The analysis of charge transport mechanism in molecular junctions based on current-voltage characteristics

Chemical Physics 528 (2020) 110514

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The analysis of charge transport mechanism in molecular junctions based on current-voltage characteristics

T

Xianneng Song, Bin Han, Xi Yu , Wenping Hu ⁎

Tianjin Key Laboratory of Molecular Optoelectronic Science, School of Science, Tianjin University, Tianjin 300072, China

ARTICLE INFO

ABSTRACT

Keywords: Molecular junctions Charge transport mechanism Hopping transport Tunneling transport

We report here a theoretical model study of the current-voltage (I-V) characteristics of tunneling and hopping transport in molecular junctions. We found that the I-Vs of the two types of transport have very different shape characteristics. The I-V in tunneling transport is in near-linear relation at low bias, while is exponential for the hopping transport. The current in hopping transport thus can span a much wider range at semi-logarithm scale than tunneling transport, which can be used as an intuitive guide in analyzing the transport mechanism. The idea is supported by series of experimental studies reported before by Frisbie et al. where a clear I-V characteristic change can be seen when transport mechanism changed from tunneling to hopping. The two theoretical models were further used to fit the reported experimental I-V data, and we found that the transport mechanism obtained by the model is in consistence with the experimental conclusion. Moreover, our method also revealed the coexistence of two transports during the tunneling-to-hopping transition, and the contribution of the two channels to the current is bias dependent. Our research thus provides a new powerful theoretical method for the study of charge transport mechanism in molecular junctions.

1. Introduction The study of the charge transport mechanism across molecular junctions is of great importance, not only for the development of the charge transport theory at microscale, but also for designing novel molecular-based electronic devices [1–3]. There have been extensive studies on the charge transport mechanism in molecular junctions, and two main mechanisms are well acknowledged [4,5]. One is tunneling, where electrons tunnel coherently through energy barrier of the molecule [4]. The other is the hopping mechanism [5]. There, the charge transferred from electrode to molecule, interacts with the molecular vibration and the environment, gets relaxed and trapped temporarily on the molecule, and then hops to the next site initiated by the thermal excitation until finally to the other electrode [5,6] (Fig. 1). Therefore hopping is a successive multi-step and thermal activated process. The mechanism of the charge transport depends on the interaction of the charge with molecule and the dielectric environment. A longer molecule, a weaker coupling between molecule and electrode, and a lower energy barrier of charge transport can elongate the traversal time and so the chance for the charge to interact with molecule and environment, which in turn facilitates the hopping transport. To distinguish the two mechanisms experimentally, conductance measurement against experimental variables, like temperature and ⁎

molecular length are generally required. The tunneling transport is temperature independent and its conductance decreases exponentially with molecular length [7–11]. The hopping transport, on the other hand, is Arrhenius-type temperature dependent and decreases linearly with molecular length [7–12]. The current-voltage (I-V) characteristics also enveloped the key information of the charge transport and have been used in analyzing the transport mechanism. For example, in very early age of molecular electronics study, Reed et al. analyzed the tunneling I-V in nanopore junctions using Simmons model [13], which simplified the molecule as a rectangular barrier without considering the chemical details of the molecule, and became a popular analyzing method due to its simplicity since then [13]. The conductance-voltage plot, also well recognized as the tunneling spectroscopy [14,15], was utilized widely especially in scanning tunneling microscopy to tell the electronic states in the junction. Frisbie et al. [7,10,16] found that when re-plotting the I-V into ln(I2/V) vs. 1/V, i.e. the the Fowler–Nordheim (F-N) plot, an inflection can be observed. The voltage at this inflection point, called the transition voltage, is related to the energy barrier of the tunneling transport [17]. Nowadays, Landauer formula with single electronic state has become a well acknowledged method in analyzing tunneling IV for molecular junctions [4]. For example, Frisbie et al. [18] fitted the experimental I-Vs of oligophenylene dithiol (OPD) molecules connected

Corresponding author. E-mail address: [email protected] (X. Yu).

https://doi.org/10.1016/j.chemphys.2019.110514 Received 9 July 2019; Received in revised form 27 August 2019; Accepted 2 September 2019 Available online 06 September 2019 0301-0104/ © 2019 Elsevier B.V. All rights reserved.

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Fig. 1. Tunneling and hopping charge transport in molecular junctions.

Fig. 2. Single-level model diagram (a) tunneling (b) hopping.

to different metal electrodes using single-level tunneling model, and the results showed that the fitting parameters were consistent with the experimental values. Erbe et al. [19]. explored the role of anchoring groups in scattering rate by fitting experimental I-Vs of tolane molecules linked by different anchor groups in conductive probe atomic force microscope (CP-AFM) devices. In contrast, the I-V analysis for hopping transport is not as well utilized as tunneling though the hopping model which has been studied by many researchers. For example, Ulstrap and Kuznetsov developed the hopping transport model based on single redox level, and the model is used mainly in explaining the current in response to gating voltage in EC-STM measurement [6,20]. Nitzan [5] also derived the analytical solution of I-V in single-level model to account for the hysteresis and negative differential resistance phenomena. Kotlyar and Porath et al. [21] have ever analyzed the hopping transport I-V across DNA utilizing the hopping transport model. However, more detailed comparative analyses of I-V characteristics in the tunneling and hopping transport have never been reported. In this work, we have studied the I-V characteristics of the tunneling and hopping transport based on the simplified single-level model. We found that the two transports possess very different I-V characteristics. The tunneling I-V is near-linear at low bias, while the hopping current is exponentially dependent on voltage. The I-Vs of the two transport are more distinct in semi-logarithm plot, where, the charge transport mechanism in the molecular junctions can be intuitively distinguished by the I-V shape. We further used two models to fit the experimental results of tunneling and hopping transport. The theoretical models are well reproduced the experimentally measured I-V, and the predicted mechanism is consistent with the experimental conclusion. More interestingly, we found that the two transport processes can coexist during the transition from tunneling to hopping, and the contribution of the two mechanisms to the current is bias dependent. Our study here provides a new perspective for the study of charge transport mechanism in microscopic systems.

Landauer-Büttiker formula [4,22]:

I=

2q h

dE [fL (E )

fR (E )] Tr (E )

(1)

wherein, fL and fR are the Fermi distribution of the left and right electrodes respectively,

fL, R (E) =

exp

(

1 E

µL, R k BT

)+1

(2)

µL, R is the chemical potential of the left (right) electrode. In the single-level model, the transmission function Tr is written as follows [22] L R

Tr = D (E )2

(3)

where = L + R , L and R are the coupling strengths of the molecules with the left and right electrodes, respectively. D (E ) is the broadened state density of the energy level ε due to the coupling, and its distribution is a Lorentz function centered at the energy level ε.

D (E ) =

/2 )2 + ( /2)2

(E

(4)

So, in the single-level model, the expression of the current in the molecular junctions can be written as follows.

I=

2q

L R

dED (E ) L

+

R

[fL (E )

fR (E )]

(5)

In the situation where coupling is relatively larger than the electron Fermi broadening of the electrode, so that the effect of temperature on the Fermi distribution can be ignored, Fermi function can be approximated into step function, and above integration produce an temperature independent analytical solution [4]

I= 2. Theoretical methods

2q h

R L R

+

arctan L

+ eV /2 + L

R

arctan R

eV /2 + L

(6)

= µ represents the energy offset (energy barrier) of the where junctions at zero bias. We can see that the variables affecting the I-V characteristics in the tunneling mechanism are mainly the energy offset and the coupling strength . The effect of the energy offset on the current characteristics in the molecular junctions is due to the relative position between the Fermi level and the molecular level that determines the bias required to reach resonant tunneling. The coupling strength mainly affects the broadening of the energy level, which affects the distribution of electron transmission coefficient.

The tunneling is generally described by Landauer formula [4] and the hopping transport is usually depicted by rate equation combing the Marcus theory to account for the elementary charge transfer step [5]. We consider the simplest situation here only, i.e. the single level model, where for tunneling there is only one electronic state and for hopping there is only one redox active state in the junctions, see Fig. 2. We will see that these simple models are enough to present the main idea on the difference between tunneling and hopping without introducing too much complexity due to multi-states.

2.2. Hopping

2.1. Tunneling

In the hopping transport, charge transfer from one electrode to the redox active state and then to the other electrode so that the redox active state switch between reduction and oxidation state continuously. We use A and B to represent the oxidation and reduction state

A general method to take care of tunneling transport is to relate the transport properties to the transmission and reflection probabilities. The current is related to the transmission probability Tr by the 2

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respectively. RAB is the rate at which electrons flow from the electrode to the molecule (so the state change from oxidation into reduction state), and RBA is the rate of the reverse process (electron flows from the molecule into the electrode). The charge transfer rate at the interface between the molecules and the different electrodes are given by Marcus theory [5] K RAB = K RBA =

+

+

dEkK (E ) fK (E ) F (E dEkK (E )[1

)

fK (E )] F (

value found in experiment [23,24]. In addition, to make the two comparable in the same order of magnitude, we multiply the hopping current by a certain constant. The I-V relations in linear and semi-log scale are shown in Fig. 3. It can be seen that the I-V curves of tunneling and hopping transport exhibit different characteristics as follows. Firstly, in linear coordinates, the tunneling I-V is linear at low bias and nonlinearity at high bias. The I-V of hopping transport looks asymptotic, the current is low at low bias, and increases rapidly after certain threshold. It is not easy to tell the difference between the two, because in the hopping transport the characteristics of the current at low bias is hard to recognize. However, the difference of the two is quite obvious in semi logarithmic scale as shown in Fig. 3b. We can see that I-V in hopping transport is in near exponential relation and the current spans a significantly larger range of magnitude than tunneling. The near exponential dependence of the current on bias in hopping transport actually originates from the Marcus theory, where the F-C factor (Eq. (10)) is related to bias in a logarithm manner. We can also recall that the Tafel plot in the electrochemistry, where the logarithm of current is plotted against over-potential, also produced similar shape [25,26]. This should not be surprising since in hopping transport the I-V relations is controlled by the cross-interface charge transport which has the same nature as that in electrochemistry. In contrast, the tunneling current is quite linear in low bias region, while takes on nonlinearity at high bias. Extensive study on the character on the I-V in tunneling transport has been done by many researchers [27,28]. The linear relation at low bias is the result of the transmission at the foot part, i.e. far away from the resonance center, of the broadened density of state (Eq. (4)). When bias increases and so that the electronic level of the electrode gets closer to resonance, the current will increase more rapidly in a nonlinear fashion. In logarithm plot, the tunneling I-V presents as a “ϒ” shape. So, it is much easier to distinguish between tunneling and hopping in log(I)-V plot. We further studied the effects of different parameters on the I-V characteristics under tunneling and hopping mechanism. The main parameters we choose are barrier, reorganization energy, transition rate, and temperature and all the I-V are shown in semi-log plot. As shown in Fig. 4(a, b, c, d), the default parameters are set to the energy difference E = 0.3eV , the reorganization energy = 0.5eV , and the coupling term kL = kR = 105s 1. For the tunneling transport, we selected the coupling strength, and barrier height. As shown in Fig. 4(e, f), the = 0.5eV and coupling strength L = R = 0.1eV . barrier was set to be As can be seen from Fig. 4(a, b, c, d), each parameter has a different impact on the I-V characteristics of hopping transport. First, for ΔE, as described above, ΔE is the energy change of the electrode-molecule system before and after the charge transfer, which can be roughly related to the energy difference between the electrode and the redox active state, E = E . Reorganization energy reflected the degree of molecular configuration change before and after the reaction. The larger the reorganization energy, the bigger the change of the configuration, and the more difficult the charge transfer is. As can be seen from Fig. 4(a, b), as the energy difference ΔE and the reorganization energy increase, the current decreases and the bias required for the saturation current also increases, indicating that both of them increase the difficulty of charge transfer, and thus a larger bias driving force become necessary [29]. However, the change in I-V shape character for ΔE and are still different. In terms of the effect of temperature on the hopping transport, the current drops significantly as the temperature decreases, which is consistent with the experimental observation. In addition, it can be seen that the dependence of the current on temperature is higher at low bias than the high bias, indicating a decreasing activation energy with bias. It is well-known from Marcus theory, the charge transport activation energy will decrease with the energy change ΔE (and will increase again if ΔE keep increase, which is the Marcus inverted region [30,31]). In the hopping transport discussed here, as bias increases (in both negative or positive polarity), ΔE increases and the corresponding activation energy decreases. This finding

(7)

E)

(8)

where f is the Fermi distribution of the electron in the electrode, ε is the molecular orbital energy level, and K = LorR represent the rate at left or right electrode interface. k is the electron coupling term that determines the transition rate between electronic states of the electrode K and molecule. It is determined by the coupling strength VAB between the electronic states in the molecule and that of the electrode and the electronic density of state K (E ) in the metal electrode by

kK (E ) =

2

K 2 |VAB |

K

(E )

(9)

F is the thermally averaged Franck–Condon term that accounts for the nuclear and environment relaxation. The expression is F ( E) =

1 2

kB T

exp

( E )2 4 kB T

(10)

where λ is the reorganization energy, kB is Boltzmann's constant. E is the energy change of the electrode-molecule system before and after the redox reaction, which can be roughly related to the energy difference between the electrode and the redox active state, E = E . The Fermi distribution function of the electrons in the electrode of the system is

fK (E ) =

exp

(

1 E

µ+e K kB T

)+1

(11)

The whole charge flow rate in this series hopping process follows a simple rate equation R L RL R R RAB RBA I = AB BA q RAB + RBA

(12)

So, the I-V relation can be obtained when incorporating Eqs. (7) and (8) into Eq. (12). The difficult part is the integration in Eqs. (7) and (8). Nitzan et al. have derived analytical expression of the charge transfer rate [5], and our study here will be based on their result. In the hopping mechanism, the influence of each variable on the I-V characteristics is mainly the effect on the activation energy required for the reaction, which in turn affects the total electron transfer rate. The energy difference between the electronic state and the molecular orbital energy level on the electrode ΔE and reorganization energy λ, affect the characteristics of the I-V curve by changing the activation energy, and the thermal energy provides the activation energy for the reaction. The influence of coupling strength on the system is mainly manifested by k (E). 3. Results and discussion 3.1. I-V characteristics We first comparatively studied the I-V characteristic of the two types of transport by plotting the I-V using Eqs. (5) and (12) respectively, as shown in Fig. 3. In the tunneling transport, the parameters are = 0.5eV , coupling strength L = R = 0.1eV ; In set to energy barrier the hopping mechanism, the parameter is set to the energy difference E = 0.3eV , the reorganization energy λ = 0.5 eV, and kL = kR = 105s 1. These parameters are set according to the common 3

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X. Song, et al.

Fig. 3. I-V plots of tunneling and hopping transport under linear scale (a) and logarithmic scale (b).

is consistent with the experimental observation by Selzer et al. [32] in single molecule junctions. Compared with the other three parameters, the coupling term k reflects the transition rate of electrons between two electronic states with the same energy, and it only change the magnitude of current without impact on the shape character of I-V. In the tunneling transport, both the barrier and the coupling strength have a significant effect on the shape of the I-V curve, as shown in Fig. 4(e, f). The effect of the barrier on the current in the molecular junctions is due to the fact that as the barrier increases, the voltage required to reach resonant tunneling increases gradually. So that the conductance decreases with tunneling barrier and the saturation current at resonance is harder to reach. Coupling strength mainly affects the magnitude of the current only. However, in the case of weak coupling, the shape of the I-V changed due to the smaller broadening of the molecular energy level and the narrower distribution of the transmission spectrum, which affects the transport of electrons in the molecular junctions (Eq. (3)). Through the above analysis, it can be seen that although the shape characters of tunneling and hopping transport change with their intrinsic physical parameters, the main features of the log(I)-V remain and the

distinction between the two are still quite obvious. The magnitude of the current in hopping transport generally spans a much wider range than tunneling transport and the log(I)-V is in a “V” shape in hopping transport while it is “ϒ” shape in tunneling. In the situation of smaller energy barrier and higher temperature, although the current rang in the hopping transport will be smaller, for example top black color curve in Fig. 4a, which looks close to the range of tunneling transport in Fig. 4e, their shapes are still distinct from each other when plotting in the same range of magnitude (Fig. S1 in Supplementary information). This distinction can be used as an intuitive guide for quick discrimination between the two mechanisms, and the idea can be proved by the experimental log(I)V plot by Frisbie et al. [7,8,10] where the transport mechanism changed from tunneling to hopping in the junctions was observed following increase of the molecule length. In Fig. 5, we have reproduced the I-Vs of ONI (oligonaphthalenefluoreneimine) molecules of different lengths from Ref. [7]. It can be seen that I-Vs of ONI2 with tunneling transport spanned only 2 order of magnitude, while for ONI10 it is 4 order of magnitude. And we can see when ONI molecules become longer, the current range also increased gradually. Similar trend can also be observed in Fig. 3 of Ref. [8] and Fig. 4 of Ref. [10].

Fig. 4. The effect of different parameters on the I-V characteristics of hopping and tunneling transport. (a, b, c, d) are the I-Vs of hopping transport with different barriers, reorganization energy, temperature and coupling strength, respectively. (e, f) are the I-Vs of the tunneling transport with different barrier and coupling strength, respectively. 4

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Therefore, the fitting shown here is more a proof a concept rather than quantitative. The fitting results are shown partially in Fig. 6, and the rest of the results and parameters obtained by fitting are given in Supporting information Fig. S2 and Table S1. Here in the molecular junctions with probe of CP-AFM as the top electrode, the number of molecules involved in charge transfer are claimed to be about 80 [37]. So, we have multiplied the current function by 80 in the fitting. We can see that the tunneling model can fit ONI2 quite well, while ONI7 and ONI10 can be fitted by the hopping model, both of which are consistent with the experimental conclusions. Meanwhile, the hopping transport cannot fit the ONI2 and neither tunneling model can account for ONI7 and ONI10, especially at low bias region. The hopping model underestimated the current than experimental result for ONI2, and the tunneling model overestimated the current than experimental data for ONI7 and ONI 10. This phenomenum is quite typical in our fitting on many other experimental results. It clearly indicates the fact that the ranges of current that tunneling and hopping can span is so different, no matter how much the parameters changes, the two types of transport cannot crossover with each other. Another prove of this idea is the fitting result of ONI3, which is on the transport mechanism transition point according to the experimental study. As shown in Fig. 6(b, f), we can see neither the tunneling nor the hopping can fit the results. It implies that the tunneling and hopping might coexist in this junction, so both of the models failed. The coexistence of tunneling and hopping has been proposed by many researchers based on temperature dependent measurement, where temperature dependence of the conductance changed into independence at certain low temperature. So, the tunneling was believed to coexist with hopping, and it won’t dominate until hopping become very weak at low temperature. The coexistence of the two transport channels was also evidenced by the stochastic currents found in the junctions, which was found to be due to the interaction of the two channels [23,24,38,39]. Our study here implies that the coexistence of the two may be able to tell from the I-V character. In our previous work in the study of ferrocene derivative junctions, we ever used a model based on the linear combination of the two transport models and we found it works for our ferrocene derivative junctions data [40]. Recently, Valianti et al. [41] also used this method in their model study of the charge transport in azurin based junctions to account for the coexistence of hopping and tunneling transport. Here we tried this mixed model again. In this model, the currents from both tunneling It and hopping Ih transport coexist in a simple linear manner

Fig. 5. The I-V plots of different lengths ONI molecules in semi-logarithm scale (reprinted from Ref. 7).

3.2. Fitting the experimental results To further verify our idea, we have attempted to fit the experimental I-Vs shown in Fig. 5 using both tunneling and hopping model, i.e. Eqs. (5) and (12) respectively. We are aware of that for the ONI molecules, there might be more than one state in both tunneling and hopping transport. However, our single level models can still work as proof of concept tools considering the facts that the main distinction between tunneling and hopping is the same in the multiple states relative to single state. In tunneling transport, the linear I-V in low bias won’t change no matter it is one state or multistate as shown in many experiments and model study [17,33,34]. The single state tunneling model has been used in the case of possible multistate and proved to work decently. For hopping transport, the exponential dependence of the current on bias originates from the elementary Marcus type charge transport, which won’t change in multi sites hopping process. We have roughly reproduced the I-Vs based on consecutive multi-state hopping model following the idea from Ref. [21,35,36] and found exponential dependent I-V as well (see Supporting information). A more detailed analysis of the multi-state model will be the next step of our study.

Fig. 6. The I-V fitting of ONI molecule junction using tunneling and hopping model. 5

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X. Song, et al.

Fig. 7. Mixed model fitting results (ONI3) and components.

I = It ·nt + Ih· nh.

(13)

We are still looking for more generalized models that may help to explain more experimental data. For example, we have expended our study to multi-state hopping model which might help to explain the experimental results where the present single state model fails (see Supplementary information for details). The study is in process and will be reported in a due course.

where nt and nh are the number of the transport channels, and It and Ih are from Eqs. (5) and (12). It is very hard to find the number of channels involved in the transport from this model since it affects only the magnitude of the current in a linear way. We have discussed in previous section, the coupling term in tunneling ( ) and hopping (k ) mainly determine the magnitude of the current. So, it is impossible to find the number of transport channels from the I-V fitting. Therefore, we totally gave up nt and nh , and simply express the current as

I = It + Ih

Acknowledgement The work was supported by the National Natural Science Foundation of China (21773169, 51633006, 51733004).

(14)

The effective number of channels all went into and k , while, on the other hand, we can overlay I , It , and Ih in the same figure to show the respective contribution of the tunneling and hopping from their magnitudes. The fitting results of ONI3 are shown in Fig. 7a, and It and Ih are shown as well in panel b. We can see that the mixed model fitted the experimental data much better than single transport model. Panel b indicated that the current magnitude of the two transports become reconciled in the way that the tunneling current dominate at low bias while hopping current dominate at high bias. This result is also in line with the experimental observation that the low bias conductance of ONI3 is temperature independent tunneling transport. However, we can see indeed the high bias current fitted hopping transport better.

Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.chemphys.2019.110514. References [1] N. Xin, J. Guan, C. Zhou, X. Chen, C. Gu, Y. Li, M.A. Ratner, A. Nitzan, J.F. Stoddart, X. Guo, Concepts in the design and engineering of single-molecule electronic devices, Nat. Rev. Phys. 1 (2019) 211–230. [2] D. Xiang, X.L. Wang, C.C. Jia, T. Lee, X.F. Guo, Molecular-scale electronics: from concept to function, Chem. Rev. 116 (2016) 4318–4440. [3] M. Galperin, M.A. Ratner, A. Nitzan, Molecular transport junctions: vibrational effects, J. Phys.: Condens. Matter 19 (2007). [4] A.R. Garrigues, L. Yuan, L. Wang, E.R. Mucciolo, D. Thompon, E. Del Barco, C.A. Nijhuis, A single-level tunnel model to account for electrical transport through single molecule- and self-assembled monolayer-based junctions, Sci. Rep. 6 (2016) 26517. [5] A. Migliore, A. Nitzan, Nonlinear charge transport in redox molecular junctions: a Marcus perspective, ACS Nano 5 (2011) 6669–6685. [6] A.M. Kuznetsov, J. Ulstrup, Mechanisms of in situ scanning tunnelling microscopy of organized redox molecular assemblies, J. Phys. Chem. A 104 (2000) 11531–11540. [7] S.H. Choi, C. Risko, M.C.R. Delgado, B. Kim, J.L. Bredas, C.D. Frisbie, Transition from tunneling to hopping transport in long, conjugated oligo-imine wires connected to metals, J. Am. Chem. Soc. 132 (2010) 4358–4368. [8] L. Liang, F.C. Daniel, Length-dependent conductance of conjugated molecular wires synthesized by stepwise “click” chemistry, J. Am. Chem. Soc. 132 (2010) 8854. [9] L. Qi, L. Ke, Z. Hongming, D. Zhibo, W. Xianhong, W. Fosong, From tunneling to hopping: a comprehensive investigation of charge transport mechanism in molecular junctions based on oligo(p-phenylene ethynylene)s, ACS Nano 3 (2009) 3861–3868. [10] H.C. Seong, K. Bongsoo, F.C. Daniel, Electrical resistance of long conjugated molecular wires, Science 320 (2008) 1482–1486. [11] W.B. Davis, W.A. Svec, M.A. Ratner, M.R. Wasielewski, Molecular-wire behaviour in p-phenylenevinylene oligomers, Nature 396 (1998) 60–63. [12] D. Segal, A. Nitzan, W.B. Davis, M.R. Wasielewski, M.A. Ratner, Electron transfer rates in bridged molecular systems 2. A steady-state analysis of coherent tunneling and thermal transitions, J. Phys. Chem. B 104 (2000) 3817–3829. [13] T. Lee, W. Wang, M.A. Reed, Mechanism of electron conduction in self-assembled alkanethiol monolayer devices, Ann. N.Y. Acad. Sci. 1006 (2003) 21–35. [14] J.G. Simmons, Low-voltage current-voltage relationship of tunnel junctions, J. Appl. Phys. 34 (1963) 238–239.

4. Conclusion We studied the characteristics of tunneling and hopping transport in molecular junctions using theoretical models. It was found that the current in hopping transport is exponentially dependent on voltage while it is linear and polynomial dependence for tunneling. Thus, the hopping current can span a much wide range than that of tunneling. Based on these finding, we proposed a method to distinguish the tunneling and hopping mechanism by the shape of the I-V plots in semilogarithm scale. We also theoretically reproduced the experimental I-V plots using corresponding tunneling and hopping model, which verified the feasibility of our proposed method. More interestingly, we found the I-V experimental data from junctions that is in transition from tunneling to hopping can be explained using a simple mixed model of the two transports, which revealed a coexistence of tunneling and hopping transport, and their relative contribution was bias dependent. Our strategy can be of great help in transport mechanism study in the molecular junctions and molecular device. Meanwhile, it deserves to mention that we still have problem in fitting many of the experimental I-V results of hopping transport reported by Frisbie and other researchers, which might be due to the limitation of the model we used. 6

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[30]

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[38]

[39]

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7

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