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Polarizability as a Molecular Descriptor for Conductance in Organic Molecular Circuits Shobeir K.S. Mazinani, Reza Vatan Meidanshahi, Julio L Palma, Pilarisetty Tarakeshwar, Thorsten Hansen, Mark A. Ratner, and Vladimiro Mujica J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b06241 • Publication Date (Web): 21 Oct 2016 Downloaded from http://pubs.acs.org on October 25, 2016

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The Journal of Physical Chemistry

Polarizability As A Molecular Descriptor For Conductance In Organic Molecular Circuits. Shobeir K. S. Mazinani,† Reza Vatan Meidanshahi,‡ Julio L. Palma,¶ Pilarisetty Tarakeshwar,† Thorsten Hansen,§ Mark A. Ratner,k and Vladimiro Mujica∗,† School of Molecular Sciences, Arizona State University, Tempe, Arizona 85287-1604, USA, School of Electrical Computer and Energy Engineering, Arizona State University, Tempe, Arizona 85287-8806, USA, Department of Chemistry, The Pennsylvania State University, Fayette, The Eberly Campus. 2201 University Drive, Lemont Furnace, Pennsylvania 15456, USA, Department of Chemistry, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen,Ø, Denmark, and Department of Chemistry, Northwestern University, 1145 Sheridan Road, Evanston, Illinois 60208-3113, USA E-mail: [email protected]

To whom correspondence should be addressed School of Molecular Sciences, Arizona State University, Tempe, Arizona 85287-1604, USA ‡ School of Electrical Computer and Energy Engineering, Arizona State University, Tempe, Arizona 852878806, USA ¶ Department of Chemistry, The Pennsylvania State University, Fayette, The Eberly Campus. 2201 University Drive, Lemont Furnace, Pennsylvania 15456, USA § Department of Chemistry, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen,Ø, Denmark k Department of Chemistry, Northwestern University, 1145 Sheridan Road, Evanston, Illinois 60208-3113, USA ∗ †

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Abstract We explore a connection between the static molecular polarizability and the molecular conductance that arises naturally in the description of electrified molecular interfaces and that has recently been explored experimentally. We have tested this idea by using measured conductance of few different experimental design motifs for molecular junctions and relating them to the molecular polarizability. Our results show that, for a family of structurally connected molecules, the conductance decreases as the molecular polarizability increases. Within the limitations of our model, this striking result is consistent with the physically intuitive picture that a molecule in a junction behaves as a dielectric that is polarized by the applied bias, hence creating an interfacial barrier that hinders tunneling. The use of the polarizability as a descriptor of molecular conductance offers significant conceptual and practical advantages over a picture based on molecular orbitals. To further illustrate the plausibility of this idea, we have used Simmons' tunneling model that incorporates image charge and dielectric effects to describe transport through a barrier that represents the molecular junction. In such a model, the barrier height depends on the effective dielectric constant of the electrode-molecule-electrode junction, which in turn can be approximately expressed in terms of the molecular polarizability via the classical Clausius-Mossotti relation. Despite the simplicity of our model, it sheds light on a hitherto neglected connection between molecular polarizability and conductance, and paves the way for further experimental, conceptual and theoretical developments.

INTRODUCTION Organic molecular electronics have garnered tremendous interest 1–5 due to the possible ability to operate as verstaile circuit elements. 5–7 This is outshone by the recently explored interface of molecular electronic devices with biochemistry 8 and the study of various phenomena such as charge transfer at the bio-nano interface. 9 One of the key properties for an 2 ACS Paragon Plus Environment

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efficient design is the ability of these molecular structures to transport electrons effectively. There are endless numbers of possibilities for the molecular structures. 10 While having a plethora of options for design gives scientists and engineers many opportunities, it can become overwhelming in the absence of a physical parameter that can be used as a screening factor that can a priori discern between desirable and unattractive candidates. 7,11 For example, such a factor can be very important in finding an optimized reader molecule in a recognition junction since synthesis and measuring or computing the conductance of these systems is very time consuming. 12,13 Traditionally, some rules of thumb have been used to make fast predictions about the conductance in a single molecule junction, and parameters such as length, 14 temperature and the energy gap between HOMO and LUMO have been considered in the literature. 15 But due to the intrinsic complexity of transport and its relation to molecular electronic structure, these rules do not always hold. 16,17 In this context, molecular polarizability can be used as a guideline for designing systems with desired transport properties.As important as it might be, this type of relatively simple correlation must be taken with caution because conductance depends on some other factors, e.g. the detailed coupling of the molecule to the electrodes and non-linear voltage effects that cannot possibly be captured in a molecule-only description. The physical plausibility of the connection between conductance and polarizability can be inferred from Mujica and coworkers’ work that described the spatial profile of the electrostatic potential in a junction by solving Schrödinger and Poisson equations self-consistently by connecting the quantum electronic density to the electrostatic potential. 18 The picture that emerges from this model is that if many-body and inelastic effects, such as charging and electron-phonon coupling, are neglected molecules behave to a large extent as a dielectric whose polarization response counteracts the driving field (See figure 1). 18



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Figure 1: The figure displays the spatial voltage profile across a junction. For a vacuum junction (left panel) the voltage V drops linearly between the contacts, as predicted by Poisson equation. For a molecular junction, assuming that the molecule behaves as a dielectric, which polarizes in opposite direction to the applied voltage causing the onset of molecular charges (δ − δ + ) at the interface. This is accompanied by a voltage drop essentially at the interface.

This leads to a spatial profile that differs substantially from that of a vacuum junction between two electrodes, which for a one dimensional system, is a linear function of the inter-electrode separation as found by solving Poisson equation for zero charge density, and corresponds rather to a S-shape function associated with a spatial profile characterized by the fact that the potential drop occurs essentially at the interfaces between the molecule and the electrodes. 18 Once the crucial role of the molecular bridge in determining the local dielectric properties of a junction has been accepted, the next conceptual step is to connect the dielectric constant to the molecular polarizability, which we introduce by assuming the validity of the ClausiusMosotti relation from electromagnetism, a very important aspect of our model that will be discussed below. Once this is achieved, the missing link is the connection between the conductance itself and the molecular polarizability. A strong theoretical support for this connection for quasi-metallic junctions has been addressed previously. 19,20 . In this work, we approach this problem in its simplest form, assuming that a molecular junction can be described as a simple tunneling junction and use Simmons’ model, which connects the barrier properties, that in turn determine the current and the conductance, to the junction’s dielectric constant. 4 ACS Paragon Plus Environment

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The main goal of this study is to explore the robustness of this connection for different design motifs and discuss the domain in which it is applicable. To this end, we have investigated the correlation between calculated molecular polarizabilities and the experimental zero voltage conductance of different families of molecules used in junctions, and our results are remarkably consistent with the qualitative and even quantitative predictions of our simple model. We finish by discussing the limitations and possible applications of the use of polarizability as a molecular descriptor for conductance in helping the design of next generation nano-electronics components.

METHODS Geometry optimizations and polarizability calculations were carried out at the Density Functional Theory level by using the Becke gradient-corrected exchange functional and Lee-YangParr correlation functional with three parameters (B3LYP) and the 6-31G* basis set by ORCA electronic structure package. 21–26

RESULTS AND DISCUSSION As mentioned in the Introduction, we will approach the rational design of molecular electronics materials by studying the correlation between polarizability and conductance. It is worth mentioning that since polarizability is a tensor property and conductance is a scalar quantity, we use as a molecular descriptor the isotropic molecular polarizability which is defined as: 1 1 α ¯ = tr(α) = (αxx + αyy + αzz ) 3 3 We have investigated systems whose experimental conductance has been reported and tested the correlation between calculated polarizabilities and such measured conductance. A first important result is that an increase in the static polarizability of the molecules results in a



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decrease in the conductance values, as should be expected from the simple model of molecules as dielectrics. Although the functional mathematical representation of this correlation is fairly general, each family of molecules has its own fitting parameters. We classify our results based on different molecular groups and show that the correlation holds within each group, thus stressing the potential of polarizability as a molecular descriptor that could provide valuable guidance for the design of new molecular electronic devices. It should be noted that the isotropic polarizability is clearly the simplest metric to use and other important properties of the polarizability tensor, particularly anisotrpic effects in the direction of orientation of the molecules can be of interest.

Halobenzene, The Effect of Substituents As an initial step, we investigated the effect of halogen substitution on benzene’s conductance. Due to their electronegative nature, halogens are electron withdrawing groups. Hence they reduce the electron density on the benzene’s π-system which results in a decrease in the energy of the HOMO orbital in comparison to HOMO energy of benzene. 27 The conductance values reported by Venkataraman and coworkers are plotted against calculated molecular polarizability; Figure 2 indicates that the conductance of the molecule decreases, although not significantly (< 5%), as its static polarizability increases due to the effect of the substituents. Further systems and design motifs are studied in order to find more conclusive results regarding the connection between polarizability and conductance in molecular junctions.

Amine-Gold Linked Molecular Motif The thiol group has been used extensively as an anchoring group between the molecular bridge and gold electrodes. 4,14,28,29 The strong bond between gold and sulfur is the reason for this general interest towards the Au-S anchoring groups. The main issue with sulfur-gold motif is the dramatic dependence of the conductance on the anchoring groups’ geometrical 6 ACS Paragon Plus Environment

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Figure 2: The correlation between conductance (milig0 ) and the static molecular polariz3 ability (Å ). In these systems the halogens affect the conductance, not very significantly, through modifying the HOMO of the molecule. It can be inferred that polarizability can be used to predict the effect of this substituents on conductance of benzene. 27 parameters. 27,30,31 Nitrogen-based anchoring groups have been used as an alternative to S-Au bonding interfaces. 32 Due to the reproducibility and smaller effect of the geometry of the linker on conductance; using this amine-based motif in designing the molecular circuits as well as thermoelectric systems seems to be a sensible path. 27,33,34 To investigate the robustness of our conjecture, we calculated the polarizabilities of the diamine family and the bipyridine family and plotted them against the measured conductance values. 27,33,34 Figure 3 summarizes the results.



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(a)

(b)

(c)

Figure 3: The structures of corresponding compounds are presented. Carbons are represented by black, hydrogens with grey and nitrogens with blue spheres (a) Structures 1, 2 and 3 belong to the diamine family whereas 4, 5 and 6 are from the bipyridine family of molecules (b). (c) Measured conductance (milig0 ), obtained from Venkataraman and 3 coworkers work, are plotted with respect to calculated polarizability (Å ). The figure suggests that for a family of molecules we can use polarizability as a descriptor of conductance to predict the electron transport properties of a molecule.

We attribute the difference between the conductances of the two families to the different nature of their individual anchoring group.


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Thiophene and Furan Oligomers Five-membered heterocycles, in particular thiophene and furan, have been used extensively in organic electronic devices such as organic field effect transistors (OFET) and organic photovoltaics (OPV). 35–40 The oligomers can be considered as a cis π-conjugated (CH)x system that are stabilized by a heteroatom. 41 These systems are aromatic. Breslow and coworkers showed that an increase in the (negative) aromatic stabilization energy correlates with the conductance values in systems with thiophene and furan motifs. 42 We have selected the aromatic systems in Breslow’s work, to put the polarizability-based design guideline through a strict test. Compound 7, 8, 9 and 10 (named in accord with Breslow’s work) are 2,5-bis-(4-aminophenylethinyl)furan,. 2,5-bis-(4-aminophenylethinyl)thiophene, 2,5-bis(4-aminophenyl) furan and 2,5-bis-(4-aminophenylethinyl) thiophene respectively (Figure 4-a) . The calculated polarizability and measured conductance are presented in Figure 4-b. There are two sets of correlations for these systems. First, the correlation between 7 and 9 (8 and 10) that are furan (thiophene) based compounds that are tweaked by the addition of an alkyne group. Second, the furan versus thiophene (7, 8 and 9, 10). It can be seen that for both sets of molecular wires, an increase in polarizability is followed by a decrease in conductance of the heterocyclic molecules. It is worth mentioning that polarizability has been used in other studies as an indicator of aromaticity, which establishes a connection between electron delocalization and conductance for this family of compounds. 43,44 Increasing the length of a molecular wire by increasing furan/ thiophene units mainly modifies the LUMO and it has been shown to have a negligible effect on the energy of the HOMO, which again points out to the limitations of schemes based on molecular orbitals to predict the behavior of the conductance . 42,43,45



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(a)

(b)

(c)

Figure 4: (a) and (b) show the structure of the compounds 7, 8 and 9, 10. In the structures Nitrogen, oxygen, sulfur, carbon and hydrogen are blue, red, yellow,black and grey. (C) The calculated polarizabilities versus experimental conductance shows that for the families of molecules, we observe how the change in the polarizability correlates with the changes in the conductance of the junction.

Hydrogen Bond Motif A ubiquitous motif in organic molecular electronics, in particular bio-inspired designs, whose use permits taking advantage of the unique properties of hydrogen bonding to build low cost, bottom-up organic materials with desirable properties. 46–52 Nishino et al have measured the conductance through hydrogen bond and showed that at short distances hydrogen-bonded wires have higher conductance than alkane chains. 53 Based on their results, we investigate the connection between polarizability and conductance for the family of carboxylic acid alkane thiols. Figure 5 shows the robustness of the correlation between conductance and


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static molecular polarizability. We have previously shown for a set of biologically-relevant hydrogen-bonded systems, that polarizability can be used as a guideline to qualitatively predict conductance through the hydrogen bonded system. 54

Figure 5: The correlation between measured conductance and calculated static polarizability of three hydrogen bonded carboxyl groups is shown in this plot. The labels show the total number of carbons in the molecular junction in a manner consistent with the work of Nishino et al . 53

Barrier Model of Conductance To explore the connection between polarizability and conductance, we consider a model of molecular conductance as a tunneling process. This model essentially ignores all the manybody, vibronic interactions, and inelastic aspects of the transport process and assumes that the molecule acts as a one-dimensional tunneling barrier where the barrier is specified by two parameters: the height and the width of the barrier. Simmons’ model, which includes image charges and dielectric effects, has been extensively used for the description of tunneling through metal-molecule interfaces with remarkable success. 55 We are particularly interested ¯ which has been shown in the in the expression for the height of the tunneling barrier, φ, Simmons’ model to be related to the dielectric constant as: 55 s1 + s2 1.15λx s2 (x − s1 ) 1 φ¯ = φ0 − qV − ln( ). 2x ∆s s1 (x − s2 ) r


(1)


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with λ defined by: λ=

q 2 ln2 8π0 x

(2)

Where q is the electron charge, V is the bias voltage, φ0 is the barrier height, s1 and s2 are the turning points in the barrier shape, ∆s = s2 − s1 , x is the nominal width of the barrier, 0 is the vacuum permittivity and r is the dielectric constant. Since we are not using an atomistic representation of the contacts, the specific values of the turning points have to be taken as parameters of the model. The relation between the current and the voltage in the tunneling junction can be recast in the following form: J = J0 (φ¯ e−A J0 =

√

φ¯

− (φ¯ + qV ) e−A

√

¯ φ+qV

q 2πh(σ∆s)2

) (3)

√ A = ( 4π∆s 2me ) h where me is mass of electron, h is the Plank’s constant, and σ is a correction factor as described by Simmons and is usually around 1. 55

The simplest connection between the dielectric constant r of an electrified interface and the molecular polarizability α of the intervening medium is given by Clausius-Mossotti equation: r = γ=

0 +2γα 0 −γα

(4)

NA d 3M

with NA being Avogadro’s number, M is the molar mass of the material and d is its density. It applies to the dielectric constant of a bulk dielectric material that is homogeneous and isotropic, and it connects the static polarizability of a single molecule with the susceptibility of a three-dimensional molecular material. 56,57 The basic microscopic premise of this relation is that in a uniform electric field, each molecule is represented as a polarizable point dipole


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that experiences the external field inducing a polarization response. Although these conditions might seem too simplistic to describe a molecular junction, it has been shown that the Clausius- Mossotti equation can accurately represent the relation between polarizability and the dielectric function even for covalently-bonded bulk semiconductors. 58 More interestingly for our current purposes, the nanoscopic validity of the Clausius- Mossotti equation has been systematically explored by Natan et al. 57 and has been shown to correspond to a well-defined limit that provides us with a physically reasonable starting point, that is only applicable to simple junctions where tunneling is the dominant transport mechanism. The inclusion of hopping or hybrid transport mechanisms, like those present in DNA, is well beyond the validity of our simple model. The combined use of Equations 4 and 1 results in the desired connection between the effective barrier’s height and the polarizability: 0 − γα s1 + s2 − B( ) φ¯ = φ0 − qV 2x 0 + 2γα where B =

1.15λx 1) ln( ss21 (x−s ). ∆s (x−s2 )

(5)

The differential conductance g is defined as:

g(V ) =

∂J ∂V

(6)

And the differential conductance in the limit of zero voltage can be obtained straightforwardly from Equation (3). √ lim g(V ) = −qJ0 e−A

V →0



q 

A φ¯ φ¯   1− 2

(7)

In deriving Equation 7, we assumed that the molecular polarizability is not dependent on the bias voltage. Expanding the function representing the barrier height in terms of polarizability and inserting it into equation 7 gives:

g = g1 e

−β1 α

β1 C − α + ... 2

!


(8)


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where β1 =

ABγ √ 20

φ0 −B

, g1 = −J0 qe−A

√

φ0 −B

and C = 1 −

A 2

√

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φ0 − B. It can be seen from

equation 8, polarizability adds more corrections to a normal exponential decay behavior. Also we have shown that other underlying effects such as the role of substituents or the aromaticity that cannot be justified by length dependence can be predicted by polarizability as a molecular descriptor. Inspired by the correlation between polarizability and the classical analogue of the molecular "volume", we write α ' L3 , where L is a typical molecular linear dimension, obtained adding the values of bond lengths. Figure 6 plots Ln of the experimental conductance versus α1/3 and compares it with the conductance values calculated via equation 8 for the diamine and bipyridine systems 3. As it can be seen in the Figure 6, this correlation holds for families of structurally related compounds. For instance, comparison of the polarizability of a substitued-cyclohexane with benzene derivatives will not follow the aforementioned discussion since the modeling parameters for benzene is expected to be different for these two systems.

(a)

(b)

Figure 6: Plot of Ln of conductance versus third root of polarizability for diamine (a) and bipyridine (b). We used the following values and fitted our equation 8. We used the following values to fitting. (a) For diamine we used g1 = 11.26, C = 0.01 and β1 = 3.76. (b) For bipyridine family, we used g1 = 2.75, C = 0.01 and β1 = 4.31 A theoretical formulation beyond a barrier model can be cast in terms of Green’s functions, as is often done for the Landauer theory for molecular conductance. Response properties, such as the molecular polarizability, have been connected to single particle Green’s functions in the literature 59 , also they can be derived invoking the Keldysh contour techniques of non-equilibrium Green’s functions. 60 14 ACS Paragon Plus Environment

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The field-induced dipole moment as a function of time is given by

hµ(t)i = i

X

µij µkl

Z

n

o

< < adv dτ Gret jk (t − τ )Gli (τ − t) + Gjk (t − τ )Gli (τ − t) E(τ ),

(9)

ijkl

where Gret (t), Gadv (t), and G< (t), are the retarded, advanced, and lesser molecular Green’s functions respectively. This implies the familiar expression for molecular polarizability:

α(ω) = −

X

n

o

adv µij µkl Gret jk (ω)%li (−ω) + %li (ω)Gjk (−ω)

(10)

ijkl

Here % = −iG< is the single-particle reduced density matrix. Using this equation, together with the Landauer expression for the conductance, n

g = g0 T r ΓL Gret ΓR Gadv

o

(11)

where g0 is the quantum of conductance and Γ is the spectral density describing the coupling between the molecule and the L,R electrodes, it is possible to derive an expression equivalent to 8. A detailed exposition of this derivation will be reported elsewhere.

CONCLUSIONS In this work we have explored the rather substantial evidence of an existing correlation between the static isotropic molecular polarizability and the molecular contribution to the zero-voltage conductance of a molecular junction for certain families of molecules. We have also examined the physical origin of such a correlation via a model that connects the local dielectric properties of the junction to its transport behavior via changes in the effective shape of the associated tunneling barrier. While our model is clearly not a first-principle theory, it does incorporate strong physical plausibility arguments and provides a rationale for the apparent correlation between



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experimental conductances and calculated molecular polarizabilites. The use of a molecular property such as polarizability, has clear advantages over correlations based on an orbital picture of conductance and information about the molecular energy spectrum, e.g. the HOMO-LUMO gap, because these models are based on a strong assumption about the nature of conductance channels that breaks down in many important cases. It also relates to recent efforts by Jackson et al to use the Kirchhoff constant as a single variable characterizing the extent of electric network connectivity at the molecular level. 61 Finally, our findings strongly suggest, together with the already demonstrated connection between conductance and electron transfer rate 62–64 , that it should be possible to reformulate Marcus theory of electron transfer in terms of response functions associated to the frequency-dependent polarizabilities, a subject we are currently working on and that is related to the original approach taken by Marcus to this subject.

Acknowledgement V.M. thanks Prof. Abraham Nitzan for interesting discussions about the subject. The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. The authors declare no competing financial interest.


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