Molecular Junctions: Control of the Energy Gap Achieved by a Pinning

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Molecular Junctions: Control of the Energy Gap Achieved by a Pinning Effect Colin Van Dyck, and Mark A. Ratner J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b07855 • Publication Date (Web): 17 Jan 2017 Downloaded from http://pubs.acs.org on January 26, 2017

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The Journal of Physical Chemistry

Molecular Junctions: Control Of The Energy Gap Achieved By A Pinning Effect Colin Van Dyck* and Mark A. Ratner

Address and affiliation [email protected], phone: +1-847-467-4985 [email protected] Northwestern University, Chemistry Department 2145 Sheridan Road Evanston, IL 60208 USA

Abstract Single Molecule junctions are the constitutive components of Molecular Electronics circuits. For any potential application, the energy gap in the junction, i.e., the accumulated energy difference between the electrode Fermi level and the two frontier energy levels of the molecule, is a key property. Here, using the NEGF-DFT method, we show that the gap of the molecule inserted between electrodes can differ largely from the gap of the same molecule, at the isolated level. It can be widely compressed by tuning the alignment mechanism at each metal/molecule interface. In the context of molecular rectification, we show that this mechanism relates to the pinning effect. We discuss the different parameters affecting the compression of the gap and its efficiency. Interestingly, we find that both the structure of the molecule and of the anchoring group play an important role. Finally, we investigate the evolution of these features out-of-equilibrium.

I) Introduction Among the many new avenues that are opened by nanotechnologies, Molecular Electronics is one of the most exciting ideas.1-2 It proposes to realize an electronic device by contacting a single (or a few) molecule(s) between metallic electrodes. Although we are currently able to create such molecular junctions,3 we are still facing an important limitation: we do not know how to select a proper molecular structure to obtain a desired electronic device. This is especially true if we aim at creating a molecular electronic device as efficient as its solid-state counterpart. In this context, we show here

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how the structure of a molecule and its anchoring group can affect a key property of a molecular junction: the energy gap in the junction. The gap in the junction corresponds to the accumulated energy distance between the electrode Fermi level and the frontier energy levels (HOMO and LUMO) of the molecule. Noticeably, we demonstrate that this property is largely affected by the interaction between the molecule and the electrodes, leading to an electrode-induced gap renormalization.

Importantly, the electrode-induced gap renormalization that we discuss here applies to HOMO and LUMO levels that are spatially separated. This relates to Donor-Bridge-Acceptor molecular structures, which usually possess frontier orbitals that mainly localize on opposite sides of the molecule. In this isolated molecule context, the energy gap can easily be renormalized by varying the Donor and/or the Acceptor strengths (internal substitutions). In the molecular junction context, the Fermi level pinning effect, which is described below, largely limits this energy gap tuning. We show here that this limitation can be overcome. The energy gap in the junction can also be renormalized, by tuning the interaction and the Fermi level alignment at each metal/molecule interface (external substitutions).

A nice example of a device that highlights the relevance of the energy gap for Molecular Electronics is the molecular rectifier.4-8 In this work, we focus on this elementary electronic device. It is characterized by an ability to be conducting for an applied bias polarity and insulating for the opposite polarity. The efficiency of the device is given by the rectification ratio (RR), defined as the ratio between the currents crossing the junction in conducting and insulating polarities, at the same applied bias in absolute value. Currently molecular rectifiers are not competitive with solid-state diodes as ratios for the single molecular wires are usually lower than 10.9-14 This is generally true but major improvements have been achieved recently. A proper control of the liquid environment surrounding a symmetric molecule led to a molecular junction with a rectification ratio up to 200.15 Moreover, a well-designed single molecule diode16 recently exhibited ratios up to 600.17 Higher ratios are achievable, if we consider an ensemble of molecules, i.e., a monolayer, for which ratios of more than three orders of magnitude are observable.18-19 In a previous work,20 we introduced a set of three design rules leading to a new rectification mechanism, efficient at the single-molecule level. The mechanism is extensively described and placed in the context of previous works on rectification in this previous study. It applies in the coherent transport regime and leads to a theoretical ratio of up to two orders of magnitude. The rules are:


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(1) “Donor/Acceptor anchoring rule”: Use asymmetrical anchoring groups to chemically bind the molecule to the electrodes: an accepting anchoring group (promoting a LUMO alignment) and a donating anchoring group (promoting a HOMO alignment) on opposite molecular extremities. (2) “Pinning rule”: 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.21-22 (3) “Decoupling rule”: 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.

Figure 1. Representation of the three different types of molecule characterized in this work. The first compound, with asymmetric anchoring groups, is a molecular rectifier following the set of design rules. The second compound violates the pinning rule, by adding a saturated wire between the conjugated fragments and the anchoring groups. The third compound violates the decoupling rule, by removing the saturated wire between the two fragments. Both the asymmetrical and symmetrical anchoring group cases are studied. In this latter symmetrical case, the impact of electroactive substituents in R position is evaluated in order to assess the pinning strength.

This efficient rectification mechanism, with a high degree of liberty for chemical design, has been shown by considering an elementary candidate molecule following the design rules (see molecule (1) in Figure 1). We observe a larger electron current in the direction flowing from the Donor-substituted fragment toward the Acceptor-substituted fragment. As already pointed out in our previous work,20 the rectification at low bias is due to two main features that are induced by the rules: (i) the compression of the HOMO/LUMO gap of the isolated molecule, as soon as this molecule is sandwiched between gold electrodes; (ii) the HOMO/LUMO gap in the junction is controlled by the



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applied bias, i.e., the respective shift between the left and right electrode Fermi levels. The first feature leads to two resonant transmission peaks, associated with the HOMO and LUMO orbitals, which lie in the vicinity of the left and right electrode Fermi levels. The second feature leads to a further renormalization of the energy gap when a bias is applied between the electrodes. This promotes the entrance of the HOMO and LUMO transmission peaks into the transmission window for the conducting polarity, i.e., a resonant tunneling mechanism. The first feature ensures that this happens for reasonably low biases. For the opposite polarity, the energy gap increases and the associated resonant transmission peaks remain out of the transmission window, allowing only for a non-resonant tunneling mechanism. This opposition of resonant and non-resonant tunneling is of high interest and at the origin of the predicted high rectification ratio.20

While these observations are made in the context of rectification, we believe that a deep understanding of these two features may also be of interest for other molecular electronic devices. Indeed, the energy gap is a key property that is of great importance for other examples of molecular electronic devices such as molecular photo-cells23-26 or transistors.27-29 Importantly, in this latter context for which a molecule is gated between electrodes, it has been demonstrated that the energy gap was renormalized by the interaction with the electrode.30 In this case, the responsible interaction was the dynamical image charge interaction, which does not require any chemical contact at the metal/electrode interface and should apply for any type of molecule. This renormalization, also of great interest for the design of molecular junctions,31 is different from what we discuss in the present work. We note that this renormalization due to the dynamical image charges cannot be described at our level of theory and is absent from our calculations.32-34 Finally, this image charge renormalization does not require spatial separation of the HOMO and LUMO orbitals.

The pinning and decoupling rules are both fundamental and at the origin of the two observations we made. We aim here at explicitly demonstrating this point, relying on the Non-Equilibrium Green’s Function coupled to the Density Functional Theory framework (NEGF-DFT).35-39 For this purpose, we study two new molecules possessing similarities with our proposed rectifier, but each violating one of the two last design rules, see Figure 1. More particularly, we study how their frontier energy levels respond to the contact with metal electrodes. In total, our study is based on the computation of 15 different molecular junctions. For the sake of clarity, the present work is structured in two main parts. First, we address the feature (i), described above. For this purpose, we introduce a first efficiency parameter to quantify the compression of the gap, and relate it explicitly to the strength of


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the pinning effect. This is done for each of the molecules reported in Figure 1. We then discuss the required conditions to get a high compression of the gap.

We study next the behavior of each molecule under out-of-equilibrium conditions, to address the second feature, (ii). For this purpose, we introduce a second efficiency parameter and further compare the impact of violation of one of the two last design rules on this control. We show here that this control is actually related to the trend of the molecule to screen the electric field in the junction, and thus the pinning effect. This trend is strongly affected by the design rules. We particularly discuss the fundamental role of the decoupling rule. This latter allows for lowering the resistance of the molecule in the junction against the gap compression. Finally we report the rectification ratios of each derivative, and discuss their relative performances in terms of the two features. This study will clarify the meaning of the three design rules.

II) Methodology

We proceed in the same way for the simulations of conductance and alignment properties of each molecule studied in the present work. First, we start by optimizing the geometry structure using Density Functional Theory (DFT) at the B3LYP level,40-42 as implemented in the NWchem-6.3 package,43 with a 6-31G(d,p) basis set. This functional has been widely used and proven to be accurate for predictions of the geometry of short organic molecules.44-46 Then, we compute a set of electronic properties (Ionization Potentials (IP), Electron Affinities (EA) and the frontier orbitals topologies) of the isolated optimized molecules, using the GGA.revPBE functional47-48 and a DoubleZeta+Polarization (DZP) numerical basis set,49 as implemented in the Atomistix ToolKit (ATK) 2008.10 program. This choice of functional and basis set is justified by the fact that our theoretical description of the molecules between the gold electrodes is done with the same program, at this precise level of theory. Therefore, we make sure that our conclusions, obtained by comparison between the molecules in gas phase and in the junction, are not affected by the choice of the theoretical method.

To create the junctions, the optimized molecules are contacted to semi-infinite (111) gold electrodes. In continuity with our previous work,20 we use nitrile and thiol groups for anchoring. We assume that both thiol and nitrile anchoring groups are lying on top of an atom of the gold surface. For thiol (on the left side), we use an Au-S distance about 2.42 Å and for nitrile (on the right side), we use an Au-N



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distance about 2.23 Å. These parameters are reasonable as on-top geometries and such distances have been suggested in previous works at both experimental and theoretical levels for these anchoring groups.50-57 This reasonable contact geometry is chosen for the whole set of molecules. This ensures that the contact geometry does not interfere with the principles and analysis described in the results section. Indeed, this latter solely focuses on the interpretation and understanding of the interplay between the Fermi level alignment and the renormalization of the gap in the junction.

The electrodes are periodic in the x,y directions, transverse to the transport axis, with a unit cell made of 4x4 gold atoms. In the scattering region, we include five gold screening layers on each side of the molecule. The electronic structure of the junctions is computed using the NEGF-DFT scheme,35-39 at the GGA.revPBE level, with a SingleZeta+Polarization basis set to describe the gold atoms and a DoubleZeta+Polarization basis set for the molecular atoms. We use a k-sampling made of 7x7x200 kpoints. The mesh cut-off for the Poisson equation solving is about 300 Ry. From the converged Green’s function, we get access to the transmission spectrum of the molecular junction.1 The transmission is the probability for an electron to transmit through the junction at a given incident energy. This is the most important quantity to characterize the transport through short molecules, as the electrons transmit coherently through the junction.58 This spectrum can be computed away from equilibrium, i.e. with a different chemical potential for the left and right electrodes and an electric field in the molecular region. In our work, the Fermi level alignment values in junction are extracted from the transmission spectra HOMO and LUMO peak positions.59 The Landauer formula allows for evaluating the current through the molecular junction, after integration of the transmission spectrum:58 +∞

  2e eV  eV  I (V ) = T ( E ,V ) n F  E , µ +  − nF  E, µ − h −∞ 2  2   

∫

  dE , 

(1)

where T(E,V) is the transmission spectrum (as calculated out-of-equilibrium), E is the incident electron energy, V is the applied bias, µ is the equilibrium chemical potential and nF is the FermiDirac population. The sign of the potential in our calculation relates with the left electrode, i.e., the thiol contact side for asymmetric junctions. By doing so for several biases, we can simulate a theoretical I(V) characteristic. The NEGF-DFT scheme gives access to other meaningful quantities


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such as the total electron density in the scattering region. It is of interest to observe how the steady state electron charge evolves with the applied bias.

Our focus in this work is mainly to get a physical insight into the mechanisms associated with the design rules and at the origin of the gap compression. Therefore, we do not include a post-correction scheme to our DFT calculations, as suggested by several studies.60 Indeed, our level of theory is sufficient to capture the bonding of the anchoring group at each interface, i.e., the creation of an interface dipole. A post-DFT correction, such as a scissor operator, would essentially lead to a corrected value for the S-parameter. However, the interplay between the S-parameter and the gap in the electrode, as described by the equation (6), is not expected to be modified by a post-DFT correction scheme. Moreover, the DFT+Σ correction schemes cannot be applied to strong gold-thiol bonds as used in the present work.60 Finally, the S-parameters as computed by DFT tend to be underestimated in comparison with GW methods.59 It is not an issue as a large range of S-parameters is covered in this work, without affecting its relation with the gap compression, as long as the decoupling rule is satisfied.

III) Results and discussion

III.A. Relation between the design rules and the gap compression effect We showed in our previous work20 that a first important consequence of our design rules is the compression of the HOMO/LUMO gap of the molecule as soon as this latter is placed between the electrodes, ∆Ejunction, in comparison to the isolated molecule gap, ∆Eisolated. Indeed, while this gap was about 1.79 eV for the isolated molecule, at the DFT/GGA.revPBE level of theory, this latter was compressed to 0.47 eV for the same molecule sandwiched between the gold electrodes, as computed from the transmission spectrum in Table 1. This compression effect is illustrated by a red arrow in Figure 2. Here, we choose to quantify the efficiency of this process by introducing the following parameter:

η = 1−

∆E junction ∆Eisolated

.

(2)



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This η parameter equals 100% if the HOMO/LUMO gap is reduced to zero and 0% if there is no compression at all and the gap remains unchanged. For the simple molecule we chose in our previous work (see molecule (1) in Figure 1), this efficiency is about 74%. We attributed this feature to the pinning effect. In this section, our aim is to precisely highlight and explain the origin of this mechanism, in terms of the pinning effect. At the same time, we highlight two requirements to obtain a high efficiency.

As a fragment is connected to the electrodes with a thiol bond, the HOMO tends to be aligned in the vicinity of the Fermi level (EF), at a certain energy offset below the latter, EF-εHOMO. An intuitive way to increase and modify this offset, referred to in the following as Fermi level alignment, is to increase the Ionization Potential (IP) by a certain amount of energy, ∆IP. We define the pinning effect as the insensitivity of the Fermi level alignment to this IP modification. The same goes for the pinning of the LUMO aligned above the Fermi level (εLUMO-EF) with a nitrile anchoring. This latter is insensitive to the modification of the Electron Affinity, ∆EA. This effect can be quantified by the S-parameters, defined as follows: S HOMO ≡

∆ ( E F − ε HOMO ) , ∆ IP

(3)

S LUMO ≡

∆ (ε LUMO − E F ) . ∆ EA

(4)

These parameters are both equal to 0 in the perfect pinning case. In reality, it lies between 0 and 1, being close from 0 in the case of a strong pinning effect. The pinning has been observed in previous theoretical works on molecular junctions.21-22, 61-63 Importantly, it is also a behavior established at the experimental level.64-68 We note that the S-parameter definition comes from the field of metal/organic and metal/semiconductor interfaces, in which the Fermi level pinning phenomenon has been widely studied.69 Essentially, the origin of this effect is the interface dipole that creates upon chemical contact on each side of the molecule. This charge reorganization leads the Fermi level alignment mechanism and can absorb the change of IP and EA observed at the isolated molecule level. We note that these parameters are strictly defined as derivatives.70 However, as it is practically impossible to modify the ionization potential and electron affinities in a continuous way, we replaced the derivation by finite differences in equations (3) and (4).


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Figure 2. Alignment of the frontier levels in respect with the Fermi level of the electrode, as the molecules are contacted to semi-infinite electrodes with the NEGF-DFT procedure. On the left, in black, are the HOMO/LUMO alignments of the three asymmetric derivatives sandwiched between electrodes. The red arrows illustrate the compression of the gap for each species, starting from the gap of the isolated molecule. The alignment of the frontier levels for the symmetrically anchored and non-substituted molecules, allowing for establishing the maximal compression of the gap, are represented in blue (-CN) and green (-SH). On the right, illustration of the pinning strength for each derivative, going from the nonsubstituted species (R=H) to the substituted species, denoted as primed values. The orange arrows illustrate the pinning effect which modifies the alignment that could be expected from the change of IP and EA, as represented by dashed lines. This allows for computing the S-parameters. The exact corresponding values are reported in Tables 1 and 2.

We now consider the symmetric versions of our rectifying molecule, see (1) in Figure 1, with equivalent A and B substitutions. We report the corresponding transmission spectra in Figure S1, and the extracted alignments in Figure 2 and Table 1. In this case, the gap is not being compressed, see Figure S1. This means that we can compute the Fermi level alignments of the fragments independent of this effect. With symmetric thiols, the HOMOs of each fragment are aligned around nearly the same energy below the Fermi level: -0.17 eV and -0.30 eV. The same goes for the LUMOs with symmetric nitriles above the Fermi level: +0.12 eV and +0.15 eV. This difference between fragments may be due to subtle contact differences, as the geometries are not optimized in junction. We consider in the following that -0.17 eV and +0.12 eV are the Fermi level alignments that a fragment connected with a thiol or a nitrile aim to achieve, as represented in Figure 2. This means that, if the fragments were totally independent and asymmetrically connected, the gap in junction, between the aligned HOMO and LUMO, would be about 0.17 + 0.12 = 0.29 eV. This value is the minimal gap that could



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be achieved in the junction, ∆Emin. However, we know that the isolated molecule made of the two coupled fragments has a HOMO/LUMO gap about 1.79 eV. Therefore the coupling between the fragments tends to modify the alignments (-0.17 eV and +0.12 eV), pushing both the HOMO and LUMO levels away from the Fermi level. To recover the gap of the isolated molecule, about (∆Eisolated-∆Emin) higher in energy, we need to shift both the HOMO and LUMO away from the Fermi level by (∆Eisolated-∆Emin)/2, assuming a symmetric perturbation of the HOMO and LUMO alignments, see Figure 3. The effect of this coupling is then equivalent to a modification of the ionization potentials and electron affinities of the fragments.

Figure 3. Scheme illustrating the mechanism of the gap compression effect, assuming an independent alignment of the left and right fragments. This compression is due to the interplay of two different HOMO/LUMO gaps and the Fermi level pinning mechanism. In blue is the minimal gap, determined by the symmetric molecule computation. This corresponds to the Fermi level alignment that each fragment aims at achieving. The gap of the isolated molecule, in red, is then a perturbation, requiring modifying this alignment. We assume this perturbation, given by the black arrows, to be equal for both the HOMO and the LUMO. The pinning effect then comes into play to modulate this perturbation with the Sparameters, in purple, at the origin of the resulting compressed gap in green (∆Ejunction).

However, assuming that both fragments are perfectly pinned (SHOMO=SLUMO=0), it is impossible to modify the HOMO and LUMO alignments and so the gap would stay at the minimal value, ∆Emin=0.29 eV. This determines the maximal efficiency, ηmax, of the gap compression, which only depends on the ability of each fragment to align closely near the Fermi level:


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η max = 1 −

∆E min . ∆Eisolated

(5)

In the particular case of our rectifier, the maximal efficiency computed from equation (5) is about ηmax=1-0.29/1.79=84%. A high maximal efficiency is the first condition to get a high compression of the gap in junction. To understand the discrepancy between this maximal efficiency and the actual efficiency computed from equation (2), η=74%, we have to compute the S-parameters defined by equations (3) and (4).

Table 1. Alignment values (in eV with the Fermi level set to zero) extracted from the transmission spectra that are also reported in Figure 2. In the first line of the table are reported the corresponding anchoring groups. The prime exponent indicated for the energies corresponds to the alignment with an electroactive substituent added. These are a donor (amine) for the SH/SH case and an acceptor (cyano) for the CN/CN case, pushing the HOMO or the LUMO away from the Fermi level. These values allow for computing the actual gap in junction with asymmetric coupling and the minimal gap in junction from symmetric coupling calculations. It can then be used for computing the compression efficiencies, from equations (2) and (5), as well as the S-parameters, from equations (3) and (4), that are reported in Table 2. SH/CN

SH/SH

CN/CN

Molecule

εHOMO

εLUMO

εHOMO

εHOMO’

εLUMO

εLUMO’

∆Ejunction

∆Emin

(1)

-0.22

+0.25

-0.17

-0.21

+0.12

+0.16

0.47

0.29

(2)

-1.21

+1.01

-0.81

-1.13

+0.46

+0.58

2.22

1.27

(3)

-0.32

+0.86

-0.04

-0.08

+0.05

+0.07

1.18

0.09

This is feasible by adding electroactive substituents, as we did in our previous work.21-22 To compute SHOMO, we add one accepting cyano group on each fragment of the symmetrically thiol-connected molecule. From an isolated molecule calculation, we compute a ∆IP of about 0.41 eV. To compute SLUMO, we add one donating amine group on each fragment of the symmetrically nitrile-connected molecule, leading to a ∆EA about 0.19 eV, for the isolated molecule. Connecting these substituted symmetric molecules to the electrodes, we can compute the Fermi level alignments, see Figure 2, and observe the induced modification in Fermi level alignments, reported in Table 1. The substitutions shift the HOMO/LUMO away from the Fermi level by 0.04 eV/0.04 eV for thiol/nitrile anchoring groups. This translates into the following S-parameters: SHOMO=0.04/0.41 ≈ 0.10 and SLUMO=0.04/0.19



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≈ 0.21, according to equations (3) and (4). These parameters are close to zero, attesting to a significant pinning effect in the alignment of the two fragments, for both thiol and nitrile anchoring groups. This pinning effect is illustrated as an orange arrow in Figure 2.

However, these parameters are not exactly zero. This means the pinning is not perfect, which rationalizes the discrepancy between the maximal and actual gap compression efficiencies. Indeed, considering that the desired HOMO and LUMO alignments (given by the symmetric molecules connected to the electrodes) are perturbed by SHOMO/LUMO*(∆Eisolated-∆Emin)/2, we obtain the following relationships among the three relevant gaps of our systems: − ∆E min   ∆E ∆E junction = ∆E min +  isolated  * (S HOMO + S LUMO ) , 2    

η = η max  1 −

S HOMO + S LUMO  . 2 

(6)

(7)

These equations summarize our interpretation of the gap compression effect and are illustrated in Figure 3. Using equation (7), taking into account the S-parameters, we can introduce the S-parameters and ηmax that are reported in Table 2, a lower efficiency of the compression, about 71%, in better agreement with the actual 74% is then obtained. The remaining discrepancy may arise from a nonlinearity of the S-parameter in the case of strong perturbations. Moreover, there is some uncertainty in the computed alignments due to rounding errors or difficulties to extract the energy position of the maximum of the transmission peaks.

The procedure that we just followed relates the properties of the two decoupled fragment/metal interfaces, the gap of the coupled fragments at the isolated level and the actual gap of the coupled fragment in junction. Such a procedure could also be followed and verified at the experimental level by a careful measurement of the set of quantities that appear in Equation (6). This latter can then be used to predict the gap of molecules made of two π-decoupled fragments, after contact to electrodes.

We can now use the same procedure to observe the consequences of a violation of the pinning rule. We showed in a previous work that the S-parameter is modified by adding a saturated spacer between the anchoring group and the conjugated fragment.21 This rationalizes our choice of the molecule (2) in Figure 1, including two saturated carbon atoms on each side of our rectifying molecule, in order to


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violate the pinning rule. In this case, the same electroactive substitutions in the fragment backbones lead to computed shifts of the IP and EA by 0.50 eV and 0.20 eV, at the isolated molecule level. After connection to gold, we can compute the Fermi level alignments, (see Figure 2 and Table 1) and get the following S-parameters: SHOMO = 0.64 and SLUMO = 0.60, from equations (3) and (4). The pinning effect is then significantly reduced after introduction of the saturated spacer and we can now investigate the impact of the pinning rule on the gap compression.

The computed isolated gap of this asymmetric molecule with gold decoupling substituents is about 2.69 eV. After connection to gold, we calculate a HOMO aligned at -1.21 eV and a LUMO aligned at 1.01 eV, see Figure 2 and Table 1. This translates into a small efficiency of the gap compression effect, only about 17%, according to eq (2). We can rationalize this value using equation (7). Upon introduction of the saturated spacer, the symmetrically anchored case leads to the alignment of the HOMO about -0.81 eV below the Fermi level and the LUMO about 0.46 eV above the Fermi level, see Figure 2 and Table 1. These values are much further from the Fermi level and significantly reduce the maximal efficiency of the gap compression, which is about 53%, from equation (5). Since now the anchoring groups do not align the fragments in the vicinity of the Fermi levels, the maximal efficiency is much lower. This is the reason why in the second design rule we suggest using anchoring groups aligning the fragments in the vicinity of the Fermi level.

This maximal efficiency is further reduced by the weaker pinning effect, as indicated by the Sparameters. Taking these into account through equation (7), we obtain efficiency about 20%, in good agreement with the observed 17%. This nicely shows the importance of the pinning effect to create a significant compression of the gap. Also, it clearly explains the origin of the second design rule, as it precisely targets an optimization of the gap compression effect. This effect is important to get an efficient rectifier, as we will show in next section. Finally, this validates our interpretation of the gap compression in terms of the pinning effect, as equation (7) correctly reproduces the efficiencies at a quantitative level, as soon as the first and third rules are satisfied. We see that the compression can be tuned significantly by modulating ηmax and the S-parameters. We now investigate the relation between the decoupling rule and the compression of the gap. For this purpose, we study the molecule (3), represented in Figure 1. This corresponds to the rectifier without any saturated bridge. This means the resulting molecule is a single fragment well conjugated from one anchoring group to the second one. This clearly violates the decoupling rule. Relying on the same



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procedure as in the previous paragraphs, we first compare the gap of the isolated molecule to the gap of the molecule in the junction. We compute these to be about 1.59 eV and 1.18 eV. This translates into an efficiency of the gap compression of about 26%, according to equation (2).

Such low efficiency may be surprising at first sight. Indeed, considering the symmetrically anchored molecules, we observe that the alignment of the HOMO and the LUMO happen extremely close from the Fermi level, at -0.04 eV and +0.05 eV, see Figure 2 and Table 1. This means that the maximal efficiency is extremely high, about ηmax=94%. Moreover, the pinning effect is really strong. After the addition of electroactive substituents, we compute the following S-parameters: SHOMO = 0.11 and SLUMO = 0.06. This means that, according to equation (7), we could obtain a compression of the gap up to 86%, a value in complete disagreement with the observed 26%.

At the origin of this discrepancy is the fact that both the HOMO and the LUMO lie on the same unique fragment in this derivative. Therefore, despite the high potential to compress the gap, the levels do not align independently and the compression scheme illustrated in Figure 3 does not apply. This suggests that a strong conjugation from the left anchoring group to the right one leads to a resistance to the compression of the gap. The compression must then be viewed as a balance between the pinning effect and a trend to preserve the gap and structure of the isolated molecule, i.e, a resistance to the compression effect. This resistance is proportional to the coupling between the fragments. The decoupling rule is then required to annihilate this resistance and get a high gap compression, described by the mechanism of Figure 3. We show in next section that such a resistance can also be observed when we consider the response of the system to an applied bias.

III.B. Relation between the design rules and the control of the gap under bias

In the previous section, we explicitly showed how the pinning rule leads to the compression of the gap, as soon as the two fragments are decoupled. In this section, we study the behavior of the rectifier and these two other derivatives under bias. We characterize how the design rules, after leading to gap compression, further allow for controlling the gap in the junction via an applied bias. Finally, we relate this control to the rectification ratios of the derivatives.

An idealization of the rectification mechanism is to view the HOMO and LUMO orbitals as perfectly pinned to their respective electrode Fermi level.20 This is represented in Figure 4. Then, the bias


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applied between the electrodes is transformed into a modulation of the HOMO/LUMO gap in the junction. In the insulating polarity, the pinning effect leads to a gap decompression, keeping the resonant levels out of the transmission window. In the conducting polarity, the pinning effect puts the frontier levels even closer from each other, inside the transmission window, leading to an additional gap compression. In the perfect case, an applied bias about ∆V would be completely transformed into a reduction or augmentation (depending on the polarity) of the HOMO/LUMO gap about e∆V. We can quantify how close we are from this ideal case by introducing the following efficiency parameter:

γ ( ∆V ) =

∆E junction ( ∆V) − ∆E junction (0V) e∆V

,

(8)

which is the ratio between the difference of HOMO/LUMO gap induced by the bias and the applied bias. The ideal case (a perfect pinning of the HOMO and LUMO) would lead to an efficiency about 100%. In the following, we choose to focus on the ratio with applied biases about ±1V: γ ± = γ ( ±1V ) . This is shown in Figure 5, reporting the transmission spectra under bias for all the asymmetric molecules. From these spectra, we extract the evolution of the energy gap out-of-equilibrium, allowing for computing the efficiency parameter using equation (8).

Figure 4. Scheme illustrating the efficiency of bias-induced control of the HOMO/LUMO gap in the junction. In black are the alignments of the frontier levels at equilibrium. In blue we see the evolution of this situation for a positive applied bias, assuming the perfect pinning case in the figure (SHOMO=SLUMO=0



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and γ=100%). This perfect situation guarantees that the frontier levels stay out of the transmission window in the insulating polarity, with the efficiency indicating how far we are from this situation. As indicated by the green arrows, a linear potential drop cannot accommodate with this perfect pinning situation. For this reason, the electron density must reorganize, in order to amplify the drop in between the fragments and reach a drop as represented by the green dashed line. This fact relates the efficiency with the accumulated charge integration profile.

We start by characterizing our rectifier obeying the whole set of design rules. At positive bias, we obtain γ + = 87% . This indicates an excellent conversion of the applied bias into the modulation of the gap, consistent with a strong pinning of each fragment. This is important for the rectification mechanism as it keeps the frontier orbitals out of the transmission window, as represented in Figure 4. On the other hand, the efficiency falls to γ − = 22% at negative bias. Under this polarity, the gap is actually more compressed, achieving a really small value about 0.25 eV, with the two levels inside the transmission window. This value is significantly different from the isolated gap which was about 1.79 eV. This corresponds to a huge compression, about η=86%, under a -1V applied bias.

Figure 5. Transmission spectra of each compound with an applied bias (in blue), together with the transmission at equilibrium (in black). These spectra allow for computing the efficiency of the gap modulation under bias. The transmission window, i.e. the part of the spectrum that is integrated using the Landauer formula, is indicated in blue. Obeying the design rules provides an efficient rectification


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mechanism, opposing resonant to non-resonant tunneling for the first compound. This clear opposition disappears as the pinning or decoupling rules are violated. For the second molecule, this is because the levels are too far from the Fermi level at equilibrium and not controlled efficiently enough. For the third molecule, this is because the levels are not controlled at all and they can enter the transmission window for both polarities. We note that the black reference (the zero in the calculations) is the Fermi level at equilibrium and the averaged Fermi level out of equilibrium.

We already pointed out in the previous section that there is a resistance in the compression of the gap, in the case of the rectifier violating the decoupling rule. We consider this lower efficiency obtained in this polarity as another manifestation of this resistance, due to the actual coupling between the two fragments. Indeed, if the efficiency were higher in this polarity, the gap would finally cancel and we would observe a crossing between the HOMO and the LUMO. Such a scenario only makes sense for two strictly independent fragments, i.e., without any coupling between the frontier orbitals on each fragment.

To further show that the electronic charge of the molecule reorganizes in a way that resists the compression of the gap in this polarity, we can compute the electron reorganization occurring under bias:

δρ ( z , ∆V ) = ∫ ( ρ ( x, y , z , ∆V ) − ρ ( x, y , z ,0V ))dxdy ,

(9)

cell

where ρ is the electron density in the scattering region. This latter is integrated along the axes transverse to the transport direction (x and y), in order to obtain a profile. In consistency with our previous work, we then integrate this profile along the transport direction (z) to obtain the accumulated charge integration profile:

z

δQ( z, ∆V) = ∫ δρ ( z' ,∆V)dz' . 0

(10)

The advantage of these integrated profiles is that the concavity allows us to directly observe whether the charge reorganization under bias is leading to an amplification (the electron charge moves toward the negative electrode) or a screening (the electron charge moves toward the positive electrode) of the



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applied electric field within the molecular region. Indeed, as the profile goes up a partial electron charge has been added in the region, and as the profile goes down, a partial electron charge has been removed. The concavity is then directly associated with the polarization of the electron density along the transport axis. Another advantage is that a partial electron charge generally moves away from the molecule under bias, in our calculations. This leads to charge density profiles, as given by equation (9), which are hard to interpret. The integration procedure in equation (10) reduces this loss of electron charge to a general decreasing trend which can then be used as a reference (dashed line in Figure 6) to investigate the concavity. We showed in our previous work21 that, to preserve the equilibrium alignment of two decoupled fragments, at each interface, the profile trend must amplify the electric field in between the two fragments. Indeed, the voltage drop modifies this alignment,10, 71 as represented by the green arrows in Figure 4. Then the drop needs to be amplified in the area between the fragments to recover the equilibrium Fermi level alignment at each interface. For the sake of completeness, we provide the voltage drop associated with each charge reorganization profile as part of the Supporting Information, see Figure S2.

The charge reorganization profiles are reported in Figure 6 for all the studied derivatives and each polarity, with |∆V|=1V as an applied bias. We observe that for the positive polarity, the trend is to amplify the electric field in the molecular region. This is consistent with a strong pinning of each fragment and at the origin of the high γ efficiency in this polarity. However, in negative polarity, the trend is to screen the electric field. This is inconsistent with a strong pinning effect on each fragment and tends to modify the alignment at each interface. This is evidence for resistance to further compression of the gap in the junction. As the gap becomes really small, the charge density tends to resist by screening the electric field which is the origin of a low γ parameter (about 22% in this polarity). This behavior, together with the absence of gap compression for the well-conjugated species in the previous section, shows that the coupling between the two fragments limit the efficiency of the compression of the gap from the pinning effect, i.e., the molecule tends to resist this compression.

We now characterize the influence of the pinning rule on the γ efficiency. After computing the transmission spectra under bias for the molecule (2) in Figure 5, a derivative of the rectifier with a limited pinning effect, we obtain the following efficiencies: γ + = 40% and γ − = 32% . This indicates that we only have a limited control of the gap in both positive and negative polarities. This is


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consistent with the lower S-parameters, indicating a weaker pinning, reported in the previous section. Actually, the control in the negative polarity is a bit higher than for the rectifier. We attribute this to the fact that the gap is not being compressed as strongly as in the rectifier case for this polarity. Indeed, the remaining gap at -1V is still about 1.90 eV, which corresponds to a rather small gap compression (about η=29%). This situation is then pretty far from a HOMO/LUMO level crossing.

In Figure 6 are given the accumulated charge profiles for this second molecule that violates the pinning rule. The corresponding profile is really flat in both polarities with neither a trend to amplify nor to screen the electric field. As a result, we conclude that the fragments are split in energy due to the electrostatic potential drop, which explains that the gap is still modulated despite a weak pinning, but not as efficiently as for the rectifier. Again, we observe here a better efficiency of gap modulation for a positive bias, the polarity that does not compress the gap. This is consistent with the resistance of the electron density to the compression, as we suggested in the previous paragraphs.

Figure 6. Accumulated charge integration profiles of the three asymmetric compounds given in Figure 1, for positive (blue) and negative (red) applied biases. The polarity of the electrodes is indicated in the legend box. The dashed line in grey indicates the general decreasing trend, meaning the molecule can lose a partial electron charge under bias. The important property is the concavity, showing the displacement of the electronic cloud toward the positive or negative electrode. In the case of a positive bias applied to the first compound, the trend is to amplify the electric field, leading to a really high control of the gap under bias. For the second derivative, there is no trend to screen or amplify as the profile is basically flat. In all other cases, the trend is to screen the electric field, translating into low controls of the HOMO/LUMO gaps under bias.



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Proceeding in the same way with our derivative violating the decoupling rule, we obtain the following efficiencies: γ + = 10% and γ − = 2% . These are extremely low values, meaning that, in this case, there is essentially no control of the gap by the electrodes. This is because both of the frontier levels are localized on the same fragment. In Figure 6 are given the accumulated charge profiles. As for the other profiles, there is an excellent correlation between the γ efficiencies and the trend to screen or amplify the electric field. It is clear that the aim is here to screen the electric field, for both polarities. This trend was also observed in our previous work21 for a flat conjugated molecule with symmetric thiol anchoring groups. The screening concavity is actually more marked for the negative bias, which is well correlated with a lower γ efficiency. Again, the control is worse for the negative polarity, as this polarity is associated to an extended compression of the gap. The resistance to this compression is here maximal, as indicated in both the η and γ efficiencies. This clearly shows the high importance of the decoupling rule.

We now report the rectification ratios of the rectifier derivatives, each violating a single design rule. The quantities important for the discussion are all reported in Table 2, and plotted in Figure S3. As we reported in our previous work,20 the rectifier obeying the rules rectifies by one order of magnitude at 0.2V. The ratio increases with the magnitude of the applied bias and reaches two orders of magnitude at a bias around 0.75V.20 For the derivative violating the pinning rule, we get a ratio around 5 for an applied bias of 1V. This derivative clearly shows lower performance. There are two reasons for that. First, since the gap is not so compressed, the tails of the HOMO and LUMO orbitals cannot enter into the transmission window at low bias, see Figure 5. Second, the bias control is much less efficient for this derivative. This means that the entrance of the resonant levels in the transmission window, increasing the conductance, is harder to achieve in the conducting polarity, see Figure 5. It also means that these levels are more likely to enter inside the transmission window in the insulating polarity. As a result, the rectifier offers poor performance. Still, there is a significant rectification ratio. This is because the voltage drop still allows for getting significant γ efficiencies, despite being reduced by the weaker pinning effect. As a result, we need to follow the pinning design rule to get an efficient rectifier.

Table 2. Series of quantities relevant to characterize the impact of the design rules in the rectification mechanism, as introduced through this work in equations (2) to (5) and (8). The γ and RR values are computed for an applied bias of 1V.


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Molecule

SHOMO

SLUMO

η

ηmax

γ+

γ-

RR(1V)

(1)

0.10

0.21

74%

84%

87%

22%

114

(2)

0.64

0.60

17%

53%

40%

32%

4.7

(3)

0.11

0.06

26%

94%

10%

2%

2.1

The derivative that violates the decoupling rule is even a worse rectifier. From the Landauer formula, we compute a maximal ratio about RR=3.15 at 0.6V and decreasing to RR=2.1 at 1V. This is because there is no control at all on the entrance of the frontier levels, as the γ efficiencies are extremely low. Then, as the tail of the LUMO level enters in the transmission window for a given polarity, it also enters in the transmission window at a similar bias for the opposite polarity, see Figure 5. This suggests that the decoupling rule appears as the most important design rule. Indeed, while the violation of the second rule limits the rectification, the violation of the last rule completely annihilates the rectification mechanism. It can be noted that the decoupling rule has also been pointed out as crucial in other recent theoretical work for getting an efficient rectifier.16

Finally, we can compare the conductance of our rectifier to its symmetric counterparts, relying on Landauer formula. Very interestingly, the conductance of the asymmetric molecule can be higher than the conductance of the symmetric molecules, for both thiol and nitrile groups. This is because our particular design rules promote a resonant tunneling with the resonant levels right in the transmission window in the conducting polarity. For a symmetric molecule, the fragments are split in energy, which reduces the transmission by an orbital polarization effect.21-22, 72-73 This corresponds to the localization of an initially delocalized orbital toward a fragment of the molecule, reducing the associated transmission probability. This has been shown experimentally to lead to the Negative Differential Resistance effect.74 As a result, the currents crossing the symmetric junctions are about 20 nA for nitriles and 112 nA for thiols at 1V. In the asymmetric junction, the current increases to 636 nA in conducting polarity, a current 6 to 32 times higher than the symmetric case. This shows that the current crossing the asymmetric molecule is far from being a simple balance between a less conducting contact (nitrile) and a more conducting contact (thiol), for which a current in between 20 and 112 nA would be expected.

IV) Conclusions



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In this work, we investigated and explained how the gap of an asymmetrically anchored molecule can be compressed, (i), and further controlled, (ii), under bias, in the context of molecular rectification. We showed that these two interesting effects are intrinsically related to the design rules we introduced to get an efficient rectifier in a previous work. In this context, these effects control whether or not the frontier levels enter in the transmission window and thus the entrance into the resonant tunneling regime, at the origin of rectification. We introduced two quantities allowing for quantifying the efficiency of these effects, and each of these two has been separately studied in the results section. To assess the role of the design rules, we compared the rectifier to two other similar molecules, each violating a design rule.

We argued that the compression of the gap is unambiguously due to the pinning effect. For this purpose, we explicitly showed the relationship between the S-parameter and the efficiency of the gap compression. This latter is first controlled by the ability of the fragments to align a frontier level in the close vicinity of the Fermi level. This defines a maximal efficiency for the gap compression, achievable in the perfect pinning case (S=0). The magnitude of pinning effect does then come into play as a second factor, reducing this maximal efficiency. We finally showed that the coupling between the two fragments promotes a resistance to this compression of the gap.

We then analyzed the degree of control that we get on the gap of the molecule as bias is applied. The degree of control is directly related to the trend of the molecular electron density reorganization to screen or amplify the applied electric field. Amplification is clearly increasing this control, as it leads to the preservation of the Fermi level alignment of each fragment, in accordance with a strong pinning effect. However this amplification is lost in several cases. First, if the pinning effect is weak, the control is limited with no clear trend to screen or amplify the electric field. Still, a limited control subsists, as there is a voltage drop splitting the fragments. Second, if the gap is too much compressed, the trend to screen the electric field starts. This ensures that the crossing of the HOMO and LUMO levels is avoided. This leads to a resistance to the compression, significantly reducing the control of the gap under bias. Because of this resistance, the control in the compressing polarity is always lower than for the decompressing polarity. Third, if there is no decoupling between two fragments, the control is completely lost. We then consider that the resistance to the gap compression is always happening in our molecules, and is proportional to the degree of π-coupling through the molecular backbone. This resistance prevents the crossing of the HOMO and LUMO in the junction.


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As a result, the role of each design rule and how it leads to the rectification mechanism is now clarified. It is clear that the decoupling rule cannot be violated and is really fundamental. If the pinning rule is violated, the rectification mechanism can still happen but with much lower efficiencies, so that significant ratios are expected only for much higher voltages. In such a case, we suggest the use of short molecules to increase the voltage drop and separate the fragments in energy. The introduction of the two efficiency parameters opens the way to fine tuning, design and characterization of new rectifying derivatives, based on the set of three design rules. Finally, we hope this present work will motivate new studies addressing the major issue of Fermi level alignment and how this latter is affected by an applied bias. If the principles that we discussed here are in the context of rectification, we believe they also may be of high interest for designing other efficient devices in Molecular Electronics.

V) Acknowledgments

The work was primarily supported by the Center for Bio-inspired Energy Science (CBES), an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award No. DE-SC0000989. C.V.D. was also supported by a Gustave Boël – Sofina Fellowship of the Belgian American Educational Foundation (BAEF). C.V.D. acknowledges the Laboratory for Chemistry of Novel Materials at the Université de Mons in Belgium for their computational support and resources.

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Molecular Junctions: Control of the Energy Gap Achieved by a Pinning

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