Charge Transport Across Insulating Self-Assembled Monolayers: Non-equilibrium Approaches and Modeling To Relate Current and Molecular Structure

This paper examines charge transport by tunneling across a series of electrically (insulating molecules with the structure HS(CH 2 ) 4 CONH(CH 2 ) 2 R) in the form of self-assembled monolayers (SAMs), supported on silver. The molecules examined were studied experimentally by Yoon et al. (Angew. Chem. Int. Ed., 51, 4658 − 4661, 2012), using junctions of the structure AgS(CH 2 ) 4 CONH(CH 2 ) 2 R // Ga 2 O 3 / EGaIn. The tail group R had approximately the same length for all molecules, but a range of diﬀerent structures. Changing the R entity over a range of diﬀerent structures (aliphatic to aromatic) does not inﬂuence the conductance signiﬁcantly. To rationalize this surpris-ing result, we investigate transport through these SAMs theoretically, using both full quantum methods and a generic, independent-electron tight-binding toy model. We ﬁnd that the frontier orbitals HOMO, HOMO-1, HOMO-2 and HOMO-3, have similar structures for all of the diﬀerent R-groups, and that the HOMO, which is largely responsible for the transport in these molecules, is always strongly localized on the thiol group. The relative insensitivity of the current density to the structure of the R group is due not only to these similar frontier orbitals but also to a combination of energy levels ( ε ) and tunneling amplitudes ( t ), which lead to similar conductance for diﬀerent tail groups, i.e. the large diﬀerence between ε ’s in conjugated and saturated groups is compensated by tunneling amplitudes inside the tail group. In addition, the coupling of the tail group to the alkane chain is much weaker for conjugated groups than for saturated ones. Conductance change in these molecules is not inﬂuenced by the broadening of the molecular levels connected to the left and right electrodes, since in all these molecules the eﬀective coupling to the silver substrate is the same, whereas the other eﬀective couplings are determined by the weak connection to the oxide Ga 2 O 3 layer. All these factors combine to produce currents largely insensitive to the R group. in molecular structure. This work indicates that signiﬁcant control over SAMs largely composed of nominally insulating groups may be possible when tail groups are used that are signiﬁcantly larger than those used in the experiments of Yoon et al. 13


I. Introduction
The putative field of 'Molecular Electronics' involves charge transport by processes in which the structure of the molecule forming the junction influences the conductance. Ultimately it hopes to design molecules whose electrical conductance can be tuned rationally through organic synthesis. [1][2][3][4][5][6][7] One of the motivations for this field has been the expectation that small changes in the structure or the environment of the molecules would change the characteristics of charge transport through them in ways that might be useful in practical electronics, sensing, or controls. In single molecule devices with weak electrode/molecule interaction, Coulomb blockade enables switching the current on and off by a gate field. 8,9 This type of control is, however, not the only way to control the current. Fracasso et al. 10 showed that the conductance in anthracene derivatives of approximately the same thickness can be influenced by the type of conjugation. The results in that experiment 10 were attributed to quantum interference, but the origin of the effect remains to be validated in other experiments. There are now many reports of substantially different conductance values measured in break junctions, see e.g. ref. 11 Another example is the exponential decay of the conductance in alkane self-assembled monolayers (SAMs), which does not exist in n-polyene chains where a chain of C-C single bonds (CH 2 ) 2n is replaced by an extended conjugated chain (CH = CH 2 −) n . 12 Yoon et al., 13 reported a systematic experimental study in SAMs which suggested that large changes in molecular structure (e.g. changing a cyclohexyl group for a phenyl group) need not induce significant changes in the conductance of molecular monolayers. Yoon's study also indicated that replacing 13 −CH 2 CH 2 − in the interior of the molecules making up the SAMs by −CONH− had no effect on tunneling current. More specifically, they measured the current densities through a series of SAMs based on different molecules located between a silver electrode and Ga 2 O 3 /EGaIn electrode, where EGaIn denotes eutectic gallium and indium (a liquid metal alloy 14 ) and Ga 2 O 3 is a spontaneously formed, electrically conducting surface oxide layer (normally 0.7 nm thick) on the EGaIn electrode. The structure of the molecules making up the SAMs is HS(CH 2 ) 4 CONH(CH 2 ) 2 R where R is one of the tail groups Figure 1: Structure of the molecules used by Yoon et al. 13 to form the self-assembled monolayers on the Ag surface. The structure of all molecules 1-13 is HS(CH 2 ) 4 CONH(CH 2 ) 2 − R where R is the tail group. In the measurements, molecules C12 and C18 were presented as calibration standards. All current densities were measured through Ag-molecule-Ga 2 O 3 /EGaIn structures.
shown in Fig. 1. In all these molecules, the tail groups have almost the same length. The tail structures can be divided into two groups: (1) aromatic structures, and (2) saturated aliphatic groups.
It is not obvious why SAMs of these very different molecules do not yield very different conductivities. 1 The measured current densities for all molecules at V bias = 0.5 V are shown in Fig. 2 on a logarithmic scale, with simple alkane chains of length 12 and 18 alkane units (e.g. S(CH 2 ) n−1 CH 3 ) included. Alkane chains are standard molecules in which electron transport has been studied extensively. [15][16][17][18][19][20][21][22] The main result of those prior studies is that the current decays exponentially with chain length with a decay constant β ∼ 1.0 per CH 2 unit, with some variation across different experiments. [15][16][17][18][19]23 This dependence has been also successfully addressed computationally using Density Functional Theory (DFT) combined with non-equilibrium Green's function (NEGF) studies. 16,22,24 Yoon et al. measured β ∼ 0.9 per CH 2 unit -a value that is in agreement with these earlier results. In fact, the 12-and 18-unit alkane chains are used as calibration standards in Yoon's experiment, to compare with the results obtained with the other molecules. The green crosses (×) show the interval of the errors. Although the saturated structures (molecules 8-13) seem to have a somewhat smaller conductance than the conjugated ones (molecules 1-7), the differences are not significant within the experimental error.
Understanding the relatively constant current density observed for this complicated system is difficult, as different mechanisms may be responsible for the observed current densities.
While inter-chain tunneling of the electrons may be significant, here we focus on the effects of the molecular electronic structure on the single-molecule current. The standard way to address this subject is by performing DFT-NEGF calculations. 25,26 We did such calculations, which confirm that varying the tail group does not induce dramatic differences in the molecular conductance. To understand this result, we describe the transport using a simple tight-binding model, with parameters inferred from a series of ground-state DFT calculations. We call this the tight-binding toy model (TBTM). The procedure we follow is in the spirit of semi-empirical models which aim to explain experiments using parameters obtained via fitting to ab initio calculations or to experimental data. In particular, one of the criteria we have used to adjust the parameters is that the structure of a few of the highest occupied frontier orbitals, as obtained by full DFT calculations, is essentially reproduced in TBTM.
The transmission of the model is then analyzed again using Green's function methods. This provides insight into why the current varies only weakly across the set. These results may

II. DFT calculations (A) DFT-NEGF based transport calculations
We start with the transmission for molecules 1-13 obtained from DFT-NEGF. We have calculated the transmission through these molecules with two different methods: (I) Gas phase-NEGF: In this method, DFT calculations for molecules in gas phase (thiolended, without electrodes) are performed and then the contribution of the molecule to transport is calculated by computing |G 1N 2 | between the p z − orbitals (perpendicular to the plane through S, C and H) of the S(1) and the R groups (N ) where G denotes the Green's function and S is the sulfur atom of the thiol linker. 27 G 1N can be understood as the quantity which measures the tunneling amplitude from site 1 to site N. Once the Green's functions are known, the transmission can be obtained using the Landauer-type equation. Here we include self-energies within wide-band limit approximation. 28 The coupling of the tail groups to the Ga 2 O 3 oxide layer (on EGaIn electrode) is an unknown parameter which is not included in the gas-phase calculations. This method does not yield the correct location of the transmission peaks (of the occupied orbitals) since the molecules are considered in gas phase -shifts induced by interface dipoles and image charges are therefore not taken into account. To obtain better insight into these shifts, we have used another method.
(II) Extended molecule-NEGF: In this method, an extended molecule is used. The extended molecule is made by connecting the molecule to 3 × 3 × 3 metal clusters from right and left.
Then we connect this extended molecule to two electrodes which are 3 × 3 × 9 clusters. For silver, we need to incorporate 19 electrons per silver atom for the largest frozen core, as opposed to 11 electrons per gold atom. In view of the similarity between the two contact types, 29 and since that we are looking for trends which are expected to be the same for silver and gold, we have chosen to perform our calculations using gold contacts. Experimentally, with a metal alloy and a conductive oxide layer on one side of the molecule, the screening cannot be expected to let surface effects decay within the width of the contacts. We present the results for a molecule between two gold contacts as another reference calculation, with the awareness that the actual system (Ag on one side and oxide/metal on the other side) is somewhere in between that of the gas phase and the extended molecule with gold contacts.
To calculate the self-energies for each 3 × 3 × 9 cluster, we divide the cluster into three layers where each layer consists of three sub-layers, as shown in Fig. 3. Then we calculate the transport properties of the system. This method relies on the strong screening in the contact regions, which renders the results relatively insensitive to their size.
The code used here is an in-house developed add-on to the Amsterdam Density Functional   (II) The LUMO structure and energy vary quite strongly with tail group. The shape of the LUMO furthermore depends quite sensitively on the geometrical optimization. With our optimized geometries (that may or may not agree in detail with the structures found in the SAMs, which are not known in detail experimentally, and probably depend on the topography of the electrode surface), the LUMO of the molecules with saturated tail groups is located on the thiol group just as the HOMO, whereas for molecules with a conjugated tail group, it is located on the tail group.
(III) HOMO-1, HOMO-2, HOMO-3 of molecules 1-13 are mostly located on the amide, on both the amide and the tail group R, and on the tail group R respectively; see supplementary information.
As the structure and chemical potential of the HOMO does not change substantially across different molecules, and as the HOMO is substantially closer to the Fermi energy of expect the transport to be hole-like -that is, dominated through the HOMO, in agreement with our NEGF calculations.

III. Tight-binding Model
The The guiding principle in constructing the TBTM model is that it produces reliable HOMO's, as the transport in the molecules under study is hole-like and off-resonant (see section II).
The probability for electron tunneling is low enough to neglect two electrons tunneling at the same time, and we can safely neglect spin. This junction is then represented by a simple, essentially linear tight-binding system, coupled to two non-interacting semi-infinite leads. The (extended Hückel-like) Hamiltonian 32 of the molecular region reads where H.c. denotes the Hermitian conjugate and N is the length of the interacting chain.
The parameters t i,j represent the tunneling amplitudes between the atomic orbitals φ i and φ j within the molecule. The electron creation and annihilation operators, d † i and d j satisfy the usual anti-commutation relations. The second term in the Hamiltonian represents the site energies ε i . Figure 7: A tight-binding toy model (TBTM) structure representing the molecules 1-13 and the oxide layer. We model the molecular junction with six segments S 1 − S 6 . The barriers with different site energies represent the following parts of the molecules : ε S : thiol group. ε A : alkane chain in which the energy of the central sites (ε CO and ε N ) are chosen to be higher than the rest of subunits; they represent the amide group (CONH). ε R : tail group R (which varies in molecules 1 − 15). ε Ox : Ga 2 O 3 layer.

The Schrödinger equation then takes the form HC
coefficients corresponding to φ i such that the molecular orbital ψ α = i C i,α φ i , and S is the overlap matrix. When the orbitals are normalized such that the diagonal elements of the overlap matrix are 1, then the off-diagonal elements between the neighboring sites are typically ∼ 0.3. 33 Our analysis shows that the overlap matrix does not play an important role, and the results that we present below turn out to be insensitive to such deviation from orthonormality. Therefore, we will take S to be the unit matrix (e.g, S ii = 1, S ij = 0 for i = j) in the remainder of the analysis. 34 We want the tight-binding chain in our toy model to mimic the electronic properties of molecules 1-15 (Fig. 7). These molecules consist of six segments. The first segment (S1) is the thiol linker and the next three segments are S2 (CH 2 ) 4 , S3 CONH and S4 (CH 2 ) 2 respectively. The next part (S5) is the tail group R and the last segment (S6) is the oxide layer. We find most of the parameters of the TBTM Hamiltonian of equation 1 from the results shown in Fig. 6 and Fig. 12, and from calculations on periodic chains, as we outline in the supporting information.
Our generic tight-binding chain for molecules 1-13 is shown in Fig. 7. It consists of 17 sites, divided into six subunits. The first molecular subunit (S 1 ), represents the thiol binding group connected to the silver electrode. Its site energy, ε S , is higher than that of the alkane chain. The following three groups of sites (S 2 , S 3 , S 4 ), represent the alkane chain with the amide. The amide sites (S 3 ) have energies ε CO and ε N that are higher than alkanes. It should be noted that our model presents a coarse-grained description of the molecules under study.
In particular, in our description of the amide units, the (C=O) group is considered as one site, and the (NH 2 ) as another one.
Two amide subunits (C=O and NH) are coupled to neighbouring CH 2 by t amide -we take those to be the same. The coupling inside the amide group is t amide . The fourth substructure The tight-binding parameters ε and t for each subunit are shown in Table. 1. In Hückel molecular orbital theory, these parameters are called α and β respectively. Reproducing the structures of the orbitals found (see Fig. 6 and Fig. 12) is the most important criterion for finding the parameters. We match the parameters of the TB chain to those orbital Table 1: Tight-binding parameters ε and t for each subunit (in eV). Subunits of the same kind are coupled by t. The coupling of a subunit to a saturated C neighbour is t n , provided the subunit itself is not a saturated C (including methylene). Literature parameters are presented from (a) Benkö et al. 39  It may seem impossible to get the HOMO to be localized on the sulphur in view of its value being lower than that of the amide unit, a difference which would only become more pronounced taking into account the weaker coupling of the amide unit to its neighbours compared with the strong S-C coupling. The reason is that there are two orbitals localized on the sulphur, a bonding state, which corresponds to the HOMO-5, and the state which which is anti-bonding with the first C atom, and which forms the HOMO. Also on the amide group, there are two states, one bonding and the other anti-bonding. The weak internal coupling in this group causes the splitting between these two to be quite small, therefore they both lie below the HOMO.

IV. Results for the current
Once the molecule is coupled to the electrodes, the retarded Green's function of the system can be obtained as G R = [ωS − H − Σ] −1 , where Σ is the total self-energy obtained from left (L) and right (R) self-energies, Σ = Σ L + Σ R . Within the wide band limit (WBL), Σ L and Σ R are purely imaginary and do not depend on energy. They represent the broadenings Σ L/R = − i 2 Γ L/R . Once the Green's functions are known, the current can be calculated from a Landauer-type equation: where is the transmission, µ L = E f − V /2, µ R = E f + V /2 and G a is the advanced Green's function which is found as the complex conjugate of G r . The Fermi function f (ω, µ) describes the electronic occupation of the levels, f (ω, µ) = 1/(1 + exp( ω−µ kT )). The Γ parameters describe the molecule-electrode coupling and are assumed the same for all molecules in our calculation.
The only parameters that change across molecules 1-13 are the site energy and the tunneling parameters of S 5 , i.e. ε R , t R and t R , representing the tail group. To study the transport through molecules 1-13, we calculate the current for different values of these three parameters. It should be noted that in matching the orbital structure from our TBTM to the DFT results, we have found it sometimes necessary to shift the site energies of the tail group somewhat (see Supplementary information). The variation of ε R evaluated from DFT is ∼ 1 eV. Our results for molecules with conjugated and saturated tails and in the presence of a bias voltage are shown in Fig. 8. They show that the largest difference in the current is a factor of ∼ 7.6 at bias voltage V b = 0.5 eV. This result agrees well with the results of the experiment (measured at bias 0.5 eV). In addition, the allowed amount of variation for these parameters to reproduce the main features of the orbital structure and current is ±0.3 eV.
The structures of some tail groups in Fig. 1 are not linear. Therefore, we also investigate the transport through a system with a cyclic tail group as shown in Fig. 9, which provides two pathways for the particles moving through the tight-binding chain. The current through such a chain is compared to the results of original model of Fig. 7 both for molecules with conjugated (ε R = 3.5, t R = 4, t R = 1) and saturated (ε R = −14, t R = 6, t R = 6) tail groups. We emphasize that, although the difference between the site energies for pi-and sigma orbitals seems rather dramatic (3.5 versus −14 eV), this does not imply a similar difference in chemical potential: as the pi-system is only half filled in the neutral state, the highest occupied pi-level is still below the for pi-sites, and for the R-groups we consider here they turn out to be about 1 eV below the Fermi energy of the gold. The sigma orbitals are all filled and therefore reach up to + 2τ ≈ −14 + 2 · 6 = −2 eV. As shown in Fig. 9, the current through the molecules with cyclic structure is in the same range of molecules with linear tails, and the largest overall change between the I-V curves is a factor of 8.
As shown in the supplementary information, the variation of ε R evaluated from DFT is not large (∼ 1 eV). Within this variation the highest occupied orbitals, which are largely responsible for the transport, are always localized on specific regions of the molecules. For instance, the HOMOs in all molecules are located on the thiol anchoring group. However, the shape of the tunneling barrier is determined by the entire tunneling chain which contains  Fig. 8, but with the linear segment S 5 representing the tail group R, replaced by a cyclic chain which provides two pathways for the electrons going through the tight-binding chain. For conjugated groups, ε R = 3.5, t R = 4, t R = 1. For saturated groups, ε R = −14, t R = 6, t R = 6.
the tail group at the end of the chain. Therefore, the transmission through the tail group is important. Our analysis indicates that the combination of t R and t R compensates for the variation of the site energy, ε R , which is close to the Fermi energy in conjugated groups, whereas in saturated groups it is far from the Fermi energy 2 . In other words, the gateway from a saturated orbital to a conjugated electron system on the tail group can be viewed as a narrow passage leading to an easily traversable track. Moving from the conjugated chain to a saturated tail group is easy (the t-matrix element is large), but the orbital energy is much farther away from the Fermi energy, suppressing the current through this structure.

V. Conclusions
We have investigated the transport through a series of self-assembled monolayers with varied tail groups, as measured by Yoon et al. 13 DFT-NEGF calculations confirm the modest current variation observed experimentally. DFT calculations show that the HOMO is largely located on the thiol group in these molecules, and this is the closest orbital to the Fermi energy of the electrodes. Therefore the transport is hole-like. To understand the weak effect of the tail group on the conductance, we have constructed a tight-binding toy model with site energies and tunneling parameters based on a series of DFT arguments and existing 2 Remember that the TBTM model was not designed to yield excitation energies-see section III. literature. Our model reproduces the structure of the highest occupied orbitals (HOMO, HOMO-1, HOMO-2 and HOMO-3). Our analysis suggests a few reasons for the surprisingly small differences among the currents in these molecules with saturated and conjugated tails: (1) The location and the energy of the frontier orbitals which are responsible for the transport are similar across the entire series. For instance, the HOMO in all molecules is localized on the thiol linker. (2) The transmission is therefore mainly determined by the tunneling through the rest of the molecule, which is influenced by the tail group. In the tail groups, the differences between the values of the site energies (ε) and tunneling parameters (t) in the conjugated and saturated groups largely compensate each other: Conjugated groups have smaller coupling (than saturated ones) but their ε is much closer to the Fermi energy than is that for saturated groups. The combination of both still leads to better conductivity of the conjugated groups. However, the coupling of the tail group to the alkane chain (t R ) is much weaker for conjugated groups than for saturated ones. So the combination of t R and t R compensates for the large difference between ε's in conjugated and saturated groups. (3) The conductance is also influenced by the broadening of the molecular levels connected to the left and right electrodes (Γ L,R ). In all these molecules Γ L is the same and Γ R is determined by the connection to the oxide layer. For the two molecules C 12 and C 18 (used as calibration standards) in which the length of the alkane tail group is extended, the coupling to the electrode decays exponentially, and therefore a significant change in the I-V characteristic can be observed. We found that the decay constant is β = 1.1 per methylene group in alkanethiols which in agreement with the experimental results (β = 0.9 − 1.0). We therefore expect the similarity of the current for saturated and conjugated tail groups to depend sensitively on the tail group length -longer tail groups would lead to saturated tail groups yielding lower current densities.

Supporting Information
Here we present a few details concerning the determination of the tight-binding parameters.
In particular we discuss how we obtain these parameters and we show how these parameters can reproduce the structure of molecular orbitals:

(A) Finding the tight-binding parameters
We find most of the parameters of the Hamiltonian from the results shown for molecular orbitals and from calculations on periodic chains as follows: • Alkane segment and saturated tail group: The parameters for an alkane chain should satisfy three criteria: (I) From several NEGF calculations, 4,16 it is established that the HOMO is located about 2 eV below the Fermi energy of the metal. (II) The conductance through alkane chains decreases exponentially with molecular length with the exponential rate of β ∼ 1 per methylene group, [15][16][17][18][19]23 i.e. G(L) = G 0 e −βL . (III) The energy separation between the levels, in particular the frontier orbitals, should be in agreement with that found in DFT. This separation is determined (for a homogeneous chain) by the parameter t, as the levels are distributed according to E n − ε = −2t cos(k n a) where k n = nπ/L. Here a is the inter-site distance and L is the effective chain length. From the orbitals found in DFT, we use those that have significant sigma character. The higher orbitals are easily identified as such -the lower ones tend to hybridize with s-orbitals on the hydrogen atoms. These three criteria together lead to a tunneling parameter t A = 6 eV and a site energy ε A of -14 eV.
This value for ε seems very low. However together with the coupling with a value of t about 6, this leads to a level at ε + 2t ∼ −2 eV for a long alkane chain.
Our parameters yield β = 1.1 per CH 2 unit based on the analytical formula β = ) which follows directly from the independent-electron Schrödinger equation.
• Amide group: We have performed a DFT calculation for an alternating [CH 2 − amide] n chain (with length n = 8). The results reveal that the bandwidth of the amide states is about 2 eV. We get a similar bandwidth in our model for the parameters t amide = 1.8 eV, t amide = 0.5 eV, ε CO = −1.3 eV and ε N = −1.8 eV, which can also reproduce the structure of the highest occupied orbitals, HOMO, HOMO-1, HOMO-2 and HOMO-3. The results presented below are valid for a range of values of these parameters-see supporting information for details.
In matching the orbital structure from our TBTM to the DFT results shown in Fig. 6 and

(B) Molecular orbitals
Using the parameters described in the previous section, we now present the structures of orbitals obtained for our TBTM. Changing the site energy of the tail group (ε R ) and its tunneling coupling t R and t R (see Fig. 7 ) for molecules with conjugated and saturated tails does not influence the HOMO energy located on the thiol linker. To verify this conclusion, we consider the electron population of a few of the highest eigenstates in the TBTM of Fig. 7.
For the molecular orbital ψ n = i C i,n φ i , where the sum is over the 17 different sites, the population of site i is |C i,n | 2 . These populations for the HOMO, HOMO-1, HOMO-2 and HOMO-3 are shown in Fig. 11 for molecules with conjugated (ε R = 3.5 eV and t R = 4 eV) and saturated (ε R = −14.0 eV and t R = 6 eV) tails. The structure of the HOMO orbitals is in agreement with our DFT calculations shown in Fig. 6 and Fig. 12. Furthermore, the lower levels such as HOMO-1, HOMO-2, HOMO-3 of the TBTM are mostly localized on amide (plus tail group in some conjugated molecules), the tail group R or on both. Due to a proper choice of parameters, their structure is in agreement with our DFT calculations.
We now show how the orbitals vary when we alter ε R . We take this variation within 1 eV (as implied by Fig. 10), both for saturated and conjugated tail groups. For molecules with a saturated tail group, shifting the site energy to ε R = −13 eV does not change the structure of the orbitals, but for molecules with a conjugated tail, shifting the site energy from ε R = 3.5 to ε R = 4.5 eV changes the HOMO-2 and HOMO-3 slightly. For ε R = 4.5 eV, HOMO-2 and HOMO-3 are both distributed over the amide and tail group whereas for ε R = 3.5 eV, HOMO-2 is on the amide group and HOMO-3 is localized on the R group.
These agree well with our DFT results.
These results show that with these selected parameters we can roughly reproduce the structure of the frontier orbitals. The allowed amount of variation for these parameters to reproduce the main features of the orbital structure and current, is about ±0.3 eV. 1 2 3 4 5 6 7 8 9 10 11 12 13   1 2 3 4 5 6 7 8 9 10 11 12 13   1 2 3 4 5 6 7 8 9 10 11 12 Figure 12: HOMO, HOMO-1, HOMO-2 and HOMO-3 of the molecules 1-13 are mostly located on thiol, amide, amide (plus tail group R), and tail group R respectively. These orbitals are to be compared with their counterparts from the TBTM in Fig. 11.; the TMTB indeed mimics the full DFT results very closely.