Letter pubs.acs.org/NanoLett
Molecular Rectifiers: A New Design Based on Asymmetric Anchoring Moieties Colin Van Dyck* and Mark A. Ratner* Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, United States S Supporting Information *
ABSTRACT: The quest for a molecular rectifier is among the major challenges of molecular electronics. We introduce three simple rules to design an efficient rectifying molecule and demonstrate its functioning at the theoretical level, relying on the NEGF-DFT technique. The design rules notably require both the introduction of asymmetric anchoring moieties and a decoupling bridge. They lead to a new rectification mechanism based on the compression and control of the HOMO/LUMO gap by the electrode Fermi levels, arising from a pinning effect. Significant rectification ratios up to 2 orders of magnitude are theoretically predicted as the mechanism opposes resonant to nonresonant tunneling. KEYWORDS: Molecular electronics, rectifier, diode, NEGF-DFT, single molecule, pinning
T
asymmetric Schottky barriers (S mechanism), asymmetric placement of a chromophore (A mechanism), or an asymmetric electron transfer through the frontier molecular orbitals (U mechanism). This last mechanism has been called the “true” unimolecular rectification.11,14,29 The standard paradigm to create such a unimolecular rectifier is the coupling of an electron donating fragment (Donor) and an electron withdrawing fragment (Acceptor) through a connecting bridge which can be either saturated (D-σ-A) or conjugated (D-π−A), in the spirit of the original rectification mechanism proposed in 1974.1 These fragments are then coupled to external electrodes using different principles and techniques with different success in rectification.14,29 The new mechanism that we propose in the following lies between the U and S mechanisms. Relying on non-equilibrium Green’s function with density functional theory techniques (NEGF-DFT),30−35 we recently reported that for a conjugated molecule bonded to gold electrodes by a thiol group, a Fermi level pinning phenomenon is observed.36,37 This means that the energy offset between the electrode Fermi level and the highest occupied molecular orbital (HOMO) orbital of the molecule (Fermi level alignment) is insensitive to the modification of the HOMO level energy of the isolated molecule (which can be approximated by the ionization potential). Quantitatively, this effect is characterized by an S-parameter close to zero
he concept of molecular electronics started with the idea that a molecule sandwiched between electrodes (a molecular junction) may act as a rectifying device.1 Forty years after this proposal, research in this field has evolved in several other directions2 with molecular junctions showing interesting features to create, for example, molecular wires,3,4 switches5−8 or transistors.9,10 However, despite the experimental evidence that a molecule can rectify,11,12 the molecular rectifier is still far less efficient than its solid state counterpart. Molecular rectifiers encounter two important issues: the robustness and the performance of the devices. We here address the performance aspect, proposing a new rectification mechanism at the theoretical level. The performance of these devices is characterized by the rectification ratio (RR), that is, the ratio between electrical currents through the junction in the conducting and insulating directions. While solid state diodes allow for ratios as large as 5 orders of magnitude, the ratio of molecular junctions is limited in many cases to one (or less than one) order of magnitude.11,13−17 This is especially true if we restrict the situation to single molecules contacted to identical metallic electrodes for which a ratio above 10 has never been reported.18−22 Using self-assembled monolayers (SAM) contacted by chemisorption to a bottom electrode and physisorption to a top electrode, ratios up to 2 or 3 orders of magnitude have been measured.23−27 To the best of our knowledge these systems are currently the best molecular rectifiers. However, monolayers may not be appropriate for building molecular-scale digital circuitry. We focus here on single-molecule rectification. Fundamentally, a molecular rectifier must include an asymmetric feature along the transport direction.28 There are several different ways to introduce such asymmetry, leading to a measurable rectification ratio, in a molecular junction. Recent literature11,14,29 shows that rectification can be induced by © 2015 American Chemical Society
S≡
d(E F − εHOMO) ≈0 d(IP)
(1)
where EF is the Fermi level of the electrodes, εHOMO is the energy corresponding to the HOMO level of the molecule when it is sandwiched between the electrodes (a quantity Received: October 24, 2014 Revised: February 10, 2015 Published: February 23, 2015 1577
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Nano Letters directly measurable in the photocurrent spectrum of the junction38), and IP is the ionization potential of the free molecule. When this S-parameter is close to zero, the Fermi level alignment is extremely hard to modify and tends to retain a constant offset, that is, the HOMO is pinned at a defined energy below the Fermi level in the perfect pinning case (S = 0). This effect has also been reported experimentally in the context of molecular electronics.39,40 It is widely established that this effect takes place at both metal/organic36,37,41−43 and metal/semiconductor44 interfaces and has a well-known origin: the interface dipole. This shifts the electrostatic potential in the molecular region with respect to the metal region, and the molecular levels with respect to the Fermi levels, modifying the vacuum level alignment. This creates a so-called Schottky barrier in the interface vicinity. By modulation of the magnitude of the interface dipole, the IP discrepancies of the isolated molecules can be absorbed and the Fermi level pinning occurs. We previously demonstrated that when the molecule is constituted of two conjugated fragments weakly coupled together, and each of these is contacted to the gold electrodes, the alignment of each fragment tends to be independent, promoting an independent pinning of each fragment to its respective electrode.36 This phenomenon has a strong impact on the electronic structure of a molecule connected to metallic electrodes. It lies at the origin of the localization of the initially delocalized frontier orbital when a bias is introduced between the electrodes (a relative shift between the left and right electrode Fermi levels). We referred to this feature as the polarization effect.36,37,45 Such a mechanism has been observed experimentally, leading to a measurable negative differential resistance (NDR).46 Our rectification mechanism, efficient at the single molecule level, takes advantage of this independent alignment based on decoupling between the two fragments. In this sense, it is close in spirit to the D-σ-A scheme proposed in the original rectification paper.1 On the other hand, the original mechanism relied on the tuning of the IP of the donor and the EA of the acceptor to concentrate the frontier orbitals and control the position of the corresponding levels in the molecular junction. However, we know from our previous work that when a molecule is strongly coupled to the electrodes, the position of these levels in the junction is not controlled by the properties of the isolated molecule, but rather by the chemical interaction at the metal/molecule interface.36,37 In light of this, we propose the following three design rules for a molecular rectifier: (1) Use asymmetrical anchoring groups to chemically bind the molecule to the electrodes: an accepting anchoring group (promoting a lowest unoccupied molecular orbital (LUMO) alignment) and a donating anchoring group (promoting a HOMO alignment) on opposite molecular extremities. (2) Use anchoring groups promoting alignment of the frontier orbitals in the vicinity of the electrode Fermi levels and substitute them on the molecule, preserving this feature and promoting a Fermi level pinning phenomenon. (3) Use a molecule made of two lateral conjugated fragments and couple these together in a way that efficiently prevents the π-coupling between the two fragments. In the following, we demonstrate that by applying this set of rules we obtain an efficient molecular rectifier. For this purpose, we use the NEGF-DFT approach to characterize the conductance properties of an elementary molecular junction obeying the design rules, see Figure 1. As anchoring groups, we use a thiol group on the left side of the molecule and a nitrile
Figure 1. Simple molecule used to illustrate the rectification mechanism. It is designed following the rules given in the manuscript: two conjugated fragments are coupled together through a saturated bridge, breaking the conjugation. Asymmetric anchoring atoms are substituted on each end: a thiol promoting a HOMO alignment and a nitrile promoting a LUMO alignment. On top is the chemical structure of the molecule, highlighting the different fragments. Below is the optimized geometry at the DFT/B3LYP level of theory with isosurfaces of the LUMO (−2.64 eV) and the HOMO (−4.43 eV) computed at the DFT/GGA.revPBE level for consistency with the transport computations. Note that the two conjugated planes are essentially parallel to each other.
group on the right side. Thiol strongly binds with gold electrodes and promotes the alignment of the HOMO molecular level in the vicinity of the electrode Fermi level.39,47−50 On the opposite side, the nitrile anchoring group has been shown to bind strongly to gold surfaces but to promote the alignment of the LUMO, thus rule (1) is satisfied.49,51−55 We know from our previous studies and literature that the thiol group directly substituted in the conjugated pathway can promote the alignment of the HOMO close to the Fermi level with a trend to stay pinned at a defined energy, as long as the conjugation between the anchoring group and the molecule is not broken.36,37 A good alignment is also observed in literature49,51−55 for the LUMO of the molecule connected to gold electrodes with a nitrile contact, and we show here that strong pinning can also be observed with this moiety (rule (2) is satisfied). These are substituted at the ends of two conjugated hexene fragments coupled by a saturated butane bridge to break the conjugation between the two fragments (rule (3) is satisfied). The rectification mechanism is quite efficient, allowing our example molecule to exhibit a computed rectification ratio of about 2 orders of magnitude, in a reasonable voltage regime. We discuss qualitatively the motivations in our design rules and the reasons why they are efficient. Essentially, the rectifier behavior is due to the opposition between resonant and nonresonant tunneling. We show that the origins of this opposition are first, the efficient compression of the HOMO/ LUMO gap by the metal/molecule contact and, second, the particular bias dependence. We show that taking advantage of these two effects is an efficient way to design a new generation of molecular rectifiers. Results and Discussion. We start our discussion by reporting the theoretical analysis concerning properties of the isolated rectifier molecule illustrated in Figure 1. The DFT/ 1578
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probability at the Fermi level (approximating the low bias conductance) is about 7 × 10−4. It is surprising at first sight to observe that the energy gap between the HOMO and LUMO peaks has been reduced from 1.79 eV for the isolated molecule to 0.47 eV in the molecular junction upon connection to the gold surface at exactly the same level of theory. We suggest that this gap reduction is due to the asymmetric alignment properties of the anchoring atoms together with the weak internal coupling between the two conjugated fragments. Indeed, as represented in Figure 3,
B3LYP optimized geometry (see the Methodology section for computational details) of this molecule is characterized by two planes corresponding to the left and right conjugated fragments. These two planar fragments and the saturated linker form a wire with two elbows in the structure at the intersection with the saturated bridge, as can be observed in Figure 1. The frontier orbitals as calculated at the DFT/GGA.revPBE level are also given in Figure 1. As expected, the HOMO is localized on the donor substituted fragment and the LUMO on the acceptor substituted fragment. These two orbitals are not delocalized over the two fragments, because it is (following the third design rule) prevented by the saturated bridge. The corresponding HOMO/LUMO gap is about 1.79 eV for the isolated compound. After introduction of this optimized molecule between the gold electrodes, we computed the transmission spectrum at equilibrium (see the Methodology section for details), as given in Figure 2, together with the molecular projected self-
Figure 3. Illustration of the rectification mechanism. On top is illustrated the connection of the molecule to the gold electrodes. The two fragments are asymmetrically substituted with donating and accepting anchoring groups (rule (1)). The chosen anchoring groups promote alignment of the HOMO and LUMO in the close vicinity of the Fermi levels after connection and charge transfer, illustrated by the gray arrows (rule (2)). If the fragments are well separated (rule (3)), this is done almost independently for the left and right fragments, leading to a gap compression upon connection to gold. Under positive bias, the HOMO and LUMO respectively tend to follow the left and right electrode Fermi levels and are also shifted by the electrostatic potential (blue line), become separated in energy, and stay out of the transmission window represented as dashed lines. In contrast, a negative bias leads to the presence of both the HOMO and LUMO in the transmission window at the origin of a significant conductance and the rectification effect.
Figure 2. Transmission spectrum of the molecular rectifier at equilibrium. The red crosses denote the MPSH spectrum identifying the origin of the transmission peaks around the Fermi level. The peaks below and above EF are associated with the HOMO and LUMO levels of the molecule (see Figure 1 for comparison), despite an energy gap that is much lower in comparison to the isolated molecule. This illustrates the gap compression arising from the metal/molecule interfaces.
considering the limiting case of two fragments totally decoupled from each other, the thiol group is known to promote the alignment of the HOMO in the vicinity of the Fermi level. On the other hand, the nitrile group promotes the alignment of the LUMO in the vicinity of the Fermi level. As these two fragments are almost independent because of the saturated bridge, they align almost independently according to the specific character of their substituted anchoring group, as illustrated in Figure 3. The result is a compression of the HOMO/LUMO gap by the metal/molecule contacts, which is an important feature of the molecular rectifier mechanism that we propose. This is evidence of what we explained in a previous study concerning the polarization: a pinning effect at the interface together with a low internal coupling leads to a strong control of the pinned orbitals by the electrodes.36 The pinning effect (defined by an S-parameter close to zero) is the driving force compressing the gap. If there is no mechanism to preserve the independent and different alignment on each side, this would be modified at each side when the two fragments are
consistent Hamiltonian (MPSH) spectrum (see the Methodology section for definition) to identify the resonant levels at the origin of the transmission peaks. We observe that both the HOMO and LUMO levels induce transmission peaks respectively below (about 0.22 eV in energy at maximal intensity) and above (about 0.25 eV in energy at maximal intensity) the Fermi level. We consider the energies at the maximal intensity of the resonant transmission peaks, −0.22 and 0.25 eV, as the Fermi level alignments of the frontier molecular levels in the molecular junction. In the transmission spectrum, the LUMO resonant peak intensity (3 × 10−2) is about 1 order of magnitude higher than the HOMO resonant peak (1 × 10−3). In between these two peaks, the transmission 1579
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Figure 4. On the left are the transmission spectra of the molecular rectifier, under positive biases. The spectrum at equilibrium, in black, evolves to the orange (+0.6 V) and then the blue (+1.2 V) spectra with an applied bias. The red/green arrows indicate the evolution of the HOMO/LUMO transmission peaks under bias, showing the increase of the gap. The peaks stay out of the transmission window as indicated by the dashed blue lines and do not participate to the current. On the right are the corresponding spectra under negative bias. This time, the evolution of the HOMO/ LUMO peaks points to a compression of the gap. These two peaks enter and merge in the transmission window, promote the conductance, and provide a high current for this polarity.
the approach of the HOMO and the LUMO offers an efficient transmission pathway, promoting a resonant tunneling regime at the origin of rectification. The qualitative model in this figure is actually a good description of what happens in the NEGFDFT computation. Figure 4 shows the evolution of the transmission spectrum under bias for both polarities. For a positive bias, the HOMO and LUMO peaks get separated and stay out of the transmission window, as indicated by the two arrows indicating their evolution under bias. As they separate, the transmission between the Fermi levels decreases because only a limited part of the tails of the HOMO and LUMO peaks remain in the transmission window. Indeed, the transmission at minimal intensity in the HOMO/LUMO gap goes from 7 × 10−4 at equilibrium to 1 × 10−5 at +1.2 V. The maximal transmission in the window at 1.2 V bias is about 4 × 10−4 at energies corresponding to the tip of the tail of the LUMO peak, which dominates the molecular conductance. According to the Landauer formula, only a small current can cross the junction for this polarity, as we are in the nonresonant tunneling regime. This is in good accordance with the qualitative picture given in Figure 3. For the opposite polarity, given by the right part of Figure 4, we observe the LUMO peak approaching the HOMO peak in energy and finally merging into a common transmission peak, increasing in intensity as we increase the applied voltage in the transmission window. This leads to the resonant tunneling regime, which is in accordance with the qualitative picture of Figure 3. The increase in intensity is occurring because the HOMO and LUMO are localized on their respective fragments, offering a delocalized pathway as they get closer in energy because of the gap reduction induced by the bias. As a result, the maximal transmission in the window is about 0.16 for bias
coupled by the saturated bridge and the gap of the isolated molecule would be preserved in the molecular junction. As the alignment and pinning effects in the strong coupling regime are driven by the interface dipoles, our proposed rectifier can be classified as an “S” rectifier, involving asymmetric Schottky barriers at each interface. In addition, the compression of the gap places the HOMO and the LUMO close to resonance with the Fermi levels, as intended in the 1974 paper. Our rectification mechanism involves the molecular orbitals and can be viewed as a “U” rectifier in which the tuning of IP and EA of the fragments is replaced by the tuning of the metal/molecule interaction. This interaction is mainly responsible for the position of the levels, and a supplementary tuning of the IP and EA would only have a limited impact on these positions because of the pinning effect. We suspect that the lack of alignment control may be the origin of the limited rectification ratios observed for molecular junctions in the strong coupling regime. Once a bias is applied to the molecule, we show at the bottom of Figure 3 that depending on the polarity two different scenarios can be expected. For a positive bias, the fragments separate in energy for two reasons.36 First, the electrostatic potential creates an energy offset separating the fragments on the left and right parts of the junction. This is represented by the blue line in Figure 3. Second, if the pinning effect is present the fragments tend to follow the Fermi level of the electrodes and preserve their equilibrium alignment. The result is a HOMO/LUMO gap augmentation with no orbital in the transmission window (the energy window between the dashed lines). The transport regime is dominated by nonresonant tunneling. When a negative bias is applied, the fragments get closer in energy and the HOMO/LUMO gap is further compressed. If this is happening in the transmission window, 1580
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0) does not occur, as the HOMO does not remain pinned. By extrapolation of the HOMO peak position at resolved biases, we compute the position of the HOMO peak in the merged HOMO/LUMO peak at −1.2 V bias to be around −0.87 eV below the left electrode Fermi level, while it was about −0.22 eV at equilibrium. The LUMO peak is also getting further from the right electrode Fermi level under bias, the corresponding energy difference goes from 0.25 eV at equilibrium to 0.50 eV at 1.2 V bias. As the pinning condition is not satisfied for this polarity, the HOMO/LUMO gap does not cancel under bias and remains around 0.17 eV at −1.2 V applied bias. We suppose this feature is due to the weak coupling between the two fragments. These are not fully independent, which translates into a resistance of the molecule to preserve a significant HOMO/LUMO gap. This naturally requires a modification of the alignment at each interface and counterbalances the pinning effect. We consider that a crossing, as would be expected from the qualitative picture in Figure 3, could only appear if there is absolutely no coupling between the fragments. Still, the trend to further compress the gap in the conducting polarity is understandable from this picture. Accuracy Issues. In this work, we base our reasoning on the NEGF-DFT technique. This first-principles technique inherits the limitations of the DFT Achilles’ heel: the exchange-correlation functional approximation (XCA).56 Pragmatically, this first-principles technique can be used at reasonable computational cost. It is not possible to quantify the impact of this approximation in our results but by reviewing the existing literature we can discuss this qualitatively. The best established fact concerning XCA in transport is the systematic overestimation of the transmission at the Fermi level. This is established by comparison with both experimental measurements30 and more sophisticated techniques such as NEGF-GW57,58 and NEGF-DFT+Σ.59−61 Despite allowing for a better description, the Σ correction is unfortunately not applicable for strong chemisorption such as gold/thiol interfaces. However, for such strong bonding at the molecule/electrode contact62 NEGF-DFT is qualitatively good with a systematic overestimation of the molecular conductance by a factor up to 10, which is a reasonable accuracy level for theoretical prediction. Because this error is systematic, conductance ratios are usually in good agreement with experiment. Rectification ratios predicted by NEGF-DFT are usually reliable.19,21,63,64 The NEGF-DFT scheme is widely recognized as a robust approach, providing understanding of the behavior on a good qualitative basis. There are several reasons for this overestimation. First, DFT usually predicts LUMO and HOMO alignments too close to the Fermi level. This is due to the well-known gap underestimation (somewhat compensated by the image charge effect65) and sometimes to an overly strong pinning effect, as recently reported for amine anchoring groups.57 This may overestimate the transmission in the conducting polarity of our proposed rectifier. Second, transmission far from resonance originating from two transmission channels, as is the case in the insulating polarity of our proposed rectifier (both HOMO and LUMO contribute) is also typically overestimated.66 This means that both polarities may be comparably overestimated and thus our rectification ratios might be reliable quantities. There is a previous study that compares theoretical Fermi level alignments at our level of theory with measured Fermi level alignments for both thiol and nitrile anchoring groups.49 The maximum error is 0.56 eV for thiol and 0.46 eV for nitrile.
Figure 5. Evolution of the rectification ratio with the applied bias. The ratio increases with the bias as the LUMO and the HOMO get closer in energy, increasing the maximal transmission probability, in the transmission window for the negative polarity.
maximum of 150 at 1.2 V. Moreover, the ratio already surpasses an order of magnitude at a bias around 0.2 V and 2 orders of magnitude at a bias around 0.75 V. This is excellent performance if we compare with existing literature but above all the proposed rectifier is extremely simple. This is because our mechanism opposes resonant tunneling to nonresonant tunneling, as in the original 1974 rectification mechanism for a D-σ-A structure. We observe that for the insulating polarity (i.e., decompression of the gap) the HOMO and LUMO levels stay fairly close to the Fermi level of the electrode to which they are attached. Indeed, the LUMO peak, aligned at 0.25 eV above the right Fermi level at equilibrium, is lying at 0.12 eV above the right Fermi level with an applied bias of 1.2 V between the left and right electrodes (see the left part of Figure 4). The HOMO peak, aligned at −0.22 eV below the left Fermi level at equilibrium, evolves to −0.18 eV below this Fermi level for the same applied bias. Therefore, the evolution of these two levels under bias is well described by a qualitative model assuming perfect pinning of the two fragments, as represented in Figure 3. Indeed, the HOMO/LUMO gap (ΔEgap in Figures 3 and 4) evolves from 0.47 eV at equilibrium to 1.50 eV under bias, a value close to the gap of the isolated molecule, 1.79 eV. This means that the applied bias of 1.2 V efficiently converts into a decompression of the gap by 1.03 eV. Moreover, this shows that the nitrile group leads to strong Fermi level pinning as the LUMO closely follows the right Fermi level under bias. In contrast, the conducting polarity aims at forcing further compression of the gap by the electrodes. If this process were as efficient as the decompression of the gap (previous paragraph), we would observe a crossing of the HOMO and the LUMO. As the peaks merge, it is hard to study this polarity in a straightforward way. However, an intermediate bias of −0.6 V leaves the HOMO and LUMO peaks still resolved. Moreover, the HOMO peak is not perfectly pinned to the left electrode and shifts down in energy while the left electrode shifts up in energy. This means that the HOMO/LUMO crossing (ΔEgap ≃ 1581
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Methodology. The first step of our characterization is to optimize the geometries of the isolated molecule reported in Figure 1, at the DFT/B3LYP(6-31G(d,p)) level of theory,68−70 as implemented in the NWchem-6.3 package.71 This hybrid functional is commonly assumed to obtain the geometries of organic compounds with good accuracy (in comparison with experimental data).72−74 These geometries are then used to compute the electronic spectrum at the DFT/GGA.revPBE(DZP) level of theory,75−77 as implemented in the Atomistix ToolKit (ATK) 2008.10 package. This is to ensure that the comparison between the electronic structure of the isolated molecule and the molecular junction is done at the same level of theory, which is of particular importance for highlighting the compression of the HOMO/LUMO gap by the electrodes. After the geometry optimization, the molecule is contacted between two semi-infinite gold electrodes. As we do not deal with the contact geometry issue, we arbitrary choose the same contact geometry for each derivative reported in Figure 2. The contact geometry is known to influence the transmission spectra of molecular junction78 and could possibly interfere with the rectification mechanism. For the thiol contact, we choose to position the radical sulfur atom atop a gold atom of the (111) gold surface at a distance of 2.42 Å. Such a contact choice is reasonable, as it has been reported by several experimental and theoretical studies in literature.79−81 For the nitrile contact, the current theoretical literature suggests that the nitrogen atom lying atop a gold atom is the optimal geometry.49,52−55 We then adopt this configuration with an Au−N distance of 2.23 Å as predicted by recent DFT calculations.55 The size of our molecule is rather short (about 2.2 nm) and the interaction with gold electrodes is strong, due to the chemical bond on each side. Therefore, we suppose the electrical current through the molecular junction to be in the coherent regime.4 In this hypothesis, the current is given by Landauer formula, which requires computation of the transmission spectrum.82 This is computed using the state-of-the-art ab initio NEGF-DFT/GGA.revPBE technique, as implemented in the ATK 2008.10 package.32,33,35 The spectra are computed for several applied biases, ΔV, between −1.2 and 1.2 V in 0.15 V steps. The nonequilibrium situation is set in the calculation as an explicit chemical potential shift between left and right electrodes together with a self-consistent solution of Poisson’s equation in the scattering region with appropriate boundary conditions. This region geometry and the calculation parameters are given in the Supporting Information. After computation of the transmission spectra, we use the Landauer approach to compute the current, I (ΔV), crossing the junction at each bias
If we consider the same pinning of the LUMO peak under bias, the maximal error of about 0.46 eV would lead to the delayed entrance of the LUMO into the transmission window at 0.9 V instead of 0.3 V. Thus, a high rectification ratio (about 2 orders of magnitude) is still expected around a reasonable applied bias of 1.3 V, see Supporting Information. Finally, we hope this study can motivate a scientific advance relying on other techniques to characterize the gap compression effect and the achievable control of the frontier orbital by the applied polarity. Conclusions and Perspectives. We demonstrated a useful rectification mechanism, which is efficient if a set of three design rules is followed. First, the molecule has to be asymmetrically coupled by donor and acceptor chemisorption moieties to the metallic electrodes. Second, these anchoring moieties must have energy levels aligned close to the Fermi level, and strong Fermi level pinning is required. The third rule is the structure of the molecule, containing two weakly coupled conjugated fragments, each contacted asymmetrically to one of the metallic electrodes. Following these rules, we obtained rectification ratios up to 150. This encouraging result should be substantially improvable. Indeed, the design rules are not highly restrictive and good opportunities to design more efficient rectifiers remain. First, the conjugated fragments can consist of many short conjugated species if the resulting gap is not too large to allow resonant tunneling in the conducting polarity. Second, the decoupling bridge also offers many possibilities for chemical design. Finally, both the electrodes and chemisorption moieties can be modified to tune the fragment alignment properties. This last point is important in the rectifier mechanism. It also seems to offer fewer possibilities because there are not yet many chemisorption couples allowing stable junctions. To increase the rectification ratios, modifications enhancing the dynamic range between nonresonant tunneling in the HOMO/LUMO gap and resonant tunneling in the transmission spectrum should be favored. Therefore, the addition of quantum interference features in the spectrum could potentially lead to much higher ratios involving the same mechanism.67 We hope that these rules will motivate a consequent effort, both experimentally and theoretically. Our proposed mechanism involves a HOMO/LUMO gap compression effect, qualitatively pictured in Figure 3. This is due to our particular design rules promoting independent alignment at each side of the molecule. We consider the Fermi level pinning effect to be the driving force at the origin of this effect, as it forces the alignment at each interface to be close to a certain value. This compressed HOMO/LUMO gap is further controlled by the application of a bias. Depending on the polarity, the gap can increase and recover the isolated molecule value or further decrease. We observe the decreasing effect to be less efficient than the increasing effect. This suggests a resistance of the molecular structure to this compression. This strong control on the HOMO/LUMO gap promotes a high rectification ratio, as the polarity then opposes resonant to nonresonant tunneling. This provides new insight on the relationship among the metal/molecule interface, the molecular structure, and the current/voltage characteristic of the junction. This may apply in several different molecular electronic situations (not only for a rectifier) as it allows for efficient control of the molecular orbitals and the promotion of resonant tunneling. Finally, we strongly believe that a saturated bridge is essential to attain this control by the applied bias in continuity with our previous work.36
I(ΔV ) =
⎡ ⎛ eΔV ⎞⎟ T (E , ΔV )⎢nF⎜E , μ + ⎣ ⎝ 2 ⎠ ⎛ eΔV ⎞⎟⎤ − nF⎜E , μ − ⎥d E ⎝ 2 ⎠⎦ 2e h
+∞
∫−∞
(2)
where E is the electron incident energy, T is the transmission spectrum, nF the Fermi−Dirac distribution, and μ the chemical potential of the gold electrodes. To identify the origin of the resonant peaks in the transmission spectra, we use the MPSH technique.35 This involves the diagonalization of the converged self-consistent Hamiltonian in a basis set restricted to a chosen set of atoms. In the following, we compute the MPSH spectrum and eigenstates 1582
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Nano Letters
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by projecting over the atoms belonging to the molecule. A direct comparison of the eigenstates in the vicinity of the transmission peaks with the orbitals of the isolated molecule identifies the level at the origin of the peak, and its evolution under bias with respect to the left and right electrode Fermi levels.
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ASSOCIATED CONTENT
* Supporting Information S
Additional information containing the full set of computational parameters is available. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Authors
*E-mail:
[email protected]. *E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS Colin Van Dyck is a recipient of a Gustave Boël - Sofina Fellowship of the Belgian American Educational Foundation (BAEF). This work was partly supported by the National Science Foundation MRSEC program (CDMOL-1121262) at the Materials Research Center of Northwestern University. The authors acknowledge the Laboratory for Chemistry of Novel Materials at the Université de Mons in Belgium for their computational support and resource.
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REFERENCES
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