the electrodes for both the on andoff states. We obtain a dynamical matrix defined as thesecond derivative of the total energy with respect to atomicpositions and diagonalize it to obtain its eigenfrequencies andeigenvectors. We include only real (nonimaginary) vibrationalmodes for computing the electron-phonon interaction andinelastic transport. We note that the off configuration, beinga transition state, has one vibrational mode with imaginaryfrequency. The effect of the imaginary modes was confirmedto be negligible by comparing to relevant models with stericfeatures that stabilize the twist angle between the two rings.

Having obtained the vibrational eigenmodes of themolecule we calculate the e-p coupling matrix using Eq. (4).The e-p coupling matrix is then included in the electronicHamiltonian with the use of the self-energies as discussed inthe previous section. The charge density of the device and theelectron Green’s functions with the self-energies representingthe e-p effects are iterated until self-consistency is achieved.We then calculate the electron transmission function usingEq. (18).

Figure 3 shows the transmission function versus thetunneling electron energy for both the on and off states. We findthat the effect of the e-p coupling on the electron transport forthe on and off states is significantly different. When the junctionis in the on state, adding the e-p interaction slightly diminishesthe electron tunneling, as shown in Fig. 3(a). This effect ismainly due to the electrons coupling with the vibrationalmodes that stretch the phenyl rings. When the phenyl ringsare in perpendicular planes, i.e., the junction is in the off state,the e-p interaction greatly enhances the electron transport.Such enhancement is mostly attributed to the phenyl rings’twisting mode. The rotation of the rings toward their parallelorientation creates a better overlap between the carbon π

orbitals across the two rings. That is, the parallel orientationfacilitates electron transport. All the other modes are found tobe relatively inactive in affecting the transport.

We next study the effect of temperature on the electrontransport in the junction.

Figure 4 shows the transmission function in the presence ofthe e-p interaction for both the on and off states. The increase inthe junction temperature has no effect on the electron tunneling

195415-6

EFFICIENCY OF THERMOELECTRIC ENERGY . . . PHYSICAL REVIEW B 83, 195415 (2011)

-0.8 -0.4 0 0.4 0.8 1.20

0.03

0.02

eΤ

0.2

0.4

0

eΤ

(a) On

(b) Off

0.01

0.3

0.1

e−p Coupling

No e−p coupling

No e−p coupling

e−p Coupling

-0.8 -0.4 0 0.4 0.8 1.2 (eV)fE E−

(eV)fE E−

FIG. 3. (Color online) Electron transmission function of thebiphenyl-dithiol TE junction versus energy for the on (a) and off(b) states with (dashed line) and without (solid line) e-p coupling.

when the biphenyl-dithiol molecule is in the conductive state[see Fig. 4(a)]. However, when the molecule is in the off state,the increase in the junction temperature drastically enhancesthe transmission function. This is mainly due to the inelasticnature of the transport. As the temperature goes up, more andmore vibrational modes become “activated” which effectivelyincreases the number of inelastic tunneling channels. At thesame time the junction in the off state is characterized by thelack of backscattering due to phonons, which could potentiallyreduce the electron transport.

The TE figure of merit depends on the electrical conduc-tance of the junction. The latter is computed by taking thenumerical derivative of the electric current with respect to thevoltage difference. Figure 5 shows the electrical conductanceof the junction as a function of the junction temperature forboth the on and off states at zero bias voltage. The junctiontemperature is defined as the average of the two electrode tem-peratures. For both configurations the conductance does notchange with the junction temperature when the e-p couplingis neglected. Indeed, at zero bias voltage, the conductancecan be expressed in terms of the transmission function asGe = 2e2 Te(E = Ef )/h. For the on state, the conductanceis slightly shifted by the e-p coupling but overall does notdepend also on the temperature. The off state conductance, onthe other hand, is strongly enhanced by the e-p coupling andtherefore depends strongly on the temperature.

We now evaluate the Seebeck coefficient using Eqs. (24)and (26) as follows. We first compute the electric current(J ) induced by the temperature difference ( T ) between twoelectrodes at zero voltage difference. Then we apply a voltagedifference ( V ) that produces the same current (J ) at zero

-0.8 -0.4 0 0.4 0.8 1.2

-0.8 -0.4 0 0.4 0.8 1.2

0

0.04

0.08

e

0

0.1

0.2

0.3

0.4

e

(a) On

(b) Off

T = 290 K

0

0

(eV)fE E−

(eV)fE E−

T = 290 K

FIG. 4. (Color online) Electron transmission function of thebiphenyl-dithiol TE junction versus energy for both the on (a) and off(b) states at junction temperatures of 0 (solid line) and 290 K (dashedline).

temperature difference. The ratio of the voltage differenceto the temperature difference in the limit of infinitesimaltunneling current yields the Seebeck coefficient.

Figure 6 shows the Seebeck coefficient versus junctiontemperature plotted for both the on and off states withand without e-p interaction included. The effect of inelasticscattering on the Seebeck coefficient for the on state is quitemoderate. Indeed, the transmission functions [see Fig. 4(a)]

0 100 200 3000

0.05

0.10

0.15

0.20

0.25On

Off

No e−p coupling

No e−p coupling

e−p Coupling

e−p Coupling

(K)T

oe

GG

FIG. 5. (Color online) Electrical conductance of the biphenyl-dithiol TE junction versus junction temperature for both the on[with (solid line) and without (dashed line) e-p coupling] and off[with (long-dashed line) and without (dot-dashed line) e-p coupling]states.

195415-7

SERGUEEV, SHIN, KAVIANY, AND DUNIETZ PHYSICAL REVIEW B 83, 195415 (2011)

e−p Coupling

No e−p coupling

e−p Coupling No e−p coupling

-80

-60

-40

-20

0

0 100 200 300

-80

-60

-40

-20

0

(K)T

(V

/K)

Sμ

(a) On

(b) Off

FIG. 6. (Color online) The Seebeck coefficient of the biphenyl-dithiol TE junction versus junction temperature for both the on (a) andoff (b) states with (dashed line) and without (solid line) e-p coupling.

indicate that the electron transport for this junction state isnot sensitive to temperature variations. The effect of the e-pcoupling for the off state is more pronounced, as shown inFig. 6(b). In this case, the e-p interaction increases the Seebeckcoefficient by a factor of roughly 2–4, since an increasedtemperature enhances the transmission function, as shown inFig. 4(b). Therefore, a smaller temperature difference acrossthe junction [which is in the denominator in Eq. (24)] is neededto generate the same electric current.

The calculated Seebeck coefficient for the on state at 300 Kis about 60–70 μK/V. In this paper we use Al-based electrodesto bias the molecule. The applied gate potential leads to anenhanced Seebeck coefficient (as discussed in the previoussubsection). The calculations of the TE properties are per-formed at the fixed positive gate electric field of 1.2 V/A.Reddy et al.13 report the Seebeck coefficient to be about12 μV/K at room temperatures for the biphenyl-dithiolmolecule in the junction between two Au electrodes. Wenote that our evaluated Seebeck coefficient is expected to belarger than that in a similar but not identical experiment.13

Furthermore, the sign of the Seebeck coefficient is differentbecause the transport is HOMO versus LUMO based.

Following the electrical conductance and the Seebeckcoefficient of the biphenyl-dithiol junction, the next ingredientneeded for the TE figure of merit is the electron thermal con-ductance, which is calculated using Eq. (28). Figure 7 showsthe electron thermal conductance versus junction temperatureof both the on and off states. Again we compare the resultsobtained with and without e-p coupling. As expected, the effectof the e-p coupling on the electron thermal conductance isnegligible for the on state while it is substantial for the junctionhaving phenyl rings in perpendicular relative orientation. For

0

0.5

1

1.5

2

0 100 200 3000

0.3

0.6

0.9

T (K)

e−p Coupling

No e−p coupling

(a) On

(b) Off

-10

(10

W/K

)t,e

G

e−p Coupling

No e−p coupling

FIG. 7. (Color online) Electron thermal conductance of thebiphenyl-dithiol TE junction versus junction temperature for the on(a) and off (b) states with and without e-p coupling.

example, at T = 250 K, the electron thermal conductanceobtained with the e-p interaction differs from that calculatedusing a pure elastic transport model by about a factor of 20.This change in the electron thermal conductance is crucialwhen computing the TE figure of merit since the phononthermal conductance contribution remains relatively small (asshown in the next subsection).

D. Phonon conductivity of the biphenyl-dithiol junction

Finally, the last ingredient needed for the TE figure ofmerit is the thermal conductance due to phonons propagatingthrough the junction. In order to calculate it we employ theelastic phonon transport method based on the Green’s functionformalism as discussed in Sec. II.

We start by computing the dynamical matrices of the leftand right electrodes. Here we choose a bulk cell which islong enough in the phonon transport direction so that thenearest-neighbor interaction approach can be safely applied(see Sec. II for details). In our case the bulk cell has 3(4 × 4) parallel layers of Al atoms. Having relaxed the atomiccoordinates, we compute the matrices Kl,l and Kl,l+1 andapply the iterative technique introduced by Sancho et al.50

as described above to calculate the phonon surface Green’sfunctions and the electrode phonon self-energies.

We next calculate the dynamical matrix of the centralregion. The central region contains the scattering region andthe left and right bulk electrode cells. The scattering regionshould include not only the molecule but several layers ofthe electrodes. This guarantees that the molecule is effectivelyscreened so that the left and right electrodes in the centralregion represent properly the bulk properties. In this workthe scattering region contains a biphenyl-dithiol molecule

195415-8

EFFICIENCY OF THERMOELECTRIC ENERGY . . . PHYSICAL REVIEW B 83, 195415 (2011)

connected to 3 (4 × 4) layers of Al atoms on each of the leftand right sides. We next compute the dynamical matrix Kc,c

of the optimized central region model.We calculate the phonon Green’s function using Eq. (35),

where the dynamical matrix of the central region is employed.The phonon transport properties represented by the phonontransmission function Tp, phonon current Qp, and phononthermal conductance Gt,p are computed following Eqs. (37),(36), and (39), respectively.

Figure 8 shows both the phonon transmission function andthe phonon contribution to the thermal conductance versusjunction temperature for the on and off states. We find thatthe phonon transport properties of the junction in the on stateare very similar to those in the off state. The phonon thermal

0 10 20 40 50 600

0.1

0.2

0 100 200 3000

0.1

0.2

0.3

On

Off

(a)

(K)T

(b)On

Off

-10

(10

W/K

)t,p

G

(meV)pE30

100 200 400 500300

0 100 200 3000.00

0.05

0.10

0.15

(K)T

On

Off

-1 (cm )

(c)

o,

,t

pt

GG

p

FIG. 8. (Color online) (a) Phonon transmission function of thebiphenyl-dithiol TE junction, versus the frequency of the phononsfor both the on and off states. (b) Phonon thermal conduc-tance versus junction temperature for both the on and off states.(c) Phonon thermal conductance normalized by the quantum of ther-mal conductance versus junction temperature for both the on and offstates.

conductance is obtained by numerical differentiation of thephonon thermal current with respect to the temperature differ-ence across the junction. As shown in Fig. 8(b), the phononcontribution to the thermal conductance of the biphenyl-dithioljunction is almost the same for both junction configurations.This similarity in the phonon thermal transport is in contrastto the variation of the electron thermal transport between thetwo configurations (see the electron thermal conductancescompared in Fig. 7). Slight differences in phonon transportare noted with the enhancement of the transport in the on stateby around 210 cm−1. This is likely due to vibrational modesthat involve stretching-squeezing motions of the phenyl rings.Such vibrations are less likely to be propagated through thejunction with the phenyl rings in the perpendicular planes. Wealso plot the phonon thermal conductance normalized to thequantum of thermal conductance Gt,0 = π2k2

B T /3h, as shownin Fig. 8(c). We find that the normalized phonon conductanceof the biphenyl-dithiol TE junction reaches a maximum at ajunction temperature of about 130 K.

E. Thermoelectric figure of merit

We calculate now the TE figure of merit following Eq. (21)using all the evaluated properties: the Seebeck coefficient(Fig. 6), the electrical conductance (Fig. 5), and the thermalconductances [Figs. 7 and 8(b)]. Figure 9 provides the TEfigure of merit versus junction temperature for both on and offjunction states. Similarly to effects on the Seebeck coefficientand electron thermal conductance, the effect of e-p couplingon the TE figure of merit is very moderate for the junction inthe on state. For the junction in the off state the e-p couplinggreatly enhances the TE figure of merit. For example, at

0

0.05

0.1

0.15

0 100 200 3000

0.05

0.1

0.15

(a) On

(b) Off

(K)T

ZT

No e−p coupling

e−p Coupling

No e−p coupling

e−p Coupling

FIG. 9. (Color online) The TE figure of merit of the biphenyl-dithiol TE junction versus junction temperature for the on(a) and off (b) states with (dashed line) and without (solid line) e-pcoupling.

195415-9

SERGUEEV, SHIN, KAVIANY, AND DUNIETZ PHYSICAL REVIEW B 83, 195415 (2011)

S

C1

C2

C3

φHot thermostat Cold thermostat

pQHot Al (111) Cold Al (111)

FIG. 10. (Color online) Atomic configuration used in MD sim-ulations. The thermostat on each side is three Al layers and itstemperature is controlled by the Langevin algorithm.

T = 250 K, taking into account the interaction of electronsand phonons leads to an increase in the figure of merit byabout a factor of 20. Thus, for some junction geometries thee-p coupling has a great impact on the device’s thermoelectricefficiency.

F. Torsion angle stability and anharmonicity investigatedby molecular dynamics

Next we employ molecular dynamics (MD) simulationsto investigate the stability with respect to the torsion angleand the phonon transport properties of the biphenyl-dithioljunction. Since the NEGF formalism for phonon transportemploys dynamical matrices, the phonon conductance Gt,p

calculated above does not include anharmonicity. However,anharmonicity in phonon transport is omnipresent, especiallyat high temperatures. We calculate Gt,p from the temperaturedistribution ( T ) and heat flow (Q) in nonequilibrium MD(NEMD), where the temperature is controlled by the Langevinthermostat.63 The NEMD simulations cover a temperaturerange of 280–500 K in the NV T ensemble (constant numberof particles, volume, and temperature) with the hot (Thot)and cold (Tcold) thermostat temperatures set 10% higher and10% lower than the average temperature, respectively. Thetemperature gradient is large enough to mask the effects oftemperature fluctuations.64 In our model the biphenyl-dithiolmolecule is sandwiched between two Al electrodes represented

by 324 atoms each (see Fig. 10). We impose periodic boundaryconditions in the lateral x and y directions for Al. Thecalculations run for 5 ns with 0.5 fs time steps. The interatomicpotentials used in the molecular mechanics simulations arefitted against DFT calculations. Two-, three-, and four-bodypotentials are employed and are listed in Table I. Instead ofdirect inclusion of H atoms, the C-H groups are treated eachas one entity (adding the mass of H to that of C).61

We invoke a quantum correction scheme to the MDsimulations. The correction term takes the following form:65–67

kBTMD =∫

dωDp(ω)hω

(f o

p + 1

2

),

(41)

Gt,p = Gt,p,MDdTMD

dT.

Here f op is the phonon distribution defined as f o

p =[exp(hω/kBT ) − 1]−1, and Dp is the phonon density of statesof Al (within a harmonic approximation). Figure 11(a) showsthe MD phonon conductance (Gt,p,MD), the quantum-correctedone (Gt,p), and the phonon conductance obtained via theNEGF first-principles approach. Both the MD (top axis) andcorrected temperatures are indicated in the figure.

The quantum correction of the thermal transport is generallyapplied to the bulk properties and is rather controversial.68

Nevertheless, we find that the MD thermal conductance trendis very similar to the results of the NEGF approach [seeFig. 11(a)]. We attribute this agreement to the relatively smallmodel size which results in noninteracting phonons at lowtemperatures. At this limit the phonon-phonon (p-p) effect notincluded in the NEGF approach is negligible. The MD results,which include the p-p scattering, show a slightly decreasingphonon thermal conductance at high temperatures (over 431 K,which is associated with the anharmonicity), while theNEGF results with noninteracting phonons show saturation(a plateau) at high temperatures.

Overall the phonon contribution to the thermal conductanceof the biphenyl-dithiol junction is relatively small comparedto the electron contribution (with the exception of the off statewithout the e-p interaction). Therefore, the anharmonic effect

TABLE I. Interatomic potential parameters for NEMD simulations. r , θ , and φ are the distance, bond angle, and torsion angle.

Interaction Potential model Parameters

PairAl-S ϕ0[{1 − exp[−a(r − r0)]}2 − 1] ϕ0 = 0.909 eV, a = 1.375 A−1, r0 = 2.600 AS-C1 ϕ0[{1 − exp[−a(r − r0)]}2 − 1] ϕ0 = 2.843 eV, a = 1.781 A−1, r0 = 1.823 AC1/2/3-C2 a ϕ0[{1 − exp[−a(r − r0)]}2 − 1] ϕ0 = 8.196 eV, a = 1.680 A−1, r0 = 1.388AC3-C3 ϕ0[{1 − exp[−a(r − r0)]}2 − 1] ϕ0 = 7.436 eV, a = 1.639 A−1, r0 = 1.489 AAl-Al 0.5kr (r − r0)2 kr = 3.989 eV, r0 = 2.864 AAngularS-C1-C2 b 0.5kθ (cos θ − cos θ0)2 kθ = 3.745 eV, θ0 = 120◦

C1/3-C2-C2, C2-C1/3-C2 a 0.5kθ (cos θ − cos θ0)2 kθ = 11.732 eV, θ0 = 120◦

C2-C3-C3 a 0.5kθ (cos θ − cos θ0)2 kθ = 9.599 eV, θ0 = 120◦

TorsionalC2-C3-C3-C2 kφ[1 − cos(φ − φ0)] kφ = 2.499 eV, φ0 = 31.6◦

aReference 61.bReference 62.

195415-10

EFFICIENCY OF THERMOELECTRIC ENERGY . . . PHYSICAL REVIEW B 83, 195415 (2011)

0 10 20 30 40 50 60 70 80 900.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

400

500

= 300 KMDT

φ ( )deg

Fre

quen

cy o

f oc

curr

ence

(b)

(a)

MD

NEGF (On state)

0 100 200 300 400

Quantum corrected MD

No quantum correction MD

MD

-10

(10

W

/K)

t,pG

0.0

0.2

0.4

0.6

0.8

( )MDT K300 400 450 500350

( )T K

FIG. 11. (Color online) (a) Phonon thermal conductance of thebiphenyl-dithiol TE junction, obtained via both the MD and NEGFcalculations, versus junction temperature. (b) Frequency of occur-rence of the biphenyl-dithiol molecular configurations with varioustorsion angles for three different values of junction temperature.

will only slightly enhance the TE figure of merit (ZT ) of thejunction at high temperatures.

We also use the MD calculations to analyze the torsionstability of the biphenyl-dithiol junction. In order to do sowe compute the frequency of occurrence of the biphenyl-dithiol molecule with a particular torsion angle between thephenyl rings, as shown in Fig. 11(b). Clearly, the moleculargeometries with the torsion angle of 30◦ occur most frequently.This is in agreement with the results of the conformationalstudy within the first-principles approach discussed in theprevious subsections. At higher temperatures the frequency ofoccurrence becomes broadened, suggesting that the moleculeis likely to deviate more from the stable configuration.

IV. CONCLUSION

In summary, we present both first-principles and moleculardynamics transport calculations of the TE properties ofthe biphenyl-dithiol molecular junction under an appliedgate voltage. Using DFT within the NEGF formalism wecompute the electrical and thermal conductances, the Seebeckcoefficient, and the figure of merit, taking into account thee − p interactions. The results obtained demonstrate that theeffect of the e − p coupling on the TE properties varies withthe molecular configuration. Two biphenyl-based junctionconfigurations, one with close to planar and another withperpendicular relative orientation of the molecular phenylrings, exhibit different responses to the inclusion of thee-p interaction into the model. The modeling illustrates theprospect of identifying molecular structural aspects that canbe used to tune and eventually optimize the TE properties.Furthermore, the MD results suggest a decrease in the phononconductance at high temperatures due to anharmonic effects.69

ACKNOWLEDGMENTS

This work is pursued as part of the Center for Solar andThermal Energy Conversion, an Energy Frontier ResearchCenter funded by the US Department of Energy, Office ofScience, Office of Basic Energy Sciences under 390 AwardNo. DE-SC0000957.

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