Length-dependent conductance and thermopower in single-molecule junctions of dithiolated oligophenylene derivatives : a density functional study

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Length-dependent conductance and thermopower in single-molecule junctions of dithiolated oligophenylene derivatives: A density functional study

F. Pauly,^1,2,

*

J. K. Viljas,^1,2 and J. C. Cuevas^1,2,3

1Institut für Theoretische Festkörperphysik and DFG-Center for Functional Nanostructures, Universität Karlsruhe, 76128 Karlsruhe, Germany

2Forschungszentrum Karlsruhe, Institut für Nanotechnologie, 76021 Karlsruhe, Germany

3Departamento de Física Teórica de la Materia Condensada, Universidad Autónoma de Madrid, 28049 Madrid, Spain 共Received 1 May 2008; published 16 July 2008

兲

We study theoretically the length dependence of both conductance and thermopower in metal-molecule- metal junctions made up of dithiolated oligophenylenes contacted to gold electrodes. We find that while the conductance decays exponentially with increasing molecular length, the thermopower increases linearly as suggested by recent experiments. We also analyze how these transport properties can be tuned with methyl side groups. Our results can be explained by considering the level shifts due to their electron-donating character as well as the tilt-angle dependence of conductance and thermopower. Qualitative features of the substituent effects in our density functional calculations are explained using a tight-binding model. In addition, we observe symmetry-related even-odd transmission channel degeneracies as a function of molecular length.

DOI:10.1103/PhysRevB.78.035315 PACS number共s兲: 85.65.⫹h, 65.80.⫹n, 73.23.Ad, 73.63.Rt

I. INTRODUCTION

In the field of molecular electronics, research has so far mostly concentrated on the dc electrical conduction properties of single-molecule contacts.¹By now it is known that the charge transport through organic molecules is typically due to electron tunneling. This is evidenced in particular by the exponential decay of the conductanceGwith increasing molecular length in contacts formed from oligomers with vary- ing numbers of units.^2–4Considering the statistical nature of experiments at the molecular scale, a conclusive comparison to theory is presently difficult. Nevertheless, associated decay coefficients appear to be reproduced by theoretical calculations.^5–7For a deeper understanding of molecular-size contacts, it is useful to analyze also other observables in parallel with the dc conductance. Emerging new lines of research involve the photoconductance^8–10 and heat transport.^11,12 In this paper we concentrate on the thermopower. For metallic atomic contacts, this quantity was already studied experimentally some years ago¹³but for molecular contacts only very recently.^14,15

The thermopower Q, also known as the Seebeck coeffi- cient, measures the voltage ⌬V induced over a conducting material at vanishing steady-state electric current I, when a small temperature difference ⌬T is applied: Q

=

兩−⌬V

/⌬T兩I=0. It is known that in bulk materials the sign of the thermopower is a hint of the sign of the main charge carriers. If Q⬍0

共Q

⬎0兲, charge is carried by electronlike

共

holelike

兲

quasiparticle excitations, as in an n-doped

共p-doped兲

semiconductor.¹⁶ Analogously, the thermopower of the molecular junction gives information about the alignment of the energies of the highest occupied and lowest un- occupied molecular orbitals

共HOMO and LUMO兲

with respect to the metal’s Fermi energy E_F.^17–19 In the experiment,¹⁴ Q was measured for gold electrodes bridged by dithiolated oligophenylene molecules. It was found to be positive, which indicates that EF lies closer to the HOMO than to the LUMO. Also, it was observed that Q grows

roughly linearly with the numberNof the phenyl rings in the molecule. More recently¹⁵the effects of substituents and var- ied end groups have been analyzed for the benzene molecule.

Using transport calculations based on density functional theory

共

DFT

兲

, we investigate in this paper the length dependence of the conductance and the Seebeck coefficient for dithiolated oligophenylenes bonded to gold contacts. For the molecules studied in Ref. 14, we find that the conductance decays exponentially with increasing molecular length. De- cay coefficients compare reasonably with those by previous DFT calculations^5–7and also with those from experiments,^2–4 considering differences in contact configurations, molecular end groups, and uncertainties with respect to environmental effects. In addition, the thermopower increases linearly, with a magnitude coinciding with the measurements.¹⁴ We also study how the results change as various numbers of methyl substituents are introduced to the molecules. The effect of these substituents is twofold:

共i兲

They push up the energies of the ␲ electrons as a result of their electron-donating behavior,^15,20 and

共ii兲

they increase the tilt angles between the phenyl rings through steric repulsion. The latter effect tends to decrease both G and Q due to a reduction in the degree of ␲-electron delocalization, while the former op- poses this tendency by bringing the HOMO closer to EF. A simplified ␲-orbital model is used to explain essential features of the DFT results.

The paper is organized as follows: Section II explains details of our DFT-based approach. Then, in Sec. III, we present the molecular contacts, whose charge transport properties we determine in Sec. IV. In Sec. V we show how the DFT results can be understood in terms of the ␲^-orbital model, but we discuss also effects beyond this simplified picture in Sec. VI. Finally, we end in Sec. VII with a discussion and conclusions.

II. METHODS

The general formulas for the treatment of thermoelectric effects, based on the Landauer-Büttiker formalism, are dis- PHYSICAL REVIEW B78, 035315共2008兲

Konstanzer Online-Publikations-System (KOPS)

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cussed in detail in several references.13,17,19,21–23By expand- ing the expression for the currentIto linear order in⌬Vand

⌬T and considering the cases ⌬T= 0 and I= 0, respectively, one arrives at

G=G₀K₀

共

T

兲

, Q= − K₁

共

T

兲

eTK₀

共T兲

,

共

1

兲

with K₀

共T兲

=

兰

␶

共E兲关−

⳵f共E,T兲/⳵E兴dE, K₁

共T兲= 兰共E

−␮

兲

␶

共E兲

⫻关−⳵f共E,T兲/⳵E兴dE, andG₀= 2e²/h. Here ␶

共E兲

is the transmission function, f

共E

,T兲=

兵exp关共E

−␮

兲

/k_BT兴+ 1其⁻¹ is the Fermi function, and ␮is the chemical potential,␮

⬇

E_F. At low temperature, the leading-order terms in the Sommerfeld expansions yield

G=G₀␶

共E

F

兲,

Q= −kB

e

␲² 3

␶

⬘ ^共E

^F

^兲

␶

共E

F

兲

kBT,

共2兲

where prime denotes a derivative. In our DFT-based results presented below, we calculateGandQaccording to Eq.

共1兲.

However, Eq.

共2兲

approximates the results to within a few percent at room temperature, since␶

共

E

兲

is smooth aroundE_F due to the off-resonant situation. It should be noted that the equations neglect electron-vibration interactions. These could in principle be included by adding the inelastic corrections to the expression for the current,²⁴ but we expect also these contributions to be relatively small even at room temperature.

The transmission functions are computed with the help of Green’s-function techniques. The electronic structure is described in terms of DFT as implemented in the quantum chemistry program TURBOMOLE 5.7, where we employ the BP86 exchange-correlation functional and the standard Gaussian basis set of split valence quality with polarization functions on all nonhydrogen atoms.²⁵ Geometry optimiza- tion is performed until the maximum norm of the Cartesian gradient has decayed to values below 10⁻⁴ a.u. For further details on our method, see Refs.26–29.

III. CONTACTS

The molecules studied are shown in Fig.1. Those labeled R1–R4 are the pure oligophenylenes. S2–S4 denote oligophenylenes where the hydrogen atom in one of the two ortho positions with respect to each ring-connecting carbon atom is substituted with a methyl group.^30,31 D2–D4 have substituents in both ortho positions. Here, the numbers N

= 1 , . . . , 4 refer to the number of phenyl rings in the mol-

ecule. The tilt angles for the R, S, and D molecules vary in the ranges 33.4°ⱕ␸Rⱕ36.4°, 84.8°ⱕ␸Sⱕ90°, and 88.8°

ⱕ␸Dⱕ90°, and the distances between the terminal carbon atoms of the molecules are described to a good accuracy by d=a+bN, with a= −0.154 nm and b= 0.435 nm

共Table

I兲.

For some of the molecules with two or more rings, different conformational isomers exist. Since for them the absolute values of the tilt angles remain approximately the same, we do not expect essential differences in the results.

The

共Kohn-Sham-DFT兲

HOMO and LUMO energies of the isolated molecules are shown in Fig.2. We observe that the HOMO-LUMO gaps of the S and D series are larger than those of the R series. Let us note that DFT generally tends to underestimate the gaps.^32,33

To form the junctions, each molecule is coupled to the hollow position of the tips of two gold

关

111

兴

pyramids via a sulfur atom. The atomic positions of the molecule and the first gold layers of the tips are then relaxed. This is depicted in Fig.3for S3. The relaxed part is also the “central region”

in the transport calculations.²⁹The tilt angles␸and the distances d of the contacted molecules are not essentially different from those of the isolated molecules.

IV. DENSITY FUNCTIONAL-BASED TRANSPORT The results for the transmission, its logarithmic derivative, the conductance, and the thermopower for all of the R, S, and D type molecular junctions are collected in Fig.4. In

S3 S2

R4 R3 R2 R1

S4

D3 D2

D4

FIG. 1. 共Color online兲 The molecules studied in this work.

When they are contacted to the gold electrodes, sulfur atoms re- place the terminal hydrogen atoms.

TABLE I. Geometrical data for the isolated molecules of the R, S, and D series 共Fig. 1兲. The ␸j are the tilt angles between two adjacent phenyl rings, andd is the distance between the terminal carbon atoms of the molecules.

Molecule ␸1

共deg兲 ␸2

共deg兲 ␸3

共deg兲

d 共nm兲

R1 0.281

R2 36.4 0.718

R3 35.3 35.4 1.154

R4 34.8 33.4 35.2 1.590

S2 90.0 0.715

S3 84.8 89.8 1.150

S4 85.4 89.8 89.0 1.584

D2 90.0 0.715

D3 89.1 89.2 1.150

D4 88.8 89.0 89.9 1.584

1 2 3 4

N -7-6

-5-4 -3-2 -10

E(eV) R

SD

FIG. 2. 共Color online兲HOMO and LUMO energies of the isolated molecules in Fig.1.

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order to compare with the room-temperature experiments,¹⁴ we set T= 298 K in Eq.

共1兲. The Seebeck coefficients are

displayed in Fig.4

共

d

兲

together with the experimental results, where the molecules R1–R3 were studied.

With increasing number N of phenyl rings, the conductance decays asG/G₀

⬃

e^−␤^N

关

Fig.4

共

c

兲兴

. For the R series, we find the decay coefficient␤R= 1.22,⁸in good agreement with the theory in Ref. 6.³⁴ Previous theoretical estimates^5,7 and experimental results^2,3for thiolated self-assembled monolay- ers are consistently somewhat larger than this value. In particular, the results reported in Ref. 2 vary in the range 1.5 ⱕ␤Rⱕ2.1, and for amine end groups an experimental value of ␤R= 1.5 was reported.⁴ The underestimation of ␤R may indicate an overestimation of the conductance computed within DFT.^35,36However, a comparison between theory and experiment is complicated due to the differences in the end groups used.³⁷For the S and D series, we find␤S= 3.69 and

␤D= 4.07, which are both much larger than␤R. This increase reflects the reduced delocalization of the␲-electron system.

Regarding the absolute conductance values, the molecule R1 has been studied by many groups, and the reported conductances vary mostly between 0.01G₀ and 0.5G₀.^38–43 Our value ofG_R1= 0.04G₀is close to the results in Refs. 41and 42, for example. The case of R2 is discussed in Ref.29.

Since the Fermi energy atE_F= −5.0 eV lies closer to the HOMO than to the LUMO level, the Seebeck coefficientQ

has a positive value

关

Fig.4

共

d

兲兴

. We also find thatQincreases roughly linearly withN, as suggested by the experiments.¹⁴ Indeed, assuming that the transmission around E=EF is of the form ␶

共E兲

=␣

共E兲e

^−␤共^E^兲^N, then Eq.

共2兲

yields Q=Q^共0兲 +Q^共¹^兲N, where Q^共⁰^兲= −k_B²T␲²

关ln

␣

共E

F

兲兴 ⬘

^/^3e ^and ^Q^共¹^兲

=k_B²T␲²␤

⬘ ^共E

^F

^兲/

^3e.⁴⁴Two things should be noted here. First, Q does not necessarily extrapolate to zero forN= 0, leading to a finite “contact thermopower”Q^共0兲. Second,Q^共0兲depends on the prefactor␣

共E兲, but

Q^共¹^兲does not. Since␣

共E兲

contains the most significant uncertainties related to the contact geometries, Q^共1兲 can be expected to be described at a higher level of confidence than Q^共0兲. Best fits to our results and the experimental data give Q_R^共⁰^兲= −0.28 ␮V/K, Q_R^共¹^兲

= 7.77 ␮V/K andQ_R,exp^共0兲 = 6.43 ␮^V/K,Q_R,exp^共1兲 = 2.75 ␮^V/K, respectively. Differences in the fit parameters mainly stem from the data point for R3, where the experimental value is lower than the calculated one. Considering the reported order-of-magnitude discrepancies between measured conductances and those computed from DFT,^35,36the agreement still appears reasonable.⁴⁵ However, for large enough T and N, the above-mentioned exponential and linear laws for the length dependences of G andQ should be modified due to interactions with the thermal environment.¹⁹The fact that the experimental data in Fig.4共d兲exhibit a rather good linearity suggests that molecular vibrations do not play a crucial role and that the electronic contribution toQ is dominant.

Although the conductances for S and D are very similar, their thermopowers are rather different. Furthermore, the magnitudes for N⬎2 follow the surprising order Q_S⬍Q_R

⬍Q_D.⁴⁶As we will discuss below, these observations can be understood through the two competing substituent effects:

共i兲

a change in the alignment of the ␲-electron levels with respect toEFas a result of the electron-donating nature of the methyl group^15,20 and

共ii兲

an increase in the ring-tilt angles.

For the isolated molecules, the second effect results in the opening of the HOMO-LUMO gap when going from molecules R to S or D, while the first effect causes the difference between the HOMO and LUMO energies of the S and D series

共

Fig.2

兲

.

V.␲-ORBITAL MODEL

In order to understand better the general features of the dependence of G andQ on the number of substituents, we study a simple tight-binding

共

TB

兲

model, which describes the

␲-electron system of the oligophenylenes

共Fig.

5兲. The on- site energies⑀j

共j

= 1 , . . . ,N兲are equal on all carbon atoms of phenyl ring j, the intra-ring hoppingt is assumed to be the same everywhere, and the inter-ring hopping u is param- etrized through u=tcos␸. We assume the effect of the side groups to come into play only through ␸ ^and⑀j. The leads are modeled by “wide-band” self-energies⌺L,R= −i⌫, acting on the terminal carbon atoms. Notice that sinceu is even in

␸, the model does not distinguish between conformational isomers, where tilt angles change sign.

We extract the parameters of our model as follows: For the R molecules, we set⑀j= 0. In the S and D molecules, the

⑀j’s for rings with one, two, or four methyl groups are ob- tained by performing DFT calculations of methylbenzene, dimethylbenzene, and tetramethylbenzene

共Fig.

6兲. We find

fixed relaxed

fixed L C R

(b) (a)

FIG. 3.共Color online兲共a兲The contacts to the gold electrodes are formed through sulfur atoms bonded to the hollow position of the tip of a关111兴pyramid.共b兲For the transport calculations, the tips of the pyramids are part of the extended molecule共C兲. The more re- mote parts共blue兲are absorbed into ideal semi-infinite left共L兲and right共R兲surfaces共gray兲.

-7 -6 -5 -4 -3 -2 -1 E (eV) 10^-10

10^-8 10^-6 10^-4 10^-2 10⁰

τ(E)

R1R2 R3 R4S2 S3S4 D2 D3D4

-6 -5 -4 -3 -2

E (eV) -20

-10 0 10 20

-τ’(E)/τ(E)(1/eV)

R1R2 R3 R4S2 S3S4 D2 D3D4

0 1 2 3 4

N 10^-8

10^-6 10^-4 10^-2 10⁰

G/G0

RS D

0 1 2 3 4

N 0

10 20 30 40 50

Q(µV/K)

R, exp RS D

(a) (b)

(c) (d)

E_F

FIG. 4. 共Color online兲关共a兲 and共b兲兴Transmission function and the negative of its logarithmic derivative.关共c兲 and共d兲兴The corre- spondingGandQ, including the temperature corrections in Eq.共1兲. The straight lines are best fits to the numerical results for R共solid line兲, S共dashed line兲, and D共dashed-dotted line兲. The experimental data in共d兲are from Ref.14.

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that the HOMO and LUMO energies in these molecules shift upward monotonously with the number of methyl substituents. This can be attributed to the electron-donating character of the methyl groups and the resulting increase in Coulomb repulsion on the phenyl ring. We use the averages of the HOMO and LUMO shifts relative to benzene and obtain, respectively, ⑀^共one兲^{= 0.17 eV,} ⑀^共two兲= 0.33 eV, and ⑀^共four兲

= 0.58 eV.⁴⁷ The on-site energies of the different phenyl rings of the R, S, and D molecules withN= 1 , . . . , 4 are sche- matically represented in the lower part of Fig. 5. The hoppingt is set to half of

共the negative of兲

the HOMO-LUMO gap of benzene, with the result t= −2.57 eV

共

Fig. 2

兲

. The scattering rate⌫and the tilt angles for R, S, and D molecules are chosen to reproduce approximately the minimal transmis- sions for R1, R2, S2, and D2 in the DFT results in Fig.4共a兲.

This yields ⌫= 0.64 eV, ␸R= 40.0°, ␸S= 85.5°, and ␸D

= 86.5°. The last free parameter is E_F, which should be de-

termined by the overall charge-transfer effects between the molecule and the electrodes. We set its value to EF

= −1.08 eV, close to the crossing points of the transmission curves of the S and D molecules

关cf. Figs.

4共a兲and7共a兲兴.

Although the model is not meant to reproduce all the details of the DFT results, similar features can be recognized.

In particular, as shown in Fig. 7共b兲, the correct order of the thermopowers is reproduced. In going from R to S, the increase in the tilt angle and the associated breaking of the

␲-electron conjugation dominates over other side group effects. As a result both G

关Fig.

7共c兲兴 and Q

关Fig.

7共d兲兴 decrease. In going from S to D, an interplay between the side- group-induced level shifts and a small residual increase in␸ raise Q above the value for R, whileGremains almost un- changed. We note that for an N-ring junction, the lowest- order terms in an expansion of the transmission in powers of u yield ␸ dependences of the form ␶

共E兲⬇

c₁

共E兲cos

^2共^N−1兲␸ +c₂

共E兲cos

^2N␸^{, for}^E

⬇

EF.²⁹Independently ofN, this results in Q

⬇

q₁+q₂cos²␸, where we find q₁,q₂⬎0 in our case.

ForN= 2, the neglect of thec₂term leads to the well-known

“cos²␸ ^{law” of} ^G,^4,29 but we see that for the tilt-angle dependence ofQthe presence of this term is significant, since q₂= 0 if c₂= 0. When EF is close to a resonance, further higher-order terms become increasingly important and deviations from the q₁+q₂cos²␸ law result. This is seen most clearly as the nonmonotonous ␸ dependence ofQ for molecule D3 in Fig. 7共d兲.

VI. EFFECTS BEYOND THE␲-ORBITAL MODEL Close to perpendicular ring tilts, results from the␲^-orbital model should be considered with care. At ␸^{= 90° the} ␲^-␲ coupling u vanishes, and any other couplings between the rings will become important.⁴⁸ Let us analyze this for the biphenyl molecules R2, S2, and D2. For them, we have var- ied the tilt angle between the rings and have determined the

SD R N=1 N=2 N=3 _ε_j N=4

ε^(one) ε^(two) ε^(four)

ε₁ ε₂ ε₃

ΣR=−iΓ ΣL=−iΓ

1 1 2 1 2 3 1 2 3 4 j

1 2 3

= tcosϕ t t t t t

u u t

u

FIG. 5. 共Color online兲The parameters of the TB model represented by a molecule withN= 3 phenyl rings. Also shown are sche- matic graphs of the on-site energies ⑀j of the different rings j

= 1 , . . . ,Nfor R, S, and D type molecules withN= 1 , . . . , 4 rings.

−6.267 eV −6.265 eV −1.125 eV −1.124 eV benzene

toluene −6.174 eV −5.948 eV −1.093 eV −0.982 eV

meta−xylene −5.969 eV −5.779 eV −1.016 eV −0.906 eV

para−xylene −6.088 eV −5.678 eV −0.987 eV −0.938 eV

durene −5.756 eV −5.393 eV −0.842 eV −0.699 eV molecule HOMO−1 HOMO LUMO LUMO+1

FIG. 6. 共Color online兲 Influence of methyl substituents on the level alignment of frontier molecular orbitals in benzene. From left to right the molecular structure is displayed together with isosurface plots of the HOMO− 1, HOMO, LUMO, and LUMO+ 1 wave functions. Below these plots the common name of the molecules and the energies of the molecular levels are indicated.

-1 -0.5 0 0.5 1 1.5 E / |t|

10^-10 10^-8 10^-6 10^-4 10^-2 10⁰

τ(E)

R1 R2 R3 R4 S2 S3 S4 D2 D3

0 1 2 3 4

N -5

0 5 10 15 20 25

Q/(π2 k2 BT/3e|t|)

R S D

0 30 60 90

ϕ(deg) 10^-8

10^-6 10^-4 10^-2 10⁰

G/G0

R3 S3 D3

0 30 60 90

ϕ(deg) 0

10 20 30

Q/(π2 k2 BT/3e|t|)

R3 S3 D3

(a) (b)

(c)

(d) E_F

FIG. 7. 共Color online兲共a兲 Transmission functions for the TB model in Fig.5with parameters chosen as explained in the text.共b兲 Corresponding Seebeck coefficients according to Eq. 共2兲. The straight lines are best fits as in Fig.4.关共c兲 and共d兲兴Gand Qas a function of the tilt angle for the molecules withN= 3. The vertical dotted lines indicate the “equilibrium angles”␸R,␸S, and␸D; the arrows represent changes when going from R3 to S3 to D3.

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charge transport properties for every ␸ by using our DFT- based approach. Further details of the procedure are described in Ref.29. For R2

共

and similarly for S2 and D2

兲

we observe that for most tilt angles

共

␸ⱗ80°兲the transmission is dominated by a single channel

关Fig.

8共a兲兴of ␲^-␲ ^character.

However, at large angles two transmission channels of the same magnitude are observed, which become degenerate at

␸= 90°. They arise from ␴^-␲ couplings between the two rings, whose strengths are proportional to sin␸^{. Thus the} two degenerate channels are of the ␴^-␲ ^and ␲^-␴ ^types.

These features are obviously not accounted for by the TB model, where only a single␲^-␲channel is present independently of␸^{. The}␴^-␲couplings should also modify the tilt- angle dependence of the thermopower, which is plotted in Fig.8共b兲. Due to the electron-donating nature of the methyl groups, the thermopower at fixed␸increases from R2 to S2 and D2. While the curve for R2 can be described by the law q₁+q₂cos²␸ ^for ␸ⱗ80°, a dip is observed for larger ␸^. Similar deviations in Q are also present for S2 and D2, where we have investigated a smaller tilt-angle interval be- cause of the steric repulsion of the methyl groups.²⁹

The degeneracy of the transmission channels is due to the D_2d symmetry of biphenyl when ␸^{= 90°.}⁴⁸ For the longer oligophenylenes, the symmetryD_2d can occur if and only if tilt angles are all at 90° and the number of rings is even.

Hence for S2, S4, D2, and D4, the ratio␶2/␶1of the first two transmission channels should be particularly large. Such even-odd oscillations are indeed visible in Fig. 9 for the S and D series, while they are absent for R. Owing to the fact

that tilt angles deviate from 90°, the oscillations decay. In particular ␶2/␶1 is much smaller for S4 than for D4, where the minimal tilt angles in the contacts are 85.1° and 89.1°, respectively.

VII. DISCUSSION AND CONCLUSIONS

As is well known, there are many theoretical uncertainties involved in the determination of transport properties based on DFT calculations; improvements for the methods of molecular-scale transport theory are currently being sought.^32,35,36Indeed, it is typical to find order-of-magnitude differences between measured conductances and those computed within DFT.^35,36It is also known that atomic configurations can have a strong influence on the conductance.^36,49,50 Hence a conclusive comparison with experimental data would require the statistical analysis of a large number of contact geometries. However, models predictQto be insen- sitive to changes in the lead couplings;¹⁷hence it is expected to be a more robust quantity thanG. Although the measurements in Ref. 14were carried out at room temperature, we do not expect molecular vibrations to play an essential role in the results due to the weakness of the electron-vibration coupling. Thus, the comparison we have made with our elas- tic transport theory seems justified. Nevertheless, the inves- tigation of the effect of molecular vibrations is an interesting direction for future research, also in view of optimizing the properties of molecular thermoelectric devices for potential applications. As far as gaining a basic understanding is con- cerned, low-temperature measurements would be desirable to remove uncertainties related to the thermal excitation of vibrations. The purpose of the present work was to study the general trends for a series of molecules that are similarly coupled to the electrodes. Considering all the potential uncertainties, the agreement we obtain for the thermopower appears to be quite reasonable.

In conclusion, we have analyzed the length dependence of conductance and thermopower for oligophenylene single- molecule contacts. While we found the conductance to decay exponentially with the length of the molecule, we observe that the thermopower increases linearly. For possible future applications, it is interesting to know how the magnitudes of these quantities can be tuned. We analyzed how this can be achieved by chemical substituents, which control the Fermi- level alignment and the molecular conformation. We demon- strated that a simple␲-electron tight-binding model can help to understand basic substituent effects. In addition, we observed an even-odd effect for transmission channel degeneracies upon variation of the number of phenyl rings, which we explained by molecular symmetries.

ACKNOWLEDGMENTS

We acknowledge stimulating discussions with M. Mayor.

The Quantum Chemistry group of R. Ahlrichs is thanked for providing us with TURBOMOLE. This work was financially supported by the Helmholtz Gemeinschaft

共Contract No.

VH-NG-029兲, the EU network BIMORE

共Grant No. MRTN-

CT-2006-035859兲, and DFG SPP 1243. F.P. acknowledges the funding of a Young Investigator Group at KIT.

0 30 60 90

ϕ(deg) 12

14 16 18

Q(µV/K)

R2S2 D2 q₁+q₂cos²(ϕ)

0 30 60 90

ϕ(deg) 10^-5

10^-4 10^-3 10^-2

transmission

ττ₁ τ₂ τ₃ τ₄

R2

(a) (b)

FIG. 8. 共Color online兲共a兲Tilt-angle-dependent transmission ␶

=兺i␶iresolved in its transmission channels␶ifor the molecule R2 and共b兲thermopower for R2, S2, and D2. The curve for R2 is fitted by a function of the formq₁+q₂cos²␸withq₁= 13.27 ␮V/K and q₂= 1.38 ␮V/K for the tilt-angle interval from 0° to 60°. The dotted vertical lines indicate DFT equilibrium tilt angles␸R2,␸S2, and

␸D2.

1 2 3 4

N 0

0.2 0.4 0.6 0.8 1

τ2/τ1

RS D

FIG. 9. 共Color online兲Importance␶1/␶2of the second transmission channel␶2as compared to the first␶1as a function of molecular length for the molecules of series R, S, and D.

(6)

*fabian.pauly@kit.edu

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