Theoretical study of carotene as a molecular wire

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Physica E 19 (2003) 133 – 138 www.elsevier.com/locate/physe

Theoretical study of carotene as a molecular wire Jun Li∗ , John K. Tomfohr, Otto F. Sankey Department of Physics and Astronomy, Arizona State University, Tempe, AZ 85287-1504, USA

Abstract Carotenoid molecules have important photo-chemical properties and may serve as molecular wires in a molecular electronic circuit. We have theoretically studied the intrinsic conducting properties of sulfur-terminated carotene between gold contacts using local orbital density functional theory. The dependence of the tunneling decay parameter “
” on the degree of single–double bond alternation within the polyene backbone is determined from the polyene complex band-structure. The electron tunneling current–voltage characteristics is calculated using the Landauer–B
uttiker formalism. The calculations are in qualitative agreement with experiments. ? 2003 Elsevier B.V. All rights reserved. PACS: 73.63.−b; 73.23.−b; 73.40.Gk Keywords: Molecular electronics; Carotene; Complex band; I –V characteristics

1. Introduction The carotenoids are an important class of organic molecules with delocalized -electrons. A typical carotene molecule consists of a polyene backbone of alternating single–double bonds terminated by phenyl rings at the end (refer to Fig. 1). Biological and photo-chemical properties of carotenoids are rooted in the conjugated polyene backbone and functional ends. The delocalized -electron pathway along the polyene backbone makes carotenoids excellent candidates for molecular wires. Earlier experiment [1] showed that carotenoids are orders of magnitude more conductive than non-conjugated molecules of the same length (such as alkanes). Quantitative information on their conductive properties was dicult to obtain since the ∗ Corresponding author. Tel.: +1-490-965-0667; fax: +1-480965-7954. E-mail address: [email protected] (J. Li).

molecule was not covalently bonded to contacts in these early atomic force microscope experiments. Recently, Ramachandran et al. [2] measured the current–voltage curves in a reliable and reproducible experimental setup, called the ‘contact pad’ approach [3]. Here, we complement this experimental study with theoretical prediction of the I –V characteristics. In this paper, we report a theoretical study on the conductivity of sulfur-terminated carotene between gold contacts using local orbital density functional theory. First, we calculate the complex band structure of carotene in terms of its structural components; the polyene backbone and phenyl rings. We determine the dependence of the tunneling decay parameter
on the degree of the bond length alternation in the polyene backbone. The complex band calculation allows us to estimate, in a simple manner, the intrinsic resistance of the carotene molecule. Finally, we use the Landauer–B
uttiker formalism [4] to calculate the I –V curve of the gold–carotene–gold system. The

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J. Li et al. / Physica E 19 (2003) 133 – 138

134

(A) Trans-Carotene: SH

LUMO

E

HS

β(E) e(B) Cis-Carotene:

BP HS

β bp

SH

Fig. 1. Scheme of the carotene molecules studied. There are two isomers: (a) trans and (b) cis.

detailed experiment and a comparison between theory and experiment is given in Ref. [2]. 2. Complex band structure Fig. 1 shows the carotene molecules of interest in two conformations. In both isomers, the structures consist of a polyene backbone and two phenyl rings. In this section, where we are searching for a qualitative description of the electron transport, we ignore the occasional CH3 side attachment and replace it by just hydrogen. The conjugated polyene backbone connecting the phenyl rings on the ends is the key structure element of carotenoids. Atoms in polyene are arranged in a planar zigzag chain with the formal repeating [5] unit – (CH)K(CH) –. The length of a formal C–C  is shorter than that in a single bond dS = 1:45 A  consistent with normal alkane single bond (1:53 A), delocalization of the -electrons. The formal CKC  is the same as that in ethydouble bond (dD = 1:34 A)  lene ((CH2 )2 ; 1:336 A). This gives the pronounced single–double bond length alternation. The degree of bond alternation, or “dimerization”, is characterized by a structural parameter  which is dened as the ratio of the single bond length dS to the double bond length dD ,  = dS =dD . The experimental dimerization exp is 1.082. Dimerization has signicant eects on the -electron system, which controls the conducting pathway along the polyene backbone. Tunneling current through an energy barrier generally goes like I = I0 e−
L , where L is the length of the barrier and
is the decay constant. Fig. 2 is a schematic of the energy dependence of
in the band

HOMO

β

Fig. 2. Schematic of how the decay constant
depends on the alignment of tunneling carrier energy between the HOMO–LUMO gap of the molecule.

gap region between the molecular HOMO and LUMO. The highest-occupied molecular orbital (HOMO) or the lowest-unoccupied molecular orbital (LUMO) of the molecule produces an energy barrier for a metal electron (e− ) of energy E between the HOMO and LUMO levels. The value of
goes to zero as the energy E approaches the band gap edges (HOMO or LUMO) and reaches a peak,
bp , near the middle of the gap, called the branch point. The valence band branch (HOMO) is connected to the conduction band branch (LUMO) at the branch point, which is the charge neutrality level. In general the HOMO–LUMO gap depends on the molecular geometry, which aects
. Thus the tunneling current may be “exponentially” sensitive to the molecular structural parameter . We study the tunneling characteristics of carotene by calculating the complex band of polyene backbone and phenyl ring, separately. We begin with the repeating unit segment –(CH)K(CH)– of the polyene backbone. Since the dimerization of the polyene may change due to vibrations, lattice distortions, or the external eld, we are interested in the relation between the dimerization , the chain’s HOMO–LUMO gap Egap , and the decay constant
. We assume that the CKC–C bond angle and the through bond tunneling length dCKC–C = dCKC + dC–C are xed to the experimental values, 125◦ and  respectively. With these two constraints, the 2:79 A, conformation of the periodic unit –(CH)K(CH)– is uniquely determined by the dimerization . For  = 1, there is no dimerization in the carbon chain, while as 

J. Li et al. / Physica E 19 (2003) 133 – 138 2.0

(A)

α=1.299

Energy (eV)

1.0

1.217 1.146 1.082 1.026 1.053

0.0

-1.0

-2.0

0

0.1

0.2

0.3

0.4

0.5

0.6

β (per carbon) 3.0

Egap (eV)

(B) 2.0 1.0

βbp (per carbon)

0.0

(C) αexp

0.4 0.2 0

1

1.05

1.1

1.15

Dimerization α

Fig. 3. Complex band structure of repeating unit –CHKCH–: (A) the decay constant
for dierent dimerization  (the energy reference has been set to the middle of gap for all the dierent ); (B) the HOMO–LUMO gap vs. dimerization ; (C) the peak decay constant at the branch point
bp vs. dimerization .

increases from 1 we have increased dimerization. We calculate the complex band structure of the unit cell –(CH)K(CH) – using an electronic Hamiltonian constructed by the local orbital density functional method Fireballs2000 [6–8]. The method to compute the complex band calculation is discussed in Ref. [9]. 1 Fig. 3A shows the decay constant
in the HOMO– LUMO gap region for dierent values of . The intersection of the
curves along the energy axis (
→ 0) corresponds to the HOMO/LUMO levels. 1 A computer program has been written to compute the complex bandstructure, given a local orbital Hamiltonian and overlap matrices and is available for download at http://physics.asu.edu/sankey/moltronics.

135

We show the dependence of the energy gap, Egap , on the dimerization  in Fig. 3B. With no dimerization, the HOMO–LUMO gap of the polyene chain goes to zero. Recall that a bandstructure approach assumes an innitely long chain. For carotene, the length of the chain is nite and there would exist a gap even without dimerization due to quantum connement effects. We neglect these eects here since our analysis is meant to give only qualitative estimates. When the polyene chain dimerizes ( ¿ 1), the HOMO–LUMO gap opens up. At the experimental dimerization exp (1.082), this theory gives Egap = 1:12 eV. This value is smaller than that of actual carotene molecule (described later) due to the innite length approximation and other simplication used here on the CH3 side attachment. The decay constant reaches a peak value
bp at the branch point near the middle of the gap. This represents the maximum reduction of tunneling current per unit cell along the innite chain. A simple tight-binding model for a one-dimension lattice of lattice constant
L gives a relation as
(E) = (2=L) ln((E) + (E)2 − 1) [9], where (E) = ((E − Ev )(Ec − E))=2t 2 + 1. Here Ev and Ec are the valence band maximum and conduction band minimum, respectively. For tunneling carrier at the branch point (assumed at the middle of the gap, E = (Ec − Ev )=2),
bp goes with the leading term, 2 Egap =t 2 and Egap =t, for weak (4t
Egap ) and strong (4tEgap ) hopping t, respectively. Fig. 3C indicates a linear dependence of
bp on  due to the linear dependence of Egap on  as in Fig. 3B. The value of  −1 . This means that the decay
bp at exp is 0:159 A −
L e for one pair of carbon atoms (one single and one double bond) is e−0:394 (on average, e−0:197 per carbon in the pair). A complex band calculation was also performed on a chain of ideal phenyl rings in a unit cell with a lattice  The result is shown in Fig. 4, and constant of 4:31 A. is compared to that of polyene (at exp ). The peak  −1 per unit cell (one full phenyl ring,
bp is 0:371 A refer to the inset of Fig. 4). Measuring in atom count,
bp = 0:4 per carbon, where we count four carbons along the circumference of the phenyl ring. This shows that the phenyl ring has a higher decay rate, or is less conductive, than polyene. A simple order-of-magnitude estimate of the conductance G of a carotene molecule can be obtained

J. Li et al. / Physica E 19 (2003) 133 – 138

136 1.5

3. I –V calculation

1.0

(A) Phenyl ring

1.0

(B) Polyene

Energy (eV)

0.5

Unit cell

0.0 -0.5

Energy (eV)

0.5

Unit cell 0.0

-0.5 -1.0 -1.5

0

0.1 0.2 0.3 0.4

β (per carbon)

-1.0

0

0.1

0.2

β (per carbon)

Fig. 4. Complex band structure of: (A) repeating unit –CH = CH– at the experimental dimerization (exp ) and (B) repeating one phenyl ring.

from the complex band structure using [10] G ≈ G0 e−
bp L where the quantum of conductance G0 equals 77 S and the
value is that value of
at the branch point. Assuming the total tunneling consists of three successive reductions from the two end phenyl rings and the polyene backbone, we decompose the exphenyl ponent into a summation as
bp L = 2
bp Lphenyl + polyene Lpolyene . The polyene backbone contains 9 pairs
bp of single–double bond segments, and with two phenyl rings, so that
bp L = 0:394 × 9 + (0:4 × 4) × 2 = 6:746. Using this result, the resistance (inverse of G) of a carotene molecule is estimated to be about 11 M. This value is within a factor of 100 to the low bias Ohmic resistance from scattering theory, which is 1:2 G for the trans isomer (see below). Note that our study shows that the decay constant
depends strongly on the HOMO–LUMO gap Egap . The experimental gap of polyene is about 2 eV. From Fig. 3A, the
curve (at =1:146) has a gap of about 2 eV with the
bp of 0.34 per carbon, or 0.68 per pair. Using this altered value,
bp L equals 0:68×9+(0:4×4)×2=9:32. This value gives a resistance of 145 M, an order of magnitude improvement due to the correction of Egap . These estimates, although qualitative, shows that the resistance is far from the “resonance” resistance of 12:9 k and that it is exponentially sensitive to the HOMO–LUMO gap.

The above estimates do not take into account effects of the contacts between molecules and metals. These are necessary to quantitatively improve the theory. The simple estimates also predict no dierence between isomers, i.e. between trans- and cis-carotene (see Fig. 1). In the synthesis of carotenedithiol, it was found that there is a mixture of two major isomers in the ratio 4 trans to 1 cis [2]. Thus, we calculate I –V curves of the two carotene isomers sandwiched between gold electrodes. These calculation use the “standard theory” based on the Landauer–B
uttiker formalism implemented within scattering theory [9,11–13]. We rst obtain the optimized structure of carotene. Structural optimization of the carotene molecule using density functional theory (DFT) produced far too little  dimerization ( = 1:03) resulting in 1.40 and 1:36 A for the single and double bond lengths, respectively. This results in a HOMO–LUMO gap which is too small (about 1:0 eV) compared to the experimental value which is in the range of 1.7–2:2 eV. It is known that Hartree–Fock (HF) calculations generally do much better than DFT in predicting the correct dimerization in the case of polyacetylene [14]. Our Hartree– Fock structural optimization on the carotene molecule  being averages yields  = 1:10, with 1.46 and 1:33 A for the single and double bond lengths, respectively. The resulting energy optimized carotene geometry from HF is then input to DFT, to produce a HOMO– LUMO gap of around 1.67 and 1:79 eV, for trans- and cis-carotene, respectively. These values are in accord with existing experimental data. They dier from that predicted by Fig. 3 because in general the quantum connement eect existing in isolated molecule is absent in corresponding innite system and the polyene backbone is loaded with CH3 side attachment. Calculations of the theoretical current–voltage curve for the carotene molecule were based on the Landauer–B
uttiker formalism [4] with the transmission function calculated using quantum mechanical scattering theory within DFT. The model system was an innite two-dimension lattice (mono-layer) of carotene molecules sandwiched between gold contacts (3 × 3) made up of eight ideal Au (1 1 1) layers in a super-cell structure as discussed in detail elsewhere [9]. The self-consistent Kohn–Sham single electron

J. Li et al. / Physica E 19 (2003) 133 – 138

Fig. 5. I –V curves of trans isomer for dierent contacts. The inset indicates the projection angle  of sulfur–sulfur end-groups on the gold surface. This angle uniquely denes a contact.

6 4

Current (nA)

states were obtained for this system using Fireballs2000, a local atomic-orbital DFT-based method in the pseudo-potential local-density approximation (LDA). The Green’s functions needed to compute the current–voltage curves were calculated from the Hamiltonian and overlap matrix elements. Though the electronic structure calculation is performed with gold slabs of nite thickness, we extend the Green’s functions to include semi-innite contacts using a block recursion technique. Full details are given in Ref. [10] and related descriptions are in Refs.  [9,15]. The current is calculated as I (V )=(2e2 =h) T (E; V ) [fL (E − eV=2) − fR (E + eV=2)], where T (E; V ) is the transmission function, f is the Fermi function, and V is the voltage. We have assumed that the electron energies in the electrodes are shifted symmetrically by +eV=2 and −eV=2 for the left and right electrodes, respectively. This is consistent with experiment [2] which shows symmetric (I (V ) = −I (−V )) results. The calculations of the metal–molecule–metal are fully self-consistent at zero bias, and thus include charge transfer eects at the interface; however, the eects of bias and the electric eld are approximated simply by shifts in the Fermi levels of the left and right contacts. Including the eects of the electric eld is possible [12,13,16], but we have not included these eects here. The details of the molecule–metal contact are also important. We follow the general surface chemical bonding requirement: for the trans isomer, the AuSC bond angle (here Au means the hollow site in the gold layer) was xed at 90◦ and the SCC angle xed at 110◦ resulting in an angle between the two sulfur end-groups and the normal to the gold substrate of 30◦ . As might be expected for insertion into the tilted docosanethiol matrix, because the experiment of Ref. [2] are performed by insertion of carotene within a SAM of docosanethiol (H(CH2 )22 SH). There is still an uncertainty concerning exactly how the carotene molecule ts into the docosanethiol lm. With this uncertainty in mind, we conceptually rotate the molecule about the surface normal on a xed conical surface oriented 30◦ with respect to the surface normal. The I –V curve dependence on the rotation is shown in Fig. 5. We use the projection angle of sulfur–sulfur end-groups on the gold surface as the last parameter to uniquely dene a rotation (see the inset in Fig. 5). For all studied contacts the I –V curves distribute within

137

2 0

θ=0° θ=30° θ=60° θ=90° θ=120° θ=150°

-2 -4 -6 -1.5

-1

-0.5

0

0.5

1

1.5

Applied Bias (V)

Fig. 6. I –V curves of cis isomer for dierent contacts. The contact denition is the same as in trans isomer.

a narrow range in the low bias and spread wider at higher bias. The cis isomer was more dicult to accommodate between two gold slabs, and the AuSC bond angle had to be distorted to 110◦ . I –V curves were computed for all allowed rotational arrangements as in trans isomer (showed in Fig. 6). Comparing to trans isomer, the I –V curves of cis isomer have lower values in the low bias range and increases sharper at higher bias. This overall feature may distinguish these two isomers. In the low bias, the average ohmic resistances are 2:2 G (cis) and 1:2 G (trans). Thus, the calculation shows that a cis isomer is almost only half as

138

J. Li et al. / Physica E 19 (2003) 133 – 138

conductive as trans one. The calculated resistance of trans isomer is in agreement with the measurement (which is about 5 G) within a factor of 4 [2]. For cis isomer it is a factor of two. However, the theoretical simulations suggest that dramatic dierences in the shapes of the I –V curves at higher bias should exist for the cis and trans isomers, which is not observed in experiment. This may be due to the low ratio of cis isomers and the diculty of insertion of a cis isomer into the docosanethiol matrix. 4. Conclusion We have theoretically studied conductive properties of sulfur-terminated carotene between gold contacts using local orbital density functional theory. The dependence of the tunneling parameter on the degree of single–double alternation within the polyene backbone is discussed based on results from the polyene complex band structure. We estimated the resistance of a carotene molecule based on the complex band structure calculation. We showed that the tunneling resistance of a molecule is exponentially sensitive to the HOMO–LUMO gap. The electron tunneling current–voltage characteristics is calculated by Landauer–B
uttiker formalism. We found that the detailed contact conguration plays an important role. We emphasize the importance of a real model of the contact between the molecule and metals. The theory also indicates that contribution from dierent isomers are distinguishable by their overall shape. The agreement between the I –V curves in theory and experiment is nearly quantitative. Acknowledgements This work was supported by the NSF (ECS 01101175 and DMR9986706). We have beneted

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