Quantum Interference in Molecular Wires: Electron Propagator

J. Phys. Chem. C 2008, 112, 13769–13774

13769

Quantum Interference in Molecular Wires: Electron Propagator Calculations William D. Wheeler† and Yuri Dahnovsky*,‡ Department of Chemistry/3838 and Department of Physics and Astronomy/3905, 1000 E. UniVersity AVenue, UniVersity of Wyoming, Laramie, Wyoming 82071 ReceiVed: May 30, 2008; ReVised Manuscript ReceiVed: July 8, 2008

In this work, we study the effect of electron correlations on transport in molecular tunnel junctions where a 1,4-benzenedithiol (BDT) molecule is placed between two gold electrodes. By employing electron propagator computational methods with geometry optimization in an electric field, we analyze different spatial configurations of a bridge attachment to metal electrodes and find that the configuration where the molecule is both stretched and tilted with respect to the surfaces by the angle of 28.5° provides the best agreement with the experimental data at εf ) -1.35 eV and thus determines the most probable attachment geometry. In addition, we investigate current-voltage characteristics vs Fermi energy and find strong negative differential resistance at εf ) 0.4 eV. To explain this phenomenon, we provide a detailed quantum mechanical analysis indicating that an applied field causes the breaking of aromaticity of a π molecular orbital of a benzene ring, resulting in the sharp drop in the electric current. The participation of the benzene π molecular orbital in the conductivity is surprising because one would expect that the main contribution is due to the overlap between the gold and the adjacent sulfur orbitals. Such a prediction provides an insight into how novel molecular devices can be constructed with desirable properties by a suitable modification of the surfaces. 1. Introduction Electron transport in molecular tunneling devices is an appealing phenomenon to quantum chemists who are interested in novel and at the same time practical applications of a welldeveloped quantum chemical computational methodology to control electronic molecular properties by interfaces, environments, and external parameters such as an electric (magnetic) field and temperature. Electron binding energies, molecular and atomic orbitals, ionization potentials, electron affinities, etc., play a crucial role in determining the suitability of a molecule to be a proper candidate for a device with desired properties. These parameters should be accurately determined by reliable methods. There are several computational schemes that are based on a tight binding approximation (uncorrelated electrons),1-5 density function theory,6-14 a many-body electron scattering approach,15-17 and an ab initio electron propagator approach.18-22 Electron propagator methodology,23-31 with its computational and conceptual advantages in the interpretation of experiments which probe molecular ionization energies and electron affinities,32-38 is now within the reach of the computational chemist who seeks further improvements to mean-field results on molecular wires by introducing ab initio electron correlation effects. Recently, the expression for electric current (see refs 18 and 19) has been derived in terms of outputs from a standard commercial software package, Gaussian-03,39 and then this expression was used for numerical calculations for current-voltage characteristics of a molecular wire with a 1,4-benzenedithiol (BDT) molecular bridge.20-22 A BDT molecular bridge was intensively studied by different experimental groups40,41 where monotonic current-voltage characteristics were obtained. In addition, this molecule as a molecular bridge was computationally studied by employing DFT (see the discussion in ref 8) * Corresponding author. E-mail: [email protected]. † Department of Chemistry/3838. ‡ Department of Physics and Astronomy/3905.

where the discrepancy of the factor of 50 with the experimental data41 was found. Other DFT-based electronic structure calculations differ in details; nevertheless, most of the calculations provide similar results (see the discussion in review42). Such a substantial discrepancy between the theory and experiment has motivated our investigation to use alternative ab initio electron structure computational schemes, in particular electron propagator methods.20-22 These techniques rigorously include electron correlations providing accurate ionization potentials and electron affinities.32-38 In our approach, we employ the expression for electric current in terms of bridge and electrode Green’s functions.47-49 As it follows from the formal many-body theory,43-46 the exact (Lehmann) representation of electron Green’s functions is determined by poles that are ionization potentials and electron affinities rather than electronic excitation energies. The poles can be found as outputs of numerical calculations within the Gaussian software.39 It is important to note that the electron correlation approach to tunneling in molecular junctions provides a different physical picture than the DFT approach. The lead electrons tunnel to bridge ionization levels rather than to excited states of a neutral molecule. Such a picture is natural in order for the bridge to remain a neutral molecule during a transport process. In this work we continue our investigations of electron transport in molecular tunneling junctions by using electron propagator methods, with the goal to understand a quantum mechanism of electric conductivity. In our previous studies22 we explained the experimental dependence of current-voltage characteristics at higher voltages while we found a strong Coulomb blockade effect at lower voltages that had not been experimentally observed. Our investigation included only a single d gold atomic orbital21 and was focused on a planar conformation which forced the C-S-Au angle to be 180°; however, the experimental geometry of the attachment is unknown. Indeed, it is more natural for a bridge molecule to be attached with 109° tetrahedral angle as in a gas phase.50

10.1021/jp804799g CCC: $40.75  2008 American Chemical Society Published on Web 08/08/2008

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Forcing this angle to be 180° causes the sulfur p-orbitals to overlap with benzene π-orbitals, destroying the aromaticity of the ring. Such a configuration is energetically unfavorable. Thus, additional investigation of electric current through a BDT molecule with different attachment conformations and the inclusion of all gold atomic orbitals is necessary. Moreover, we provide some unusual Fermi energy dependencies of electric current with a strong negative differential resistance at low voltages that could be useful to design new electronic devices. 2. Computational Details 2.1. Electric Current. The Hamiltonian that describes a tunnel junction is given by the following expression:

ˆ )H ˆL+H ˆR+H ˆM+H ˆ LM + H ˆ RM H

(1)

where the left lead, the right lead, and the bridge are described ˆ L, H ˆ R, and H ˆ M, respectively. The interaction by Hamiltonians H between the left (right) lead electrons with the bridge electrons ˆ LM (H ˆ RM). The expression for electric current was is given by H derived by Meir and Walgreen47,48 while a final expression which is convenient for numerical calculations using electron propagator methods was found in refs 18 and 19:

J)

e p

∑

γL(l)γR(l)al[ fL(εl) - fR(εl)]

l

|γL(l) + γR(l)|

(2)

Since the electron density in the leads is zero for εl > εF, the line-width functions, γL,R, become zero, resulting in vanishing electric current. In eq 2, γ matrices for left and right electrodes are given by nl

γL(l) )

∑

n

ci2(l)γiL(l),

γR(l) )

∑

i)nl+1

i)1

ci2(l)γiR(l)

(3)

In eq 3, the index i denotes a gold terminal orbital with the corresponding partial spectral function, γi, and the lth pole’s matrix element, ci(l), that reflects the contribution of the terminal orbital, i, in the total current. The electric current in eq 2 is expressed in terms of the pole strengths, al, the poles, εl, and the Fermi functions of the left, fL(εl), and right, fR(εl), electrodes. In the ith gold terminal orbital, the lth pole’s matrix elements for the left and the right electrodes can be defined in the following manner:18 n

ciL(l) ) 〈i|l〉L )

∑ dk(l)SikL

k)1 n

ciR(l) ) 〈i|l〉R )

∑ dk(l)SikR

(4)

k)1

Here dk(l) is a coefficient in the expansion of the lth Dyson R orbital in terms of the contributing atomic orbitals. SL, is an ik overlap matrix element between the left (right) electrode ith atomic orbital with the kth atomic orbital of the bridge. The R expansion coefficients, dk(l), overlap matrix elements, SL, ik , the pole strengths, al, and the poles, εl, are obtained as output from ab initio electron propagator calculations using the Gaussian software package.39 To find these parameters, we use the outer valence Green’s function (OVGF) computational method. The summation in eq 2 is over a minimum of 20 Dyson poles. Typically, only one or two contribute to the current. The summations in eq 3 include all of the basis functions (1s, 2s, 3s, 4s, 5s, 6s, 7s, 2p, 3p, 4p, 3d, 4d, 5d, 6d; nl ) 36) for the left (right) Gold atom in the computation. However, only a few have

Figure 1. A0 represents the gas-phase conformation with the optimized Au-Au distance of 8.842 Å. A1 depicts the molecule with the stretched Au-Au distance of 10.061 Å along the Y-axis. A2 shows the molecule with the stretched Au-Au distance of 10.061 Å placed between electrodes that are separated by 8.824 Å.

significant contributions. All γL(l)’s and γR(l)’s in eq 3 are set equal to 1 eV. 2.2. Basis Sets. All molecular orbital calculations are performed using the Gaussian-03 program39 with the 6311++G(d,p) basis sets on S, C, and H atoms and the MDF basis set and pseudopotential on gold atoms, unless otherwise noted. For one conformation, additional calculations are performed using the aug-cc-pVDZ basis set on S, C, and H. Geometry optimizations for each value of an electric field are calculated using the B3LYP51 method. All geometry optimizations are followed by frequency calculations to verify that the structure is the global minimum. Subsequent calculations to compute Dyson orbital pole strengths and poles (ionization potentials and electron affinities) are carried out using the OVGF method. 2.3. Conformational Geometry. In an initially step, a full geometry optimization is performed on the Au-BDT-Au molecule to obtain the most energetically favorable starting conformation. Three spatial conformations are considered for the computation of the current as shown in Figure 1. These conformations are selected to represent a variety of experimental conditions. The first conformation, A0, is that of the fully optimized gas-phase molecule, except that the Au atoms have been fixed on the Y-axis, with an Au-Au distance that is the same as for the fully optimized geometry (8.842 Å). An electric field is then applied along the Y-axis, and the geometry is reoptimized. This is repeated for as many as 11 values of the applied field. The field values used represent applied voltages of 0.0-2.0 V. In this conformation, the electrodes can be thought of as perfectly spaced with the Au atoms on each electrode

Quantum Interference in Molecular Wires

Figure 2. Surface plot of current vs voltage and vs Fermi energy for Au-BDT-Au molecular wire with conformation A0 (Au-Au ) 8.842 Å). γL ) γR ) 1 eV.

exactly opposite each other. In the second conformation, A1, the Au atoms are displaced along the Y-axis so that the Au-Au distance is 10.061 Å. The electric field values used in this conformation are chosen to approximate applied voltages the same as for A0. For this geometry, the electrodes are no longer perfectly spaced, but the Au atoms on each electrode remain exactly opposite each other. The third conformation, A2, has the same Au-Au distance as the second, but the Au atoms are displaced in the Z-direction by (2.4 Å, making the angle between the Au-Au axis and the Y-axis of θ ) 28.5°. This effectively reduces the field along the Au-Au axis to E| ) E cos θ and additionally applies a field perpendicular to the Au-Au axis of E⊥ ) E sin θ. Such a geometry approximates an experimental situation where the electrodes have the ideal spacing (8.842 Å), but with an imperfect alignment. The symmetry for all calculations is C1, except for the case of the gas-phase calculation with no applied field, where the symmetry is Ci.

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Figure 3. Surface plot of current vs voltage and vs Fermi energy for Au-BDT-Au molecular wire with conformation A1 (Au-Au ) 10.061 Å). γL ) γR ) 1 eV.

Figure 4. Surface plot of current vs voltage and vs Fermi energy for Au-BDT-Au molecular wire with conformation A2 (see Figure 1). γL ) γR ) 1 eV.

3. Results and Discussion In this section we present numerical calculations of electric current for three different conformations of the molecular device shown in Figure 1 and study different regions of current-voltage characteristics depending on Fermi energy, εf. 3.1. Current-Voltage Characteristics. In the current calculations, Fermi energy is a parameter because only two gold atoms model the metal electrodes. For the gas phase conformation A0 we calculate the current-voltage characteristics using an OVGF electron propagator method in two different basis sets: 6311G++(d,p) and aug-cc-pVDZ. The computational results in both basis sets are close, indicating the reliability of the OVGF calculations. Thus, in the future we present only the calculations in the 6311G++(d,p) basis set. Figures 2- 4 show the surface plots of the current through a BDT molecule for the different conformations, A0, A1, and A2, as a function of Fermi energy and applied voltage. All three figures exhibit similar dependencies in the whole range of Fermi energies. However, there are some essential differences. At lower Fermi energies, from εf ) -1.5 to 0.0 eV, the stretched conformations display a smaller peak with a fall off at higher voltages. At Fermi energies from εf ) 0.0 to +0.5 eV, there is a peak at lower voltages. At lower voltages, Figures 2-4 exhibit approximately the same values of the current within a 10% difference, but for the gas-phase configuration, A0, the molecule is still stable up to

Figure 5. Current vs voltage at εf ) -1.10 eV for configuration A0.

2 eV (see Figure 2) while at configurations A1 and A2 the bridge breaks down at V ) 1.5 V (see Figures 3 and 4). From these three configurations we choose the one that provides the best fit to the experimental data.41 For three different configurations, A0, A1, and A2, the current-voltage characteristics are shown in Figures 5-7 at εf ) -1.10 eV, εf ) -1.30 eV, and εf ) -1.35 eV, respectively. We have studied different configurations with various Fermi energies and concluded that configuration A2 with εf ) -1.35 eV explains the experimental dependences41 in the best way. Indeed, configuration A1 exhibits a small Coulomb blockade

13772 J. Phys. Chem. C, Vol. 112, No. 35, 2008

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Figure 6. Current vs voltage at εf ) -1.30 eV for configuration A1.

Figure 9. Probability contributions from different gold terminal orbitals 2 into the electric current, |cL,R i (40)| , vs voltage and vs Au atomic orbital number for the 40th Dyson molecular orbital in conformation A0 for (a) the left and (b) for the right electrodes. Orbitals 1-7 represent s gold orbitals, 8-16 are p orbitals, and 17-36 are d gold orbitals for the left electrode, while 37-43 are s gold orbitals, 44-52 are p orbitals, and 53-72 are d gold orbitals for the right electrode.

Figure 7. Current vs voltage at εf ) -1.35 eV for configuration A2.

Figure 8. Current-voltage characteristics at εf ) 0.40 eV for configuration A0.

effect while configuration A0 reveals a smaller peak (a negative differential resistance) in the current-voltage characteristics at lower voltages. Our calculations show that the main contribution to the current for these Fermi energies is due the overlap between gold d with sulfur p orbitals in molecular Dyson orbital 40. 3.2. Negative Differential Resistance. In this subsection we focus on the peak in the current at positive Fermi energies shown in Figure 2. The current-voltage characteristic is given in Figure 8. The detailed analyses reveals that at εf ) 0.4 eV the main contribution to the conductivity at voltages above 1 eV is due

to Dyson orbital 40 while the conductivity at lower voltages is determined by Dyson orbital 42. It is important to understand a quantum mechanical origin of such a peakwise behavior vs applied voltage. In general, a current-voltage behavior depends on three different factors according to eqs 2-4: (a) the difference between the Fermi functions, fL(εl) - fR(εl), (b) coefficients dk(V), and (c) overlap integrals SL,R ik (V). The difference between the Fermi functions selects the range of Dyson poles and Dyson orbitals participating in the conductivity, and the coefficients dk(V) determine the weight of a kth bridge atomic orbital in a molecular Dyson orbital, while the overlap integrals reflect the strength of the interaction between the ith gold orbital and the kth bridge atomic orbital (e.g., a sulfur orbital). The signs of dk(V) and SL,R ik (V) can be positive or negative, resulting in strong quantum interference between the electrons described by different bridge and gold atomic orbitals, while the contributions due to the gold terminal orbitals, ci2, are summed independently according to eq 3. The dependence of the Fermi function difference on voltage was analyzed in detail in ref 22. Figures 9 and 10 demonstrate the probability contributions, ci2, from different gold terminal orbitals into the electric current in accordance with eqs 2 and 3. As shown in Figures 9 and 10, the probability contributions from the left and right electrodes are not equal due to an electric field. The current-voltage characteristics at higher voltage is entirely determined by the 40th Dyson molecular orbital as depicted in Figure 9. The contribution from different gold terminal orbitals monotonically grows with applied voltage. From Figures 2, 8, and 9 one finds the sharp increase in the current at V > 1.5 V. In this case, there is a strong conformation change in the molecule rotation about the C-C-S-Au dihedral angle of 76° only on the right side of the junction. Figure 10 demonstrates that the main contribution to the peak in the current-voltage characteristics at lower voltages for εf ) 0.4 eV is due to 6dyz and 4py gold orbitals. Moreover, the probability contribution grows at lower voltages with the peak at V ) 0.227 V and then decreases almost to zero at V ) 0.727

Quantum Interference in Molecular Wires

Figure 10. Probability contributions from different gold terminal 2 orbitals into the electric current, |cL,R i (42)| , vs voltage vs and Au atomic orbital number for the 42nd Dyson molecular orbital in conformation A0 for (a) the left and (b) for the right electrodes. The numeration of the orbitals is the same as in Figure 9.

Figure 11. Molecular Dyson orbital 42 at V ) 0.227 V.

Figure 12. Molecular Dyson orbital 42 at V ) 0.227 V.

V. Such a nonlinear dependence defines the negative differential resistance in the molecular junction. This property can be used in practical devices where a gold surface is modified in a way to change the Fermi energy to εf ) 0.4 eV. The peak in current-voltage characteristics shown in Figures 2 and 8 can be explained from a quantum chemical point of view. Figure 11 depicts Dyson molecular orbital 42 where one finds the strong overlap between the aromatic π-molecular orbital of the benzene ring and the 6dyz gold atomic orbital at the peak value, V ) 0.227 V. As discussed above, the current vanishes at V ) 0.727 V. The corresponding molecular orbital picture is shown in Figure 12 where the aromaticity of the benzene π molecular orbital is destroyed, resulting in the weak overlap with the gold atomic orbitals of the left electrode. Consequently, the current sharply drops. 4. Conclusions In this work we have studied conductivity through a molecular tunnel junction with a 1,4-benzenedithiol (BDT) molecule as a

J. Phys. Chem. C, Vol. 112, No. 35, 2008 13773 molecular bridge employing electron propagator computational methods. A molecular device based on a BDT molecule was intensively studied both experimentally40,41 and theoretically (see review in ref 8). The discrepancy between experimental data and DFT calculations was found to be of the factor of 50. Such an inconsistency between the theory and the experiment has motivated us to study this molecule using an alternative approach based on ab initio electron propagator techniques. As wellknown from electronic structure calculations of molecules, these methods rigorously include electron correlations and provide remarkably correct results that are capable of the explanation of different experimental data and predict novel spectral properties of molecules.32,38 Since the expression for electric current in a Meir-Walgreen’s approach47-49 contains bridge electron Green’s function and numerical calculations within electron propagator methods provide numerical electron Green’s functions as output, then the electron propagator methods become convenient for the calculation of electric current within the Mier-Walgreen’s approach. In the exact Lehmann’s representation of Green’s function,43-46 the poles represent ionization potentials and electron affinities rather than excitation energies as follows from tight-binding and DFT approaches. In our previous studies,20-22 we used electron propagator calculations to explain experimental current-voltage characteristics of the Au-BDT-Au device. In those calculations, we found a strong Coulomb blockade effect in contrast to the smooth behavior observed experimentally at lower voltages. To better understand nonequilibrium transport properties in such devices at lower voltages, we have calculated electric current for different configurations of a bridge attachment to metal electrodes (see Figure 1). Since the gold leads are represented by two Au atoms, the value of Fermi energy remains unknown, and therefore, electric current has been studied in a wide range of Fermi energies. The surface plots of the current vs voltage and εf are shown in Figures 2-4 for different bridge-surface configurations. Some interesting features in these dependences have been revealed: (a) a smooth behavior of the current vs voltage at negative εf and (b) peaks at positive Fermi energies. We have analyzed different configurations shown in Figures 5-7 and found that configuration A2 (see Figure 7) provides the best agreement with experiment41 at εf ) -1.35 eV. The analysis reveals that the main contribution to the current is due to a complicated quantum interference from different bridge atomic orbitals with positive and negative coefficients dk and overlap integrals with gold terminal orbitals Sik for the left and right electrodes (see eqs 2-4). However, the gold terminal orbitals contribute independently. It has been found that in this region of Fermi energies the molecular Dyson orbital 40 provides the main contribution. In addition, our studies indicate a strong conformation change at V > 1.5 V in conformation A0. The conformation change is due to a rotation about the C-C-S-Au dihedral angle of 76°. At positive Fermi energies we have found a peak in the conductivity which shows a region of strong negative differential resistance. Although such Fermi energy is apparently not available for conductance in the device with the BDT molecule and gold electrodes, it gives us insight into how novel molecular devices with desirable properties can be constructed by a suitable modification of the surfaces. As follows from our analysis, Dyson orbital 42 provides the main contribution to a peak behavior of the current. Indeed, Figure 10 demonstrates the probability contributions from different gold terminal orbitals, |ci|2, into the electric current in accordance with eqs 2 and 3 where the main contribution to the peak is due to 6dyz and 4py

13774 J. Phys. Chem. C, Vol. 112, No. 35, 2008 gold orbitals. A peakwise dependence of the current vs voltage can be explained from a quantum chemical point of view where a strong overlap between the aromatic π-molecular orbital from the benzene ring and the 6dyz gold atomic orbital is observed at the peak value V ) 0.227 V, as depicted in Figure 11. If the voltage is changed to V ) 0.727 V, the current vanishes and the aromaticity of the benzene π-molecular orbital is destroyed, resulting in the weak overlap with the gold atomic orbitals of the left electrode. Consequently, the current sharply drops. The contribution of the benzene π-molecular orbital is surprising because one would expect that the main contribution to the conductivity is due to the overlap of the gold and nearest sulfur atomic orbitals. Since the electron propagator methods employ optimized conformations and include electron correlations in the most accurate manner, they provide a promising computational approach for the explanation of experimental data and the prediction of novel important properties in molecular tunneling junctions. Acknowledgment. We are grateful to the National Science Foundation that has supported this research through Grant CHE0426090. References and Notes (1) Seminario, J. M.; Derosa, J. M. J. Am. Chem. Soc. 2001, 123, 12418. (2) Joachim, C.; Vinuesa, J. F. Europhys. Lett. 1996, 33. (3) Magoga, M.; Joachim, C. Phys. ReV. B 1997, 56, 4722. (4) Tikhonov, A.; Coalson, R. D.; Dahnovsky, Yu. J. Chem. Phys. 2002, 116, 10909. (5) Tikhonov, A.; Coalson, R. D.; Dahnovsky, Yu. J. Chem. Phys. 2002, 117, 567. (6) Lang, N. D.; Avouris, Ph. Phys. ReV. Lett. 2000, 84, 358. (7) DiVentra, M.; Pantelides, S. T.; Lang, N. D. Phys. ReV. Lett. 2000, 84, 979. (8) Tomfohr, J.; Sankey, O. F. J. Chem. Phys. 2004, 120, 1542. (9) Kosov, D. S. J. Chem. Phys. 2004, 120, 7165. (10) Kosov, D. S. J. Chem. Phys. 2003, 119, 1 (2003) (11) Evers, F.; Weigend, F.; Koentopp, M. Phys. ReV. B 2004, 69, 235411. (12) Koentopp, M.; Burke, K.; Evers, F. Phys. ReV. B 2006, 73, 121403(R). (13) Arnold, A.; Weigend, F.; Evers, F. J. Chem. Phys. 2007, 126, 17410. (14) Scmitteckert, P.; Evers, P. Phys. ReV. Lett. 2008, 100, 08640. (15) Kozhushner, M. A.; Polyanskii, V. S.; Oleynik, I. I. Chem. Phys. 2005, 319, 368. (16) Kozhushner, M. A.; Polyanskii, V. S.; Oleynik, I. I. Phys. ReV. B 2006, 74, 165103. (17) Oleynik, I. I.; Kozhushner, M. A.; Polyanskii, V. S.; Yu, L. Phys. ReV. Lett. 2006, 96, 096803. (18) Dahnovsky, Yu.; Zakrzewski, V.; Kletsov, A.; Ortiz, J. V. J. Chem. Phys. 2005, 123, 184711. (19) Dahnovsky, Yu.; Ortiz, J. V. J. Chem. Phys. 2005, 124, 144114. (20) Dolgounitcheva, O.; Zakrzewski, V. G.; Kletsov, A.; Sterling, M. R.; Dahnovsky, Yu.; Ortiz, J. V. Int. J. Quantum Chem. 2006, 106, 3387. (21) Sterling, M. R.; Dolgounitcheva, O.; Zakrzewski, V. G.; Dahnovsky, Yu.; Ortiz, J. V. Int. J. Quantum Chem. 2007, 107, 3228. (22) Kletsov, A.; Dahnovsky, Yu. J. Chem. Phys. 2007, 127, 144716. ¨ hrn, Y. Propagators in Quantum Chemistry; Wiley(23) Linderberg, J.; O Interscience: Hoboken, NJ, 2004. (24) Szabo, A.; Ostlund, N. S. Modern Quantum Chemistry; Dover Publications: New York, 1996.

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