pubs.acs.org/NanoLett
The Relation between Structure and Quantum Interference in Single Molecule Junctions Troels Markussen,†,‡ Robert Stadler,*,§ and Kristian S. Thygesen† †
Center for Atomic-scale Materials Design (CAMD), Department of Physics, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark, ‡ Danish National Research Foundations Center of Individual Nanoparticle Functionality (CINF), Department of Physics, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark, and § Department of Physical Chemistry, University of Vienna, Sensengasse 8/7, A-1090 Vienna, Austria ABSTRACT Quantum interference (QI) of electron pathways has recently attracted increased interest as an enabling tool for singlemolecule electronic devices. Although various molecular systems have been shown to exhibit QI effects and a number of methods have been proposed for its analysis, simple guidelines linking the molecular structure to QI effects in the phase-coherent transport regime have until now been lacking. In the present work we demonstrate that QI in aromatic molecules is intimately related to the topology of the molecule’s π system and establish a simple graphical scheme to predict the existence of QI-induced transmission antiresonances. The generality of the scheme, which is exact for a certain class of tight-binding models, is proved by a comparison to first-principles transport calculations for 10 different configurations of anthraquinone as well as a set of cross-conjugated molecular wires. KEYWORDS Quantum interference, graphical rules, coherent electron transport, single molecule junctions, density functional theory
T
some authors been classified as Fano resonances5,33,34 in order to place them into the wider context of antiresonances in molecular wires.35,36 Most recently, the effect of side groups has also been investigated for so-called crossconjugated molecules,9-12 which appear to be promising candidates for implementing switching and rectifying behavior in single-molecule junctions.11
he interest in employing quantum interference (QI) to control the current flow through single-molecule devices has recently intensified.1-14 The idea of utilizing the electron’s wave nature for electronics was introduced in mesoscopics two decades ago,15-18 but it was also recognized early on, both theoretically19 and experimentally,20,21 that QI effects were responsible for the observed reduction of the conductance of a benzene molecule contacted in the meta configuration as compared to the para and ortho configurations. The latter phenomenon has been explained in terms of phase shifts of transmission channels or interfering spatial pathways19,22-27 with an intimate link to the nodal structure of the involved molecular orbitals. The role of QI has also been investigated for larger aromatic molecules1,6-8,28 as well as for incoherent transport in the Coulomb blockade regime.29-32 Regarding the utilization of QI as an enabling tool for single-molecule devices, two conceptually different proposals have been put forward. One is to use a local gate potential to adjust the position of QI-induced transmission nodes relative to the bias window,1,6-8 and the other is to control the electron transmission through chemical/conformational modification of side groups to aromatic molecules.2-4,13 Due to the typical shape that interference patterns caused by side groups adopt in the transmission functions, they have by
The most striking signature of QI in phase-coherent molecular transport junctions is that of transmission antiresonances. Such antiresonances have been shown to occur at energies where the phases picked up by the electron along different pathways of the molecule exactly cancel each other, thereby leading to destructive interference.19 In this Letter, we show that the occurrence of transmission antiresonances at the Fermi energy of aromatic molecular junctions is completely determined by the molecular structure, and we establish a simple graphical method for their prediction. The graphical scheme is exact for a certain class of tight-binding (or Hu¨ckel) models3 and is here shown to apply much more generally in praxis. In particular, we show that it correctly predicts the presence/absence of QI-induced antiresonances in 10 configurations of anthraquinone differing only in the position of the two oxygen atoms as well as in three different cross-conjugated molecules. Within a single-particle picture, the transmission probability of an electron entering a molecular junction with an energy E is given by38
* To whom correspondence should be addressed:
[email protected]. Received for review: 05/12/2010 Published on Web: 09/29/2010 © 2010 American Chemical Society
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T(E) ) Tr[GΓLG†ΓR](E)
(1)
where FIGURE 1. Demonstration of how the graphical QI prediction scheme is applied for electron transport through benzene in ortho, meta, and para connections. For the connections without QI, a continuous path can be drawn either with (para) or without (ortho) pairing up remaining sites. This is not possible for the only connection, which exhibits QI for benzene (meta), where the isolated site has been marked by a green spot.
G ) (E - Hmol - ΣL - ΣR)-1 is the Green function matrix of the contacted molecule, ΣL/R is the self-energy matrix due to the left/right lead, and ΓL/R ) i(ΣL/R - ΣL/R†). Let us assume that the Hamiltonian describing the molecule is given in terms of a basis consisting of localized atomic-like orbitals, φ1, φ2, ..., φN, and that only the two orbitals φ1 and φN couple to the left and right leads, respectively. In this case the transmission reduces to
T(E) ) γ(E)2|G1N(E)|2
in terms of simple diagrams constructed according to the following rules: (i) Two sites can be connected by a path if they are nearest neighbors. (ii) At all internal sites, i.e., sites other than 1 and N, there is one incoming and one outgoing path. It is now straightforward to show that the condition for complete destructive interference is fullfilled if it is not possible to connect the external sites 1 and N by a continuous chain of paths and at the same time fulfill the rules (i) and (ii). On the other hand, if such a continuous path can be drawn, then the condition (4) is not fulfilled and a transmission antiresonance does not occur at the Fermi energy. To see how the graphical rules follow from eq 4, we use the following general expression for the determinant of a M × M matrix
(2)
where G1N(E) is the (1N) matrix element of the Green function, G, and γ(E) ) [ΓL(E)]11 ) [ΓR(E)]NN, where we have assumed identical coupling strengths to the left and right leads for simplicity. The coupling strength, γ(E), is related to the electronic structure of the electrode and the overlap between molecular orbitals and electrode wave functions. On the other hand, the QI effects are contained in G1N(E). The latter can be calculated using Cramer’s rule
G1N(E) )
det1N(E - Hmol) det(E - Hmol - ΣL - ΣR)
i)M
(3)
det(A) )
σ
where det1N(E - Hmol) is the determinant of the matrix obtained by removing the first row and Nth column from E - Hmol and multiplying it by (-1)1+N. Taking the Fermi energy to be zero without loss of generality, we can then state the condition for complete destructive interference, T(EF) ) 0, as
det1N(Hmol) ) 0
(5)
i)1
where σ is a permutation of the numbers 1, 2, ..., M and sgn(σ) equals 1 (-1) for an even (odd) permutation. In our case A equals Hmol with the first row and Nth column removed and thus M ) N - 1. Note that for the Hu¨ckel model only a few of the terms in eq 5 will be nonzero due to the assumption of nearest neighbor hopping and due to the onsite matrix elements being zero (and equal to the Fermi energy). We now associate each matrix element Aij with a path connecting sites i and j and note that the external indices 1 and N appear only once in each term of eq 5 (one path starting at site 1 and one path ending at site N) while the internal indices appear twice (one incoming and one outgoing path). The graphical rules now follow by inspection, which was explicitly demonstrated in ref 3 for a set of small representative tight binding matrices with three, four and five atomic orbitals. In Figure 1 we show how the method is applied for predicting QI for the well-known case of benzene connected to two leads in ortho, meta, and para configurations.19 While for ortho all atomic sites are part of a continuous line from the left to the right lead, for para the two orbitals not on this line form a pair with a bond between them (marked by a red elliptical confinement in Figure 1). As is well-known these two configurations do not exhibit QI. The meta setup
(4)
We stress that our theoretical treatment is based on the assumption of phase-coherent transport, i.e., the tunneling time, p/γ, is assumed to be small compared to the coherence time of an electron on the molecule. This condition is expected to be fullfilled for molecules which form covalent bonds to the electrodes where γ . kBT. In the opposite limit of weak coupling to electrodes, QI has been investigated on the basis of rate equations.29-32 In general Hmol will be a complicated matrix and the above condition (eq 4) will never be exactly fullfilled. Nevertheless important insight can be obtained by considering the simple case of a Hu¨ckel model of the π electrons in conjugated hydrocarbon systems. In this model we take all the pz orbitals to have equal onsite energy (zero) and include only nearest neighbor hopping (t). In this special case eq 4 can be expressed © 2010 American Chemical Society
∑ sgn(σ) ∏ Aiσ(i)
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FIGURE 2. Configurations of the anthraquinone-based molecules studied in this article and demonstration of the graphical QI scheme. The notation I1-I5 and N1-N5 refers to “Interference” and “No interference”, respectively. For N1-N5 one can draw a continuous path connecting the left and right contact sites and pair up all remaining sites. For I1-I5 it is not possible to connect the contact sites without creating isolated sites (green spots). For the DFT-based electron transport calculations, the thiol anchors are attached to flat Au(111) surfaces after removing the protecting H atoms from the sulfur.
In Figure 3 we show the calculated transmission functions of the 10 anthraquinone molecules (note the logarithmic scale). The black lines are results of first-principles DFT transport calculations while the red dashed lines are results of tightbinding (TB) models. The DFT transport calculations are performed using the GPAW electronic structure package.39,40 In a first step, the molecules are inserted between two Au(111) surfaces and the molecules plus the outermost surface layers are relaxed. We find the most stable position of S on the Au(111) surface is a bridge site slightly shifted toward the hollow site, where only minor changes in the S-Au bonding geometry were observed for the different molecular configurations shown in Figure 3. The supercell contains seven layers of 3 × 3 Au atoms and we use a (1,4,4) k-point sampling. Electronic wave functions are represented on a real space grid and exchange correlation is described by the PBE functional.41 The zero-bias transmission function of the relaxed structures is obtained using a double-ζ polarized (DZP) basis set of numerical atomic orbitals.42 The TB models describe each C, S, and O atom by a single pz orbital with an on-site energy of zero (equal to the Fermi energy) and include a nearest neighbor hopping of t ) -3 eV. The orbitals on S are connected to wide band leads via a coupling strength γ ) 1 eV chosen to approximately fit the width of the DFT transmission peaks. Returning to Figure 3 we first observe that the TB transmission functions of molecules I1-I5 all vanish at the Fermi energy while those of molecules N1-N5 are all finite. This is just as predicted by the graphical QI scheme. The latter should come as no surprise since the graphical scheme is
on the other side has an isolated atomic site without a neighbor which is not part of the curve (highlighted by a green spot), and therefore QI is correctly predicted to occur. As a first application of the graphical QI method, we consider the 10 different anthraquinone-based molecules shown in Figure 2. Recent electrochemical experiments on the molecule denoted I2 in Figure 2 revealed switching behavior based on a reversible reduction to its hydroquinone redox state.37 Density functional theory (DFT) transport calculations showed that the oxidized state of this molecule, when contacted to gold electrodes, exhibits QI while the reduced state does not, leading to a significant on/off conductance ratio.14 This result was rationalized by inspection of the molecular orbitals; however, a direct link between the occurrence of QI and the geometrical/ chemical structure of the molecule was not pursued. Returning to Figure 2, we note that the difference between the anthraquinones lies only in the position of the two oxygen atoms. Schematic drawings of the topology of the π system is listed next to each of the molecules. Note that as far as topology is concerned it is enough to consider the aromatic core of the molecules as the anchor and spacer parts are identical in all cases. Application of the graphical QI scheme to the molecules suggests that a transmission antiresonance at the Fermi energy should be present (absent) for molecules I1-I5 (N1-N5). For molecules N1-N5 we have shown one possible path connecting the left and right contact sites, while for molecules I1-I5 we have given one example illustrating that it is not possible to connect the left and right contact atoms and still fullfill the rules (i) and (ii) given above. © 2010 American Chemical Society
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FIGURE 4. Molecular structures (top), application of the graphical QI scheme (middle), and transmission functions from TB models (bottom left) and DFT (bottom right) for three cross-conjugated molecules. The transmission function color code is as follows: CC1 (solid red), CC2 (dashed blue), CC3 (dashed-dotted black).
a σ bond with H and consequently becomes unavailable to the π-system. In this case it is easy to verify that the graphical QI scheme predicts the absence of destructive QIsagain in agreement with the first principles calculations.14 Next, we consider the three cross-conjugated molecules shown in the upper panel of Figure 4. In a recent series of papers,9-12,23,27 Ratner and co-workers demonstrated that cross-conjugation generally leads to destructive QI thus making such molecules particularly interesting for single molecule devices. Referring to the middle and lower panels of Figure 3, which shows the application of the graphical QI scheme to the three molecules and their transmission functions obtained from DFT (right) and TB (left), we conclude that also for these cross-conjugated molecules the graphical QI scheme correctly predicts the presence/absence of QIinduced transmission nodes. Furthermore, by comparing the three molecules one can derive from the graphical scheme why QI effects are so often observed for crossconjugated molecules. Whereas molecules CC1 and CC2 are chemically stable, molecule CC3 has an unpaired electron and is therefore a highly reactive free radical. Due to the conditions imposed by the cross-conjugation on the alignment of double bonds, it is impossible to design a molecule that is cross-conjugated and at the same time has an even number of C atoms in the side group. By adding a single atomic site, one would create (1) a free radical, (2) an ion, or (3) a cumulated diene. Since cross-conjugation requires the bond at the branching point to be a double bond, (1-3) can only be avoided by adding pairs of C atoms and the graphical scheme intuitively illustrates that adding pairs of sites cannot remove QI effects. The success of our graphical scheme clearly suggests that QI effects should be regarded as robust, intrinsic properties of the molecule itself or, more precisely, the
FIGURE 3. Transmission functions calculated from DFT (black solid lines) and tight-binding (red dashed lines) for the structures I1-I5 and N1-N5, respectively. For the DFT calculations the thiol anchors are attached to flat Au(111) surfaces after removing the protecting H atom from the sulfur. In the tight-binding models the on-site energy of the pz orbitals is set to zero (equal to EF) for both C, O, and S, and the nearest neighbor hopping is t ) -3 eV. The contact sites (S atoms) are coupled to wide band leads with a coupling strength γ ) 1 eV. TABLE 1. DFT Conductance Values in Units of G0 ) 2e2/h for the Molecules Listed in Figure 2a molecule
G (G0)
molecule
G (G0)
I1 I2 I3 I4 I5
7.71 × 10-8 1.82 × 10-8 3.03 × 10-4 3.55 × 10-4 8.84 × 10-4
N1 N2 N3 N4 N5
1.38 × 10-2 8.57 × 10-2 7.09 × 10-3 8.98 × 10-3 6.19 × 10-3
a Note that the conductance values in the left column are systematically lower than those in the right column. This is a result of destructive QI occurring close to the Fermi energy for molecules I1-I5.
exact for the considered type of TB models. It is, however, striking that the first-principles transmission functions reproduce this trend. In fact, as can be seen from Table 1 the conductance of molecules I1-I5 is systematically and significantly lower than the conductance of molecules N1-N5. We are therefore led to conclude that the graphical QI scheme is indeed able to predict the occurrence of transmission antiresonances for the anthraquinone-based molecules. We note in passing that a reduced form of anthraquinone does not exhibit an antiresonance at EF in general.14 This can be understood from the following argument: When the oxygen atoms are reduced, the pz orbital of O is used to form © 2010 American Chemical Society
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Danish Center for Scientific Computing. R.S. is currently supported by the Austrian Science Fund FWF, Project No. P20267.
topology of its π system. In particular QI effects should be largely insensitive to the electrode material and the details of the contact geometry. The presence/absence of a transmission antiresonance in the relevant energy range between the highest occupied (HOMO) and lowest unoccupied (LUMO) molecular orbital is a property of the symmetry of the molecular orbitals and not their precise energetic position. Consequently, the fact that DFT typically underestimates HOMO-LUMO gaps44,45 and has problems in describing level alignment at solid-molecule interfaces46 does not influence the conclusions of the present work. Indeed, we have verified that shifting the HOMO and LUMO levels of the molecules (specifically opening the gap by application of a scissors operator47) may change the exact position of a transmission antiresonance relative to EF but it does not affect its overall shape and it always remains inside the HOMO-LUMO gap. In order to clarify the range of applicability of the graphical QI scheme, we want to also assess its limitations. As already noted, it relies on the assumption of phase-coherent transport. Within the phase-coherent regime, we further assumed that the onsite energies of the relevant atomic orbitals of the molecule do not differ too much on the scale of the hopping strength t. This assumption is certainly met in all-carbon organic molecules but may also extend to more complex molecules containing heteroatoms as the results of this work indicate. Our model only considers the π electron system, and deviations could result if the σ states contribute significantly to the transport. Finally, it was assumed that the molecule is chemically coupled to each of the leads via a single atomic orbital only and with a similar coupling strength γ(E) for both electrodes. If more orbitals can couple to the electrodes, there will be correspondingly more terms in eq 2. In summary, we have assessed a graphical scheme to predict the presence of QI effects in electron transport through aromatic molecules. The graphical scheme is based on a simple tight-binding model of the π electron system but as shown here its application extends to a broad range of realistic molecular junctions. By comparing to first-principles transport calculations, we demonstrated that the scheme correctly predicts the presence/absence of transmission antiresonances at the Fermi energy for 10 different configurations of anthraquinone as well as for three different crossconjugated molecules. The graphical scheme provides a direct link between molecular structure and QI and should be a useful guideline in the design of molecules with specific transport properties.
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Acknowledgment. The center for Atomic-scale Materials Design (CAMD) receives funding from the Lundbeck Foundation. The Center for Individual Nanoparticle Functionality (CINF) is sponsored by the Danish National Research Foundation. The authors acknowledge support from FTP through Grant No. 274-08-0408 and from The © 2010 American Chemical Society
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