Activation–relaxation processes and related effects in quantum conductance of molecular junctions

We used MCBJs, fabricated at the Helmholtz Nanoelectronic Facility of Forschungszentrum Jülich GmbH, Germany, as experimental samples. The samples were fabricated from rectangular, flexible substrates made of stainless steel. The lateral dimensions were 12 × 55 mm and the thickness was 0.15 mm. Several layers of polyimide PI2561 were subsequently spin-coated onto the substrates to create an insulating layer. The samples were then covered with a stack of e-beam-sensitive polymers (polymethyl methacrylate PMMA 649.04 and PMMA 679.04). After e-beam patterning, metal deposition was used to create large contact pads and nanoconstrictions simultaneously. After the lift-off process and reactive ion etching, the samples were ready for measurements. More detailed information on the fabrication technology can be found in [33–35].

After fabrication, we introduced 10⁻⁶ M 1,4-benzenediamine solution in ethanol to the surface of our sample by drop casting. We used an immobilization time of several minutes, during which amino groups of the investigated molecule bond with the gold via covalent bonding. The excess of molecules that did not bind to the gold was later removed by rinsing in ethanol. To measure the electrical properties, we first bent our sample until the gold nanoconstriction on the top broke and a single BDA molecule bridged two electrodes (lock-in state). We obtained a resistance value of 1.2 MΩ for the system at room temperature, which agrees well with the literature data [25]. All measurements were performed in high vacuum (∼10⁻⁶ mbar) in a shielded environment. After establishing the molecular junction, we measured the conductance at different temperatures in the range 274.5–278.7 К. This range is relatively narrow because even low temperature changes alter the stiffness of the gold and break the junction. To investigate the influence of mechanical strain on the electrical properties of the system, we altered the gap width by bending our sample. This resulted in a change of system resistance at room temperature. The dependence of the sample resistance on inverse temperature at different bending points is shown in figure 1. First of all we measured the resistance change of gold-BDA-gold molecular junction with decreasing of the temperature. When resistance increases due to MCBJ break we reduced distance between two electrodes and continue to measure the temperature dependence of resistance with definite position of molecule bridged two electrodes (red points in figure 1). At some temperature, resistance increases rapidly and distance between the contacts was again adjusted to attach the BDA molecule. In this configuration the temperature measurements (green points) were continued. The dependences R_g (1/T) and data given in figure 1 are taken from a number of measurements. Each data point is obtained by the averaging of resistances, measured about 5 times with a step of two minutes at a certain temperature. The reproducibility of the data was confirmed by following experiment. After decreasing the temperature the behavior of the resistance dependence with increase of the temperature was obtained to be about the same within the experimental error. The most typical resistance dependence on temperature is used for the analysis.

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At a certain gap length, the conductance of the molecule corresponds to the state value ${G}_{a}\approx 7.8\times {10}^{-3}\,{G}_{0}$ (point 'a') and the molecule is in a low-energy state ${E}_{L}.$ Here, the quantum conductance is ${G}_{0}={\left(12.9{\rm{k}}{\rm{\Omega }}\right)}^{-1}.$ Increasing the temperature by ∼0.8 K molecule receives additional thermal energy ${\rm{\Delta }}E$ and it is excited from a low energy level (point 'a') up to a higher excited energy level ${E}_{H}$ (point 'b') and the conductance increases to ${G}_{b}\approx 9.5\times {10}^{-3}{G}_{0}.$ This is described below in section 3.2 (see also figure 3). The temperature change of 0.8 K corresponds to the change in the thermal energy of the molecule ${\rm{\Delta }}E=0.04$ me${\rm{V}}\,=6.4\times {10}^{-24}$ J. A simultaneous increase in the gap on ${\rm{\Delta }}x$ results in conductance modulation. The molecule quickly acquires additional thermal energy ${\rm{\Delta }}E\approx 0.04\,{\rm{meV}}$ to compensate the forces arising from the extension of the gap elementary length, from the molecule orientation and/or from position changes. Conductance again falls, dropping to a low value of ${G}_{c}\approx 8.1\times {10}^{-3}{G}_{0}$ (point 'c'). The molecule therefore returns to the low-energy stable state. Further temperature increases with the same temperature step of ∼0.8 K and transfer energy ${\rm{\Delta }}E$ causes the molecule to revert to a highly conducting metastable state ${E}_{H}$ at point 'd' (${G}_{d}\approx 8,9\times {10}^{-3}{G}_{0}$). Continuing increases in the gap length between the electrodes of the elementary magnitude ${\rm{\Delta }}x$ causes the molecule to return again to a low-conductance stable state ${E}_{L}$ (point 'e', ${G}_{e}\approx 8,3\times {10}^{-2}{G}_{0}$). When the temperature increases further, the molecule moves to a high-energy state level ${E}_{H}$ (point 'f', ${G}_{f}\approx 9,3\times {10}^{-3}{G}_{0}$). During all transitions from point 'a' to point 'b', 'c' to 'd' and 'e' to 'f', the change in conductance (modulation deepness) is not very large and is only (1.1–1.2 quanta). For comparison, stable contacts were observed in [36] with both low conductance in the order of 10⁻³ conductance quanta as well as with high conductance values above ∼0.5 quanta. Our data unambiguously show that the conductance of the BDA molecule is determined by a single transport channel provided by the same molecular level, which is coupled to the metallic electrodes through the whole conductance range.

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Each transition process from low to high energy levels 'a'–'b', 'c'–'d' and 'e'–'f' is accompanied by the characteristic activation energy ${E}_{a}$ (see figure 1), calculated according to following relationship:

$$\begin{eqnarray}&&R\left(T\right)={\rm{C}}{\rm{o}}{\rm{n}}{\rm{s}}{\rm{t}}\times \exp \left(-\displaystyle \frac{{E}_{a}}{kT}\right).\end{eqnarray} \tag{ 1 }$$

We used experimental data from figure 1. For the transitions, the following activation energy values were calculated: for 'a'–'b' ${E}_{a,a-b}\approx 2.1\times {10}^{-19}\,{J}=1.3\,{\rm{eV}};$ for 'c'–'d' ${E}_{a,c-d}\,\approx 0.96\times {10}^{-19}\,{J}=0.6\,{\rm{eV}};$ and for 'e'–'f' ${E}_{a,e-f}\approx 0.5\,\times {10}^{-19}\,{J}=0.31\,{\rm{eV}}.$ It should be noted that the energy is reduced almost twofold for each step. These experimental data will be explained below.

It is quite clear that observed and presented above effect of the quantum conductance modulation. At the transitions from states 'b' to 'c' and 'd' to 'e', the conductance decreases with an increasing gap on ${\rm{\Delta }}x.$ Similar behavior of the quantum conductance amplitude at 4.2 K was also observed in [37]. Features of these transitions will be discussed later. For a visual understanding of the molecule conductance (energy) temperature modulation process, figure 3 presents a schematic of the energy levels for a gold-BDA-gold junction. The low-conductance stable energy state is taken as the LUMO level, and the level with energy ${E}_{H}$ corresponds to the excited state.

The behavior of the currents through the junction was considered. The presented results (basic data in figure 1) were obtained after bond breaking in the junction. Under a low applied voltage ($V\lt {\rm{\Phi }}/e,$ ${\rm{\Phi }}$ is the metal work function) through the junction, a direct tunnel current flows (see also [20]). The current values corresponding to high-conductance and low-conductance states at the applied voltage 20 mV were calculated as⁴ :

$$\begin{eqnarray*}&&{I}_{H}={G}_{H}V={G}_{b}V\approx 9.5\times {10}^{-3}{G}_{0}V\approx 14.7\,{\rm{nA}}.\end{eqnarray*}$$

$$\begin{eqnarray*}&&{I}_{L}={G}_{L}V={G}_{a}V\approx 7.8\times {10}^{-3}{G}_{0}V\approx 12.1\,{\rm{nA}}{\rm{.}}\end{eqnarray*}$$

The average current was calculated as:

$$\begin{eqnarray*}&&{I}_{av}=\displaystyle \frac{{I}_{H}+{I}_{L}}{2}=13.4\,{\rm{nA}}.\end{eqnarray*}$$

The high and low current states differ by 2.6 nA and the maximum magnitude of the conductance modulation is equal to $\left({G}_{{\rm{m}}{\rm{a}}{\rm{x}}}-{G}_{{\rm{m}}{\rm{i}}{\rm{n}}}\right)/{G}_{{\rm{m}}{\rm{a}}{\rm{x}}}\approx \left({G}_{b}-{G}_{{\rm{a}}}\right)/{G}_{b}\approx 0.18,$ or 18%.

For further analysis, we used information obtained from random telegraph signal (RTS) measurements on gold-BDT(benzene-1,4-dithiol)-gold junctions in [38] (see figure 3 in [38] and its description). A small RTS signal was registered, and the difference between the high and low current states was about ${\rm{\Delta }}I\approx 1.5\,{\rm{nA}}$ at an average current $I\approx$ 14 nA with current fluctuations below 12%. In our experiment, the average current, I, was also used at a level of $\approx$14 nA. The conductance modulation effect was small for the BDA molecule and BDT molecules. We assume that the quantum conductance weak modulation discussed above directly affects the RTS behavior. In this case such, the emission and capture times ${\tau }_{e}$ and ${\tau }_{c}$ in the RTS process can be described as follows. Capture time is the time during which the molecule is in the low-conductance steady state corresponds to the LUMO energy level (${E}_{L}$ level). This corresponds to low-conductance states (points 'a', 'c' and 'e'). During the emission time, the molecule moves in an 'excited' metastable state ${E}_{H}$ (see figure 3), and conductance and therefore current increase (states 'b', 'd' and 'f'). The conduction modulation value is ∼18% and the depth of the current modulation is ∼12%. The difference in modulation depths in these processes is not large. It should be also noted that a relatively low value of the conduction modulation deepness (18%) can be associated with small changes in temperature. For comparison, in nanotubes, giant RTS values of 60% [37] or more than 80% [2] have been reported.

The regular recurrence of the conductance in each temperature step of 0.8 K has to be conditioned by the BDA molecule 'energy gap construction' peculiarities. We assume that the BDA molecule has a sub-band near LUMO with an energy corresponding to (0.04 eV). The conductance activation energy ${E}_{a}$ also decreases approximately twice during each temperature increase of 0.8 K (1.3 eV, 0.6 eV and 0.31 eV, respectively). Such regularity in quantum conductance and activation energy changes could be related to physical and mechanical effects and can be used to characterize the molecular junction with a BDA molecule. Assuming that there are some regularly distributed energy barriers, it is clear that with increasing temperature such barriers can be more easily overcome and the effective ${E}_{a}$ decreases. In such a case, there are quasi-levels regularly distributed in energy between HOMO and LUMO, and activation processes can be performed through those levels. The higher the temperature, the easier the electron transition from a stable state to a metastable state. However, if the electric field is high, the hot electrons may also overcome the barrier. In our case where the applied voltage is 20 mV, the resulting electric field in the junction is small (of the order of 10⁴–10⁵ V cm⁻¹), which reflects that the effect of hot electrons can be neglected.

The results in [39] show that increasing the molecular length results in a decrease in the HOMO-LUMO gap, a decrease in the electrical conductance and an increase in the value and oscillation of the thermo-power, reflecting thermo-activation processes in the structure. The RTS was observed during break junction experiments with gold binding molecules in [40]. The authors attributed the emergence of signals to the spontaneous binding and unbinding of the molecular anchor groups to and from the electrode tips. Both the formation and the breaking of the molecule–metal bonds are thermally activated and show an exponential distribution of lifetimes for the different conductance states. The difference in RTS amplitudes as well as in capture and emission times is associated with different microscopic conduction mechanisms. The large RTS amplitude observed in [40] is related to a total break between the molecule and metal contact. In our work, in contrast to [40], we studied transport phenomena with only a small configurational change in the contact with no break or disconnection between the molecule and metal.

It should be noted, that in opposite to the case considered in [27] in our measurements we demonstrate the dependence of quantum conductance versus displacement $G\left({\rm{\Delta }}x\right)$ only at the gap opening and the simultaneous action of slow temperature activation. Our measurements also show that magnitude of $G$ decreases with the increase of displacement (from point 'b' to 'с', 'd' to 'е', figure 2), but the process occurs with weak thermal activation (from point 'а' to 'b', 'c' to 'd'). With the growth of ${\rm{\Delta }}x,$ we do not see a plateau in the dependence $G\left({\rm{\Delta }}d\right),$ (see e.g. figure 1(c) [27]) because ${\rm{\Delta }}x$ is much smaller and it does not reach the formation of a plateau. As we can see in figure 2(b) [27] the decreasing of $G$ with ${\rm{\Delta }}d$ accompanied by plots of local growth of $G$ (for example, between points 1 and 2, or points 3, 4, 5, figure 2(b) [27]). In the figure 1(c) and figure 3(a) in [27] on the plateau of the dependency $G\left({\rm{\Delta }}d\right),$ nanoscale fluctuations of conductance are clearly visible. These fluctuations are probably also the result of some activation processes. Note also that these nanoscale fluctuations occur at distances of the order of 0.1 nm (figure 3(a) [27]), which is a comparable value with our ${\rm{\Delta }}x$ value. When we change the gap size on ${\rm{\Delta }}x,$ we also have nanoscale fluctuations of conductance, which we explained by thermal activation processes.

Conductance (resistance) thermo-sensitivity ${S}_{TG}$ (or ${S}_{TR}$) of the molecule can be presented as the change in conductance ${\rm{\Delta }}G$ (resistivity ${\rm{\Delta }}R$) at the temperature change in ${\rm{\Delta }}T:$

$$\begin{eqnarray}&&{S}_{TG}=\displaystyle \frac{{\rm{\Delta }}G}{{\rm{\Delta }}T},{S}_{TR}=\displaystyle \frac{{\rm{\Delta }}R}{{\rm{\Delta }}T}.\end{eqnarray} \tag{ 2 }$$

Based on the data in figures 1 and 2 for the BDA molecule, we get ${S}_{TG}\approx \left(0.8\div1.6\right)\times {10}^{-7}\,{{\rm{\Omega }}{\rm{K}}}^{-1}$ (${S}_{TR}\approx \left(2.5\div5\right)\,\times {10}^{5}\,{\rm{\Omega }}\,{{\rm{K}}}^{-1}$). This reflects sufficiently high sensitivity.

The molecule with additional energy has to lose it on some processes. The additional energy ${\rm{\Delta }}E=6.4\times {10}^{-24}$ J (corresponding to a temperature change of 0.8 K) is much smaller than the potential energy. Assuming that the molecule with additional energy ${\rm{\Delta }}E$ spends this additional energy to compensate for the force $F$ arising from the elongation of the gap length⁵ , then

$$\begin{eqnarray}&&{\rm{\Delta }}E=F{\rm{\Delta }}x.\end{eqnarray} \tag{ 3 }$$

For example, in the case of a one-dimensional elastic force in linear approximation, we use Hooke's law:

$$\begin{eqnarray}&&{F}_{H}=\left|{k}_{s}{\rm{\Delta }}x\right|,\,{\rm{\Delta }}E\equiv {k}_{B}T={k}_{s}{\left({\rm{\Delta }}x\right)}^{2},{k}_{s}=\displaystyle \frac{{\rm{\Delta }}E}{{\left({\rm{\Delta }}x\right)}^{2}}.\end{eqnarray} \tag{ 4 }$$

Here, ${k}_{s}$ is the molecular spring constant, ${k}_{B}$ is the Boltzmann constant. The stretching length of the Au−NH₂ molecular junction was 0.03 nm [25]. Assuming that the elementary enlargement is equal to stretching length ${\rm{\Delta }}x=0.03$ nm, then

$$\begin{eqnarray}&&{k}_{s}=\displaystyle \frac{{\rm{\Delta }}E}{{\left({\rm{\Delta }}x\right)}^{2}}=\displaystyle \frac{4\times {10}^{-5}}{{\left(0.03\times {10}^{-9}\right)}^{2}}\,{\rm{eV}}\,{{\rm{m}}}^{-2}=7.1\times {10}^{-3}\,{\rm{N}}\,{{\rm{m}}}^{-{\rm{1}}}.\end{eqnarray} \tag{ 5 }$$

According to experimental and theoretical results presented in [19] for Au-BDA-Au molecular junctions, the rupture force F_rupt = 0.4 nN (expt), F_max = 1.0 nN (expt fit), L_bind = 1.4 Å (expt fit), L_max = 0.46 nN (theory), L_bind = 1.09 Å (theory), E_bind = 0.37 eV (theory). Here ${F}_{{\rm{r}}{\rm{u}}{\rm{p}}{\rm{t}}}$ is the Au−N rupture force and bond strength, ${F}_{{\rm{m}}{\rm{a}}{\rm{x}}}$ is the maximum sustainable force, ${E}_{{\rm{b}}{\rm{i}}{\rm{n}}{\rm{d}}}$ is the metal/molecule binding energy, ${L}_{{\rm{b}}{\rm{i}}{\rm{n}}{\rm{d}}}$ is the binding distance.

In our case, ${k}_{s}=7.1\times {10}^{-3}\,{\rm{N}}\,{{\rm{m}}}^{-1}$ and ${\rm{\Delta }}x=0.03\,{\rm{nm}}.$ Using equation (4), we obtained ${F}_{H}=0.00021\,{\rm{nN}}.$ This is 3 orders of magnitude smaller than the rupture force in [19]. This confirms that there is no breaking in our case.

For comparison, we will consider the case when expansion forces are compensated by Coulomb forces ${F}_{C}.$ For simplicity, we assume that the ends of the molecule are singly ionized (effect of monovalent gold atoms). Then, instead of equation (3), we get

$$\begin{eqnarray}&&{F}_{C}=F,\displaystyle \frac{{k}_{e}{e}^{2}}{{\left({l}_{m}+{\rm{\Delta }}x\right)}^{2}}={k}_{C}{\rm{\Delta }}x.\end{eqnarray} \tag{ 6 }$$

Here, BDA molecule length is ${l}_{m}\approx 0.57$ nm, and ${k}_{e}=8.99\times {10}^{9}\,{{\rm{N}}{\rm{m}}}^{2}\,{{\rm{C}}}^{-2}$ is Coulomb's constant. At ${\rm{\Delta }}x=0.03\,{\rm{nm}},$ ${k}_{C}=21.31\,{{\rm{N}}{\rm{m}}}^{-1}.$ Comparison of magnitudes for ${k}_{s}$ and ${k}_{C}$ shows that the forces arising due to the activation energies are hundreds of times weaker than electrostatic forces. The forces arising due to the activation energies cannot significantly influence the current transport processes, but can affect the strength of the electrode-molecule coupling, the fluctuation of quantum conductivity and hence the RTS behavior. This result is in good agreement with the assumption that different bond states influence the strength of the electrode-molecule coupling at the tilt angle of the molecule and the binding of the end group on the top or hollow site, which is also referred to in [36]. Another study [41] also showed that the energetic position of the molecular orbital may determine the transmission conductivity.