Charge transfer plasmon resonances across silver–molecule–silver junctions: estimating the terahertz conductance of molecules at near-infrared frequencies

Institute of High Performance Computing, and Research), 1 Fusionopolis Way, 16-1 E-mail: wul@ihpc.a-star.edu.sg; baiping@ih Department of Chemistry, National Univ Singapore 117543 Institute of Materials Research and Eng Singapore 138634 Department of Materials Science and Engine Engineering Drive 1, Singapore 117575 Singapore University of Technology and Des Center for Advanced 2DMaterials, National Singapore 117542 Cite this: RSC Adv., 2016, 6, 70884


Introduction
In the past decade, there has been signicant progress in understanding the optical properties of plasmonic nanostructures. [1][2][3][4][5] A unique feature of plasmonic nanostructures is the sensitive dependence of their plasmon resonance energies on nanostructure geometry and dielectric environment. This property lays the foundation for one of its most promising applicationsplasmonic sensing. [6][7][8] In particular, there is one type of plasmon resonance, the charge transfer plasmon (CTP), which has drawn a lot of attention in recent years because it becomes important at atomic length scales. [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] This charge transfer plasmon usually occurs across an optically-conductive junction between two nanoparticles, and its plasmon resonance energy is particularly sensitive to the shape and conductivity of the junction. This property of CTPs opens the door to many new applications such as single molecule sensing, 21 optical nanoswitches, 14 probes for molecular conductance at optical frequencies 15 and so on.
Depending on the conduction mechanism of the junctions, different types of CTP resonances can be classied. The rst type comprises the metallic or semiconductor junction, where the conduction mechanism is free carrier (e.g., free electron) transport. Dimers that are linked by a thin metal bridge are able to generate a CTP with a resonance typically in the near-infrared region of the spectrum. 2,12,21 By lling the dimer gap with semiconducting materials (e.g., amorphous silicon), the frequency of the CTP can be controlled as progressive photoconductive gaploadings are achieved (theoretically) by varying the free-carrier density of the semiconductor. 14 The second type refers to dielectric junctions, including air gaps, vacuum gaps, and inorganic/organic dielectric lms. In contrast to a bridge made of a metal or semiconductor, no free carrier is available in the dielectric lms. The mechanism of charge transport between the two plasmonic resonators is quantum mechanical tunneling. For junctions with high tunneling rates (i.e., for small gaps with low tunneling barrier heights), tunneling charge transfer plasmon (tCTP) can be observed. 13,[16][17][18][19] In this work, we use a platform that could support various types of CTPs, which consists of a self-assembled monolayer (SAM) connecting two metallic nanoparticles, forming a SAMbridged plasmonic dimer. We use this platform to probe the molecular conductance at terahertz frequencies. Within a SAMbridged plasmonic dimer, the molecular junction is typically characterized by molecular frontier orbitals, i.e., the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). The properties of molecular junctions depend on many factors, e.g., molecule-electrode interactions strength, chemical structure of the molecule, nature of the molecule-electrode contacts, etc., which make it possible to control the electrical and optical properties of molecular junctions by altering the chemical structure of the molecule. For example, conjugated molecules are considered to be more conductive than aliphatic molecules due to their small HOMO-LUMO gaps; this property has been used to tune the frequency of tCTP in molecular tunneling junctions. 20 In other work, incremental replacement of the nonconductive molecules with a chemically equivalent conductive version differing by only one atom has produced a strong 50 nm blue-shi of the coupled plasmon. 25 Standard DC or low-frequency electrical transport measurements in molecular electronics 26,27 cannot be performed at near-infrared frequencies (i.e., 200-400 THz or 0.75-1.5 mm). Characterization of charge transfer plasmons through molecular junctions could ll this unexplored regime, probing molecular conductance at frequencies of hundreds of terahertz.
It is challenging to model the optical response of the SAMbridged plasmonic dimers for three reasons: (I) multiscale: the length-scales of the plasmonic metal electrodes and molecular monolayers are very different, e.g., the electrode is a cube with side-length of a few tens of nanometres consisting of heavy Ag atoms, and the SAM consists of many molecules made up of light C and H atoms with molecular length on the order of one nanometer; 20 (II) interface: the electron transport/transfer processes in metals and molecules are distinct, which makes the modelling of the continuous charge transfer through the metal/SAM interface difficult, even at DC frequency; and (III) multiphysics: the problem involves at least two non-negligible physical processes, which include plasmonic oscillation and charge transport. To solve the problem, three modelling methods are available at present. First of all, full quantum-mechanical rst-principles calculations 26,27 should be the most accurate method to capture the multiple physics in this problem, but it becomes impractical when the large number of electrons in the plasmonic electrodes needs to be taken into account. 17 Second, classical electromagnetic models are the most popular models to describe the optical response of plasmonic systems due to their simplicity, but they only deal with electromagnetics and not with charge transport. In certain circumstances where additional physics (e.g., photoexcitation 14 and quantum tunneling 17,19,[22][23][24] ) is involved, part of the nanostructure cannot be modeled by its conventional local linear dielectric function and corrected electromagnetic models are more appropriate. The corrected electromagnetic models divide the multiphysics problem into two parts: (I) a ctitious junction material (whose local permittivity is characterized using a Drude model) is introduced, representing the additional physical phenomenon other than electromagnetics; and (II) with the inclusion of the ctitious junction material, the optical/plasmonic properties are calculated within the classical electromagnetic framework. Compared to the full quantum-mechanical calculations and classical electromagnetic models, the corrected electromagnetic models are simple, fast and capable of modelling multiphysics problems. The most representative example, the quantum-corrected model (QCM) 17 that is used to model quantum tunneling junctions, has recently attracted a lot of attention and been extended to model different junction morphologies, 22 to improve its accuracy for practical situations, 23 and also to different numerical implementations. 24 However, the essence of QCM does not changethe introduction of the ctitious junction.
For the SAM-bridged plasmonic dimers, we will employ the same corrected electromagnetic models by introducing a ctitious junction material. However, different from those available in the literature, our model further breaks down the problem by separately modelling the following two processes: the 'charge injection at the metal/SAM interface' and 'charge transport in the ctitious junction material'. We will use a space-charge model (applicable to both classical 28 and quantum 29 regimes) to correlate the charge injection and charge transport parts. In doing so, a frequency-dependent ac conductivity of the ctitious junction material (i.e., the space-charge region) is used. This frequencydependent ac conductivity is critical to show the dependence of the CTP resonance energy on the conductivity of the SAM.
By using the frequency-dependent ac conductivity for the ctitious junction, we have successfully explained our experimental ndings of tCTP in our rst two sets of molecular tunneling junctions. 20 The present work will further develop the technique to a more comprehensive and generalized model, namely, the space-charge corrected electromagnetic model. Making use of this generalized model, we will conduct a thorough theoretical investigation of the charge transfer plasmons across the molecular junctions in silver nanoparticle-SAMsilver nanoparticle dimer systems, for a wide range of molecules with different lengths and optical conductivities. Our theoretical modelling of these dimer systems establishes a one-to-one relationship between the conductance of the SAM layer and the resonant energy of the CTP mode. Based on the nding, we will demonstrate in this work the application of using plasmonic oscillations for measuring the terahertz conductance of single molecules at near-infrared frequencies.

Results and discussion
Space-charge corrected electromagnetic model Fig. 1a shows the hybrid dimer system that we use to study single molecule terahertz conductance using plasmon characterization. In this dimer system, cuboidal Ag nanoparticles are functionalized with SAMs, forming metal-SAM-metal junctions through self-assembly. Through optimization of the self-assembly process, these dimeric structures with a single layer of wellaligned SAMs are useful systems to study the molecular conductance at near-infrared plasmon frequencies.
More specically, we describe our Ag-SAM-Ag systems using a space-charge corrected electromagnetic model as shown in Fig. 1b. Our model separates the 'charge injection' and 'charge transport' processes. The rationale behind this separation is that, the commonly-used Drude model (to describe the ctitious junction material) within the classical/corrected electromagnetic framework assumes that the motion of electrons is damped via collisions, but the charge injection at the interface (e.g., through quantum tunneling or eld emission) is oen a collision-free or ballistic transport process. It remains questionable to model a ballistic transport process using a Drude model. By breaking down the problem, we can consider the electron collisions in the 'charge transport' region, but not in the 'charge injection' process.
As shown in Fig. 1b, the 'charge injection' process denes the number density of free electrons that are able to escape out of the metal into the SAM layer: n sc , where 'sc' stands for space charge. Space charge is a concept in which excess electric charge is treated as a continuum of charge distributed over a region of space (usually dielectrics) rather than distinct point-like charges. This model typically applies when charge carriers have been emitted from a solid. The well-known classical Child's Law 28 that denes the space-charge limited current (SCLC) in a plane-parallel vacuum diode has been theoretically extended to the quantum tunneling regime for diodes with nanometer-scale gaps. 29 Here, we apply the same concept to model the charge injection from the plasmonic metal electrode into the SAM layer, where a space-charge region forms in the ctitious junction material. It is these n sc free electrons that transport in the space charge region with a damping frequency g sc . Their optical properties are therefore described using the Drude model: 30 where 3 0 is the vacuum permittivity and u is the frequency of the driving optical eld. The frequency-dependent ac conductivity of the space charge region in eqn (1) can be written as 30 Here s sc is the dc conductivity of the space charge region, n sc represents the number density of the injected free electrons into the SAM layer (i.e., the space charges), e (and m) are charge (and mass) of the electrons, and s sc is known as the mean free time between collisions among the free electrons transporting in the space charge region: s sc ¼ 1/g sc , where g sc is known as the characteristic collision frequency. By rearranging the terms in eqn (1) and (2) into a form that can be directly compared to the pioneer quantum-corrected model by Esteban et al. 17 that was used to describe tunneling junctions, we have: It is clear that (I) our damping frequency g sc in the space charge region (i.e., the characteristic collision frequency) is analogous to their tunneling damping parameter g g (but with different damping mechanisms); (II) they use plasma frequency u g 2 ¼ n sc e 2 3 0 m to represent the number density of free charges n sc in the space charge region, while we use conductivity s sc ¼ n sc e 2 s sc m since we are interested in conductance measurements. Up to this point, the two models are essentially the same in mathematical form, although physically different. Our model highlights the individual (but correlated) modelling of the collision-free 'charge injection' part at the interface (which determines n sc ) and the 'space charge region' part within the ctitious junction (described by the electron-collisions based Drude model eqn (3)). However, their model specically assumes that tunneling g g [ u and only dc conductivity of the ctitious junction is considered, where the frequency dispersion of the conductivity is neglected. 17 In fact, a recent theoretical study 31 has suggested that the tunneling damping g g can be signicantly reduced (up to 4 orders) by increasing the tunneling rate to reach the space-charge-limited regime. Here we aim to develop a more generalized model, so we remove this assumption and use the full expression of 3 sc (u). The two parameters n sc and g sc (or s sc ¼ 1/g sc ) are used to take care of various charge injection and transport mechanisms. Physically, within the space charge region, the relaxation time s sc is a function of the number density of free carriers n sc : more densely distributed free carriers tend to collide more frequently, giving a shorter s sc . However, the exact relation between the two is unknown (to the best of our knowledge). Previously, we predicted that the number density of the injected free electrons needs to be close to that of the metal (on the order of 10 28 m À3 ) in order to trigger the CTP mode, 19 while the relaxation time of the free electron gas in metals is typically on the order of 10 fs at room temperature. 30 Therefore, s sc ¼ 10 fs is assumed throughout this study. Numerically, the resonant energy of 0.3-3 eV corresponds to a driving frequency u of 72.5-725 THz, whereas the damping frequency of the free electron in the space charge region g sc is on the order of 100 THz. These comparable frequency scales also support the usage of frequency-dependent ac conductivity in the space charge region.
However, it should be noted here that the proposed spacecharge corrected electromagnetic model does not capture the detailed atomic-scale interactions 32 between the injected electrons and the molecule atoms in the 'charge transport' part. The critical properties of a SAM layer (e.g., HOMO-LUMO gap, alignment of HOMO-LUMO with respect to the Fermi-level of the metal, molecular length, etc.) are all taken into account by the 'charge injection' modelling part (to be discussed in latter section of Charge injection mechanisms).
Before treating any specic charge injection mechanisms, we rst classically simulate the plasmon resonances of the Ag-SAM-Ag systems for different types of SAMs by using two variables as input parameters: (I) the molecular length d of SAMs and (II) the conductivity s sc of the SAM charge transport layer, which is directly relevant to the number density of free electrons that are able to be injected from one electrode into the space charge region. Therefore, a map of plasmon resonant energy ħu CTP versus the molecular length and conductivity of the SAMs can be compiled which can be used as a benchmark for subsequent investigations. Experimental measurements of the plasmon resonance of a particular type of the SAM then allow us to estimate the conductivity of that SAM layer. By knowing the number of molecules in the SAM layer, we can subsequently estimate the conductance of each single SAM molecule.

Simulated plasmon resonance of Ag-SAM-Ag systems
The plasmon resonances of Ag-SAM-Ag systems are simulated using nite-element-method (FEM) optical simulations. The spectra of absorption, scattering and their sum (i.e., extinction) for the Ag-SAM-Ag system are calculated. Simulations are performed using plane-wave excitation by assuming that the incident light is polarized along the longitudinal axis. Here the two identical Ag cuboids are assumed to have a square cross-section with side-length w and length l. The dielectric function of Ag is taken from Palik's handbook. 33 Between the two Ag cuboids, the SAM layer is assumed to have a square cross section with area A ¼ w Â w nm 2 and the length of the SAM layer d is determined by the molecular length. As the SAM layer now functions as the electron transport medium, it should not be described by its conventional refractive index. Instead, we assume the SAM layer as a ctitious junction material, and employ the corrected electromagnetic model using eqn (1) and (2) to simulate the system. Fig. 2 shows simulated plasmon resonance spectra of one representative Ag-SAM-Ag system for d ¼ 0.5 nm. Various assumed conductivities s sc (S m À1 ) are simulated with values varying from perfectly insulating vacuum (s sc ¼ 0) to highly conductive Ag (s sc is approximately on the order of 10 7 S m À1 ). It is interesting to note that the two extreme cases of a vacuum gap (top spectrum) and Ag gap (bottom spectrum) in Fig. 2a show a very similar spectral peak close to 2.5 eV. However, the modes are distinct, as shown in Fig. 2b (top and bottom). The vacuum gap case shows a bonding dipole plasmon (BDP) mode ("À" "+" "À" "+") whereas the Ag gap case presents a charge transfer plasmon (CTP) mode ("À" "+"). When the conductivity of the SAM layer is gradually reduced from Ag to vacuum (from bottom to top in Fig. 2a), the spectra show smooth transitions: (I) the CTP modes continue to shi to lower energy with reduced magnitude (to nally disappear for s sc < 10 5 S m À1 ), (II) the BDP modes only appear for less conductive gap media with s sc < 1.5 Â 10 6 S m À1 and their resonance energies are hardly sensitive to changes in the gap conductivity, in contrast to the CTP modes.
According to the observations above, we can divide all the Ag-SAM-Ag junctions with different gap conductivities into three regimes as shown in Fig. 2a: low conductivity regime (<10 5 S m À1 ), moderate conductivity regime (10 5 to 1.5 Â 10 6 S m À1 ), and high conductivity regime (>1.5 Â 10 6 S m À1 ). In low conductivity regime, only BDP is supported; whereas in high conductivity regime, only CTP is supported. However, for moderate conductivity regime, both CTP and BDP are supported. One example of 5 Â 10 5 S m À1 is demonstrated in Fig. 2b where the mode images of BDP and CTP are shown in central panels (ii) and (iii) respectively. The co-existence of the two modes suggests that these Ag-SAM-Ag junctions with moderate conductivities behave like leaky capacitors. This can be also proved by the calculated gap eld enhancements for CTP resonance (shown in Fig. 2c), where the gap elds are enhanced by a factor up to 200 for moderate conductivities (up to 1.5 Â 10 6 S m À1 ).
The eld enhancement is found to increase with the gap conductivity till a critical value of s sc ¼ 10 6 S m À1 ; then it drops as the conductivity further increases. To explain this, capacitance model could be used (the details of the model can be found later in "Measured data" section). Briey, there are two competing mechanisms to inuence the junction elds. First, the energy stored in the capacitor can be taken as the energy of a single plasmon. As the conductivity increases, the energy of the single CTP plasmon increases or shis to right as indicated in Fig. 2a, therefore the energy stored in the capacitor (and hence the average electric eld in the junction) will be increased accordingly. Second, if the stored energy in the capacitor keeps the same, a leakier capacitor or a more conductive junction leads to a lowered junction eld. Our calculation in Fig. 2c suggests that the rst mechanism dominates for junction conductivity smaller than 10 6 S m À1 ; while the second mechanism dominates for conductivity greater than 10 6 S m À1 . In fact, for further increased junction conductivities (>2 Â 10 6 S m À1 ) that fall into the high conductivity regime, the junction elds drop quickly to zero at the CTP resonance as shown in Fig. 2c. We cannot use the capacitance model anymore, because these Ag-SAM-Ag systems behave like conductors.
This sensitivity of the CTP resonance energy ħu CTP to s sc establishes the basis of measuring the THz conductivity of SAM layers using the CTP plasmon resonance. Variation of conductivities from 10 5 to 5 Â 10 6 S m À1 would induce a change of CTP energy from 0.6 to 2.5 eV in this case, indicating a rather broad tunability. The conductivity-dependence of the CTP energy can be understood by the following simple physical picture. First, we understand that a minimum number of electrons need to participate in a charge transfer plasmon oscillation for a CTP resonant peak to appear in the spectrum. As the gap region is less capable of transferring electrons, it takes a longer time to transfer this minimum number of electrons across the gap. Macroscopically, the CTP resonance slows down and shis to lower energy (or lower frequency).
While the current study focuses on the conductivitydependence of the CTP energy (by xing the geometry of the junction), most of the studies in the literature (e.g., CTP over metallic bridges 2,21 ) in fact concentrated on the dependence of the CTP energy on the geometry of the junction via A or d (while keeping the conductivity as a constant), where A is the crosssectional area of the junction, and d is the distance the electrons have to travel across the gap. As experimentally shown, 2,21 a decreased A shis the CTP resonance to lower energy with a lower peak magnitude. The decrease of A allows fewer electrons to be transported at a given time, thus it takes longer to transfer the minimum number of electrons to form the CTP resonance, shiing the mode to lower energy. Similarly, increased d implies a longer travel distance for the electrons; thus the CTP resonant oscillation shis to lower energy.
Combining the three factors: the energy of the CTP mode should be dependent on the conductance of the junction: G ¼ s 0 (A/d), which includes both factors of conductivity and geometry. It has also been conrmed by experiments that the decreased conductance of metal bridges corresponds to a CTP resonance at lower energy. 21 This implies that a minimum threshold conductance is required to trigger the CTP resonance at low energy. A more important implication is that a plasmonic system with larger A/d ratio has smaller threshold conductivity s sc for observing the CTP mode. For example, a spherical dimer requires threshold conductivity on the order of 10 6 S m À1 for the low-energy CTP. 15 This implies that a spherical-dimer plasmon system will be more suitable to measure molecules with conductivity values above 10 6 S m À1 . On the other hand, for the currently proposed cuboidal-dimer plasmonic system, due to the increased A/d factor, the threshold conductivity to observe the low-energy CTP is lowered to 10 5 S m À1 . If the junction conductivity is high, say 10 6 S m À1 , the cuboidal-dimer plasmonic system will generate a high-energy CTP mode, as shown in our Fig. 2. This phenomenon has also been theoretically observed by comparing a spherical gap system and a at gap system, 22 where no low-energy CTP was observed in at gaps (large A/d), although it is unclear what values of s sc were used in that study. Another reason of not observing the low-energy CTP mode in that study is probably the assumed frequencyindependent conductivity, as will be discussed below.
It should be emphasized here that the dependence of the CTP resonance energy ħu CTP on s sc (Fig. 2) could only be captured by using a frequency-dependent conductivity as shown in eqn (1). If assuming a frequency-independent conductivity, the CTP resonance would not shi to lower energy when s sc is varied, as in our previous simulation model 19 and the pioneering work by Nordlander and Aizpurua et al. 15 In those two theoretical works, the CTP mode and BDP mode are wellseparated; hence, the assumed frequency-independent conductivity could still show a smooth appearance or disappearance of the two modes at their own resonant energies. However, for the current cubic-dimer plasmonic system, the CTP mode (for the Ag gap case) and the BDP mode (for the vacuum gap case) occur at similar energies (>2.0 eV as shown in Fig. 2a). This coincidence of the two resonant energies is relevant to the morphology of the gaps, as it also occurs for other dimers with at gaps. 22 The assumed frequency-independent conductivity would induce a mixture of the two modes, complicating a correct interpretation of plasmon modes and the underlying physics. Here we therefore advance our model to include frequency-dependent conductivity. In fact, this approach implies that the CTP resonance is essentially the oscillation of the space charges in the ctitious junction material, just as was the case for the BDP resonance, which is the oscillation of the free electrons in the two plasmonic Ag resonators.

Plasmon map construction
The section above and Fig. 2 only show the results of one particular molecular length: d ¼ 0.5 nm. It provides a useful means to correlate s sc with ħu CTP . However, there are many types of molecules that can be used as SAMs, and their molecular lengths may be shorter or much longer than 0.5 nm. Conversely, some SAMs have the same molecular length, but may have different conductivities. Therefore, it is instructive to investigate how ħu CTP varies as a function of both d and s sc . Fig. 3 shows the results of our calculations. For dimers functionalized with less conductive SAMs (i.e., the lower part of Fig. 3), the dependence of the CTP plasmon energy on the molecular length is relatively weak. This implies that dimers functionalized with molecules of different lengths but similar conductivity would have similar plasmon energy. However, dimers functionalized with molecules having similar lengths but different conductivities would have very different CTP energies. The trend is opposite for dimers functionalized with more conductive SAMs (i.e., the upper part of Fig. 3). Here, the CTP plasmon energy is more sensitive to the molecular length variations. This implies that the measured conductivity will have relatively higher accuracy and sensitivity for the less conductive molecules (e.g., molecules with s sc < 5 Â 10 5 S m À1 ).

Charge injection mechanisms
Up to this point, for all the results shown above, the values of s sc (or n sc ) are all assumed without considering any charge injection mechanism. It simply represents the value of n sc that is required to support a CTP mode at certain resonant energy, no matter what the charge injection mechanism is. To take into account various mechanisms, charge injection models must be developed to compute a charge-injection dependent s inj (or n inj ). For example, for quantum-tunneling-enabled charge transfer plasmons, the charge injection modelling part will calculate a s tunneling inj (or n tunneling inj ); if the tunneling-supplied n tunneling inj meets the required n sc to support a CTP at a certain resonant energy, we can conclude that this CTP is a tCTP.
Here, we develop a quantum-mechanical model to compute the tunneling charge-injection conductivity s tunneling inj for the SAM with known molecular length d and Ag/SAM interface barrier height 4, by theoretically assuming a wide range of the extraction electric eld E at the Ag/SAM interface. By doing so, we could estimate the required extraction electric eld E to reach the space-charge limited tunneling regime for a particular type of SAM (i.e., dened by d and 4), similar to the approach used to model the Au-vacuum-Au systems. 19,31 The quantum-tunneling charge injection at the Ag/SAM interface is modeled by the tunneling barrier prole F(x) of the Ag-SAM-Ag via the following ve terms: 19,20,29,31,34,35 The rst term is the intrinsic barrier height 4 at the interface of Ag and SAM (i.e., the energy level alignment of the molecular frontier orbitals of the SAMs and the Fermi-levels of the Ag electrode). This term 4 is lower than the work function of the Ag electrode which serves as the barrier height in an Ag-vacuum-Ag system and it is tunable by changing SAMs. The second term is the image charge potential 36 described by: which is gap size d dependent and is important for the subnanometer gaps. The third term is a triangular external applied electric eld potential energy F v (x) that linearly varies with the extraction electric eld E at the Ag/SAM interface: The fourth term accounts for the self-induced space-charge eld 29,34,35 of the tunneling electrons F sc (x). Lastly, the h term is the quantum exchange-correlation eld F xc (x). It is clear that the concept of space charge have already been incorporated into the tunneling charge injection process in the literature, 29,31,34,35 indicating the applicability of the space-charge model in the modelling tCTP resonance.
The coupled Schrödinger and the Poisson equations are then solved iteratively using a WKB-type approximation technique 34,35 for the tunneling barrier prole [illustrated by the red solid line in Fig. 4a] and the tunneling electron density n tunneling inj . This n tunneling inj can then be used to obtain the quantum tunneling charge-injection conductivity s tunneling inj from eqn (2). This model automatically includes direct tunneling, Fowler-Nordheim tunneling, and over-the-barrier tunneling mechanisms. In addition, since only dimeric structures with a single layer of well-aligned SAMs are assumed in this theoretical study, there is no difference in modelling the through-space tunneling and through-bond tunneling mechanisms, unlike the cases for modelling the disordered or double layered SAMs. 20 Fig . 4b-d shows the results of the quantum-calculations for three selected SAM molecular lengths d ¼ 0.5, 0.7, and 1.0 nm, and a range of practical intrinsic barrier heights (2.0-4.0 eV). For comparison, the work function of Ag is 4.26 eV. The extraction electric eld is theoretically varied from 10 7 to 10 10 V m À1 . Take Fig. 4b (for all kinds of SAMs with the same molecular length d ¼ 0.5 nm) as an example, we observe that for a large interface barrier height 4 ¼ 4 eV, the tunneling probability is low, we need the extraction eld on the order of 10 9 V m À1 to enter into the space-charge limited tunneling regime. This observation agrees well to our previous studies on Au-vacuum-Au system. 19 Such a high eld makes it difficult to experimentally access. However, in this work, when the Ag-SAM-Ag system is proposed, the interface barrier height 4 can be tuned to lower values like 2, 3 eV or even lower, and the required extraction eld is greatly reduced to below 10 8 V m À1 as suggested by Fig. 4b. To compare the quantum-calculated s tunneling inj with the values of detectable s sc (i.e., 10 5 to 10 6 S m À1 ) from our plasmon parameter map in Fig. 3, we draw a grey block in Fig. 4b, and all the data points falling in the grey block indicate the condition of the tCTP. We nd that for all the 0.5 nm SAMs with barrier height up to 3.0 eV, the quantum tunneling CTP is feasible if the extraction electric eld can be on the order of 10 8 V m À1 .
Things are different for longer SAMs, for example, d ¼ 0.7 nm as shown in Fig. 4c. As the tunneling distance is longer now, the tunneling probability decreases, leading to a strongly reduced overall tunneling conductivity. Even for a barrier height as low as 2.0 eV, the tunneling conductivity is estimated to be about 10 5 S m À1 only when the extraction eld can be above 5 Â 10 8 V m À1 as shown in Fig. 4b. For even longer monolayer SAMs such as d ¼ 1.0 nm as shown in Fig. 4d, following the same reasoning, the maximum tunneling conductivity for a range of practical intrinsic barrier heights can be on the order of 10 5 S m À1 only if the extraction eld can reach to the level of a few 10 9 V m À1 . These results imply that for smaller extraction elds, or longer SAM molecules, or larger interface barrier heights, the tunneling electrons that participate in the plasmonic oscillation is not enough. Now, we refer back to our plasmon parameter map in Fig. 3. A grey triangle was drawn to indicate the quantum-tunnelinginduced charge transfer plasmon regime assuming an extraction eld of 10 9 V m À1 . The regime of quantum-tunneling was found using the same approach as that used in Fig. 4, except that possible barrier heights are now allowed from 0.1 eV to innite. It is observed that shorter SAMs (e.g., $0.4 nm) support a wider frequency range of tCTPs (0.6-2 eV), whereas longer SAMs (e.g., $1.0 nm) only support low-energy tCTPs (0.6 eV).

Measured data
We use here previously reported experimental data 20   The measured ħu CTP and d (only monolayer SAM junctions are selected) are plotted in red (EDT) and blue (BDT) in Fig. 3, respectively. From Fig. 3, we can readily infer the conductivity of the SAM: s EDT z 1.0 Â 10 5 S m À1 , giving an equivalent conductance G EDT ¼ s EDT (A/d EDT ) ¼ 3221G 0 for the 0.55 nmthick EDT layer; whereas s BDT z 2.5 Â 10 5 S m À1 , with equivalent conductance G BDT ¼ s BDT (A/d BDT ) ¼ 6611G 0 for the 0.67 nm-thick BDT layer. Here G 0 ¼ 2e 2 /h ¼ 7.727 Â 10 À5 S is the quantum conductance.
It is now instructive to estimate the electric eld in the gap in our EELS experiments to check whether the conditions of tunneling can be met. To the best of our knowledge, there is no direct method to measure the electric eld associated with the incident electron beam. Instead, we employed a widely used LC circuit model. 2,38 We noticed from our calculations in Fig. 2 that our measured EDT (s EDT z 1.0 Â 10 5 S m À1 ) and BDT (s BDT z 2.5 Â 10 5 S m À1 ) belong to the moderate conductivity regime. For such junctions, both CTP and BDP are supported (Fig. 2a) and the gap eld is enhanced at CTP resonances (Fig. 2c). Therefore the Ag-EDT-Ag and Ag-BDT-Ag junctions can be modeled as capacitors. 2, 38 We could estimate the average electric eld in the gap based on the energy of a single plasmon Here A is the area of overlap of the two plates, 3 r is the relative static permittivity of the material between the plates, 3 0 is the vacuum permittivity, and d is the separation between the plates.
Therefore the average eld in the gap is jEj ¼ is computed to be on the order of 10 8 V m À1 at the CTP resonance. For comparison, the values of plasmonic elds determined in molecular junctions by current rectication at optical frequencies are about 10 9 V m À1 . 39 Our estimated eld on the order of 10 8 V m À1 is not able to reach the Fowler-Nordheim tunneling regime for an Auvacuum-Au dimer in our previously study 19 due to the large intrinsic barrier height dened by the work function of Au (where it is found that 10 10 V m À1 is required 19 ). However, we have shown in our Fig. 4 that the average gap eld on the order of 10 8 V m À1 is enough to induce Fowler-Nordheim tunneling for some Ag-SAM-Ag systems studied in this work, because now the intrinsic barrier is greatly lowered.
Let us take a close look at Fig. 4b for SAM of 0.5 nm thickness again, for the practical range of intrinsic barrier heights (1-3 eV), and the possible range of extraction electric eld (10 7 to 5 Â 10 8 V m À1 ) in our EELS experiments, the calculated tunneling induced conductivity is indeed on the order of 10 5 S m À1 . This is a clear indication that tCTP is one of the possible ways to explain what we have observed experimentally. 20 From the previously reported surface coverage of 8.0 Â 10 À10 mol cm À2 for these molecules on silver surfaces, 40,41 we estimate that 16 000 molecules are present in the 37 Â 37 nm 2 junction of each silver dimer. Therefore, the conductance per EDT or BDT molecule is roughly 0.2G 0 and 0.4G 0 at the CTP resonance (140 THz and 245 THz), respectively, assuming that every molecule participates in charge transport. It is found that these THz conductance values per molecule happen to be of the same order as those previous experimentally obtained values in break junction measurement at dc and low-frequencies. 42,43 Despite the similar conductivity values, the differences between our experiments and the referenced experiments 42,43 should be taken note: (I) ours is molecular monolayer junction whereas the referenced experiments are single-molecule junctions. They are clearly different transport junctions. 26 . (II) The shapes of the electrodes in both test beds are vastly different. (III) The measurement frequencies are different. Very recent work on an atomic-scale plasmonic switch 44 also suggested that a single atom can open a conductive tunneling channel with a conductance in the order of 0.13G 0 .
It is interesting to note that the two types of molecules we have chosen for our proof-of-concept experiment (EDT and BDT) both fall within the tunneling CTP regime of Fig. 3. They have quite similar molecular lengths (the difference is only 0.1 nm), but rather different CTP plasmon energy (the difference is about 0.4 eV). This is a piece of clear experimental evidence that CTP plasmon energy not only depends on junction geometry, but also on junction conductivity. We propose further experimental investigations for different SAMs and other gap dielectrics to further explore the tunneling and other charge transport regimes.

Conclusions
In conclusion, we have established a framework for measuring the conductance of single molecules at near-infrared, THz frequencies by utilizing the charge transfer plasmon resonances in hybrid Ag-molecule-Ag dimer systems. This measurement is based on the unique feature of charge transfer plasmons: their energies are sensitive to the electrical properties of the molecular junction. A wide range of molecules was theoretically explored to construct a charge transfer plasmon parameter map using a generalized space-charge corrected electromagnetic model.
The space-charge corrected electromagnetic model developed in this work treats the charge injection and charge transport separately. A Drude expression (assuming damping frequency on the order of driving frequency) is used to model the dynamics of the space-charge limited charge transport, whereas the charge injection modelling takes care of the number density of the transporting charges. It should be noted that this space-charge corrected electromagnetic model is only appropriate to model the space-charge limited charge transport problem.
In particular, the space-charge limited quantum-tunneling charge transport problem is attempted. The space-charge limited regime (i.e., small gap spacing, low interface barrier height, and large gap eld) is found, and the molecules that are able to support space-charge limited quantum-tunneling charge transfer plasmons under certain extraction electric elds are identied.
To demonstrate the feasibility of the proposed framework, previously reported experimental results from two types of molecules were analyzed: saturated, aliphatic 1,2-ethanedithiolate (EDT), and aromatic 1,4-benzenedithiolate (BDT). The measured conductance per EDT or BDT molecule is respectively 0.2G 0 (at 140 THz) and 0.4G 0 (at 245 THz). Both molecules likely support the space-charge limited tunneling charge transfer plasmons, based on the comparable magnitude between our theoretically-predicted required junction eld (for given molecular length and barrier height) and the estimated junction eld in our EELS measurement.
Our study suggests a direction for future experimental investigations, e.g., using different molecules that support other types of charge transport mechanisms. This would further rene the map shown in Fig. 3, providing a complete and comprehensive charge transport picture in the THz frequency domain.