Bachelor of Science Thesis Stockholm, Sweden 2013 TRITA-ICT-EX-2013:227
J I M M Y F R A N C H I
structures of single molecular junctions from current voltage characteristics
K T H I n f o r m a t i o n a n d C o m m u n i c a t i o n T e c h n o l o g y
Degree Project in Molecular Electronics
Investigation of the electronic structures of single molecular junctions from
current voltage characteristics
Author:
Jimmy Franchi
Supervisor:
Prof. Hirokazu Tada
A degree project submitted in fulfilment of the requirements for the Bachelors Degree in Micro Electronics, written at
Tada Laboratory
School of Engineering Science at Osaka University
August 2013
Carl Sagan
Abstract
Tada Laboratory
School of Engineering Science at Osaka University
Investigation of the electronic structures of single molecular junctions from current voltage characteristics
by Jimmy Franchi
Molecular electronics has lately shown significant progress due to rapid advances in in- vestigative methods and is vital to the future of the electronic industry since the funda- mental limits of the solid state transistors are closer than ever. To reach the application of using single molecules as electronic components, it is imperative to investigate the electronic structure of a single molecule connected to metal electrodes.
This research has focused on examining current voltage behavior to clarify the electronic structure using transition voltage spectroscopy (TVS) and a single level tunneling (SLT) model. In an approximate model, TVS can be used to determine the molecule frontier orbital-electrode energy gap (EF− ε0) and in most cases prove molecular presence in the junction. The SLT model can be fitted to experimental data to obtain molecule-electrode bond strength and molecule-electrode energy gap.
The measurement system prepared and tested was a mechanically controlled break junc- tion. It has ability not only to test current voltage characteristics but also probe exter- nal magnetic and electric field modulation effects. The current-voltage characteristics of benzenedithiol, hexanedithiol and a recently synthesized oligothiophene molecule with five thiophene rings were measured and analysed by using TVS and fitting the SLT model.
Most of the current-voltage characteristics could be fitted using the SLT model. Anal- ysis revealed that the variation in molecular conductance is mainly due to variation in molecule-electrode coupling strength and not due to the EF − ε0 energy gap. Differ- ent EF − ε0 were discovered for the three different molecules. It was shown that the transition voltage roughly approximates the EF− ε0 energy gap by comparing the TVS results to the results obtained from fitting the SLT model.
Having desperately fumbled in the dark, reaching out for someone, I would like to thank Professor Hirokazu Tada for accepting me as a student in his laboratory. Experiencing frontier science was both more difficult and more fascinating than I ever could imagine;
for this I have Professor Ryo Yamada to thank, who through invaluable help, assistance and consultation showed me what science is really about. Thanks to his help I feel I have developed more than I thought possible during such a short time.
Tosei Kanematsu made my time in the laboratory full with golden moments I will never forget. He also helped me through the hands on process of creating a single molecular junction and took SEM pictures of the junction. I want to thank Marine Fayolle for helping me understand the basics in the difficult field of organic magneto-resistance and Lee See Kei for providing information on investigated molecules and giving invaluable feed-back. I would also like to give my thanks to my International Coordinator at KTH, Rasmus ˚Aberg, for making this entire process possible.
iii
Abstract ii
Acknowledgements iii
List of Figures vii
Abbreviations ix
Physical Constants x
1 Introduction 1
2 Principles and Methods 4
2.1 Transition voltage spectroscopy and the Simmons model . . . 4
2.1.1 Transition voltage spectroscopy . . . 4
2.1.2 Simmons approximation . . . 5
2.1.2.1 Low bias region . . . 7
2.1.2.2 High bias region . . . 7
2.2 Coherent transport model . . . 8
2.2.1 Landauer model . . . 8
2.2.2 I–V characteristics of a molecular junction expressed by a single level tunneling model . . . 12
2.3 Mechanically controllable break junction . . . 13
2.3.1 Instrumental and experimental principle . . . 13
2.3.2 Preparation of the electrodes . . . 15
2.3.3 Preparing a molecular junction . . . 15
3 Development of Experimental System 18 3.1 Design of a transition voltage spectroscopy measurement system . . . 18
3.1.1 Hardware design . . . 18
3.1.2 Software design . . . 20
3.2 Test of experimental set-up . . . 21
4 Current Voltage Characteristics of BDT, HDT and 5T-di-SCN 23 4.1 Molecules used in experiment . . . 23
4.2 Measurement method . . . 24
4.3 BDT . . . 25 iv
4.3.1 Transition voltage spectroscopy . . . 25
4.3.2 Single level tunneling model . . . 30
4.4 HDT . . . 35
4.4.1 Transition voltage spectroscopy . . . 35
4.4.2 Single level tunneling model . . . 37
4.5 5T-di-SCN. . . 40
4.5.1 Transition voltage spectroscopy . . . 40
4.5.2 Single level tunneling model . . . 42
5 Conclusion and Future Work 45 5.1 Experimental system development . . . 45
5.2 Single molecular current voltage analysis . . . 45
5.3 Future investigation . . . 46
A MATLAB Code 47 A.1 Scripts . . . 47
A.1.1 analysisTVS . . . 47
A.1.2 dataFit . . . 48
A.1.3 gateSweep . . . 48
A.1.4 GGraph . . . 49
A.1.5 MFCloud . . . 50
A.1.6 usedDataCloud . . . 50
A.1.7 VtransAvgI . . . 51
A.1.8 VtransHist . . . 52
A.2 Functions called by scripts . . . 53
A.2.1 cloudPlot . . . 53
A.2.2 movingAverage . . . 53
A.2.3 getFN . . . 53
A.2.4 ITPlot . . . 54
A.2.5 IVPlot . . . 55
B Electrode Preparation Details 56 B.1 Photolithography . . . 56
B.2 Gold electrode deposition by thermal evaporation under vacuum condition 56 B.3 Cleaning and wire placement . . . 57
B.4 Gold electroplating . . . 58
C Application of External Fields and Junction Length Modulation 61 C.1 External field application . . . 61
C.1.1 External electric field application . . . 61
C.1.1.1 Principles . . . 61
C.1.1.2 Method . . . 62
C.1.1.3 Making a current-voltage amplifier . . . 63
C.1.1.4 Results . . . 64
C.1.1.5 Discussion . . . 65
C.1.2 External magnetic field application . . . 65
C.1.2.1 Principles . . . 65
C.1.2.2 Method . . . 66
C.1.2.3 Results . . . 67
C.1.2.4 Discussion . . . 67
C.2 Junction length modulation . . . 69
C.2.1 Principles . . . 69
C.2.2 Method . . . 69
Bibliography 71
2.1 Simmons model . . . 6
2.2 Landauer model . . . 9
2.3 Asymmetric junction as modeled by Matsuhita et al. . . 11
2.4 MCBJ . . . 14
2.5 Schematic junction . . . 16
2.6 Conductance quantization . . . 17
2.7 Conductance plateaus . . . 17
3.1 Schematic picture of experimental set-up. . . 19
3.2 Picture of experimental set-up. . . 19
3.3 LabVIEW environment . . . 20
3.4 Test results from the experimental set-up . . . 22
4.1 Molecule schematics . . . 23
4.2 Not ideal data. . . 25
4.3 Current-Voltage plot for BDT . . . 26
4.4 Fowler-Nordheim plot for BDT . . . 27
4.5 Summary of BDT TVS . . . 28
4.6 Histogram of BDT transition voltages . . . 29
4.7 Fitting behavior . . . 31
4.8 BDT ε0 . . . 32
4.9 BDT junction asymmetry . . . 32
4.10 ΓL,R plotted against current . . . 33
4.11 Averaged current against apparent transmission value . . . 34
4.12 Example of HDT I–V characteristics . . . 35
4.13 Cloudplot and corresponding summary of data for HDT . . . 36
4.14 Transition voltage histogram for HDT . . . 37
4.15 Curve fitting results for HDT . . . 38
4.16 HDT junction asymmetry . . . 39
4.17 Current-voltage characteristics recorded for 5T . . . 40
4.18 Cloudplot and summary of 5T measurements . . . 41
4.19 5T transition voltage histogram . . . 42
4.20 Curve fitting results for all 5T data. . . 43
4.21 5T junction asymmetry . . . 44
B.1 Junction . . . 58
B.2 Electrodeposition set-up . . . 59
B.3 Junction connection . . . 59
vii
B.4 SEM of junction . . . 60
C.1 External electric field effect . . . 62
C.2 Low-pass filter . . . 63
C.3 Representative BDT external gate voltage sweeps . . . 64
C.4 Results for magnetic field application to 5T . . . 68
5T 5–thiophene-di-SCN
AD Analog-Digital
BDT Benzene–[1,4]–dithiol
CP-AFM Conducting Probe Atomic Force Microscopy
DA Digital-Analog
DFT Density Functional Theory FET Field Effect Transistor
HDT Hexanedithiol
HMDS Hexamethyldisilizane
HOMO Highest Occupied Molecular Orbital IETS Inelastic Tunneling Spectroscopy LUMO Lowest Unoccupied Molecular Orbital MCBJ Mechanically Controlled Break Junction NEGF Non-Equilibrium Green’s Function OMAR Organic Magneto-Resistance SEM Scanning Electron Microscope STM Scanning Tunneling Microscope TVS Transition Voltage Spectroscopy
ix
Euler’s number e = 2.71828
Elementary charge e = 1.60218 × 10−19C Electron rest mass me = 9.10938 × 10−31kg Planck’s constant h = 6.62607 × 10−34Js Boltzmann’s constant kB = 1.38065 × 10−23JK−1
x
Introduction
The never-ending chase of ever smaller and faster transistors at a lower power con- sumption manically drives the industry forward in shrinking the inorganic silicon-based technology. However, there are physical limits to the top-down method1 of creating components and this is where molecular electronics shows its greatest promises. Molec- ular electronics has the possibility of breaking the inherent limitations in the size of a silicon-based transistor2 by using a single or a few molecules to create the ultimately small transistor. Molecular electronics is a field of science that will continue to expand from now on. Due to the fundamentally challenging nature of molecular electronics, both front end theorists and experimental researchers are involved in the hunt for expanding the limits of technology.
Interest in the field of molecular electronics began back in the 1970s but the techniques required to analyse single molecular characteristics have only lately been developed [1].
There are now several ways of preparing and measuring properties of molecular junctions coupled with external magnetic, thermal and electric fields. Techniques used for creating molecular junctions include conducting probe atomic force microscopy (CP-AFM), cross- wire method, scanning tunneling microscopy (STM) and mechanically controlled break junction (MCBJ). Of course, each technique has advantages and disadvantages and some serve better for some situations than others. The two main advantages of the STM is the ability to make thousands of measurements on molecular junctions in a short amount of time and obtaining the precise knowledge of the junction length. However, drawbacks of the STM are that the molecular junction will have asymmetric electrodes and there have been reports that a molecular junction prepared with the STM detaches quickly
1The top-down method of creating electronics usually involves using a pattern (such as a photo mask) in order to create the components, rather than self-assembly.
2The minimum width of a silicon based transistor is believed to be about 9 nm due to quantum tunneling effects at shorter lengths.
1
[2]. MCBJ on the other hand uses a mechanical reduction mechanism coupled with a piezoelectric actuator in breaking the atomic electrodes to create a junction, providing more stability than the STM; however, this leads to a weaker knowledge about junction length and electrode structure [3].
Many groups have recently measured conductance of molecular junctions using the STM break junction method [2, 4–13]. In essence, the STM tip is repeatedly brought in contact to a substrate with adsorbed molecules and subsequently retracted until the junction is broken. Current-time transients are recorded for hundreds∼thousands of times in order to create conductance histograms. It has been discovered that molecular conductance does not occur in well-defined values, but rather scattered around several peaks in conductance histograms [14].
The variability found in molecular junction conductance might be due to the microscopic molecule-electrode bonding as discussed by Reichert et al. [15]. This was also shown in the theoretical work performed by Ulrich et al. [11]. One of the most important questions in the field of molecular electronics is how the variability of the molecular junctions can be controlled or taken advantage of. A possible application using the intrinsic variability and non-uniformity of the molecular junction is stochastic resonance.
Statistical break junction experiments show the most probable regions for molecular conductance; however, this is not good enough for electronic applications [14]. Making static molecular junctions in electronic applications require a way of determining whether the observed conductance of a junction is due to a molecule or a vacuum tunneling gap.
In order to overcome the problems faced, several techniques to confirm the presence of a molecule in a junction and to analyse molecular behavior have recently been developed:
transition voltage spectroscopy (TVS) [16], inelastic tunneling spectroscopy (IETS) [17], junction distance modulation [18] and measurement of static I–V characteristics such as field effect transistor (FET) type behavior [19]. At this point in time, it is imperative to investigate the possible functions of different molecular junctions in order for the field of molecular electronics to reach its possible applications.
TVS and detailed current voltage analysis using a single level tunneling model have lately attracted attention as they can be used to investigate the electronic structure of single molecular junctions. These methods give information on the energy gap between the electrode Fermi level and the frontier molecular orbital level (EF−ε0) and molecular orbital broadening (Γ), which are key parameters in understanding molecular junction charge transport.
In this study, a reliable set up for measuring single molecular properties using the MCBJ was constructed and the current voltage characteristics of molecular junctions was mea- sured for benzenedithiol (BDT), hexanedithiol (HDT) and a recently synthesized olig- othiophene. The obtained current voltage characteristics were analysed by TVS and a single level tunneling model. External electric and magnetic field application and junction distance modulation set-up were also developed, but the results obtained were primitive. These methods are discussed in AppendixC.
Principles and Methods
There are two main approaches for interpreting the current voltage characteristics of a molecular junction: i) the Simmons model and ii) the coherent transport model. There have lately been several articles from different groups challenging the Simmons approach [20–24]. However, I will discuss it since it gives a good introductory understanding of a molecular junction and TVS. Next, I will explain the coherent Landauer model and how to use it to extract key parameters from experimental results. Finally, I will explain the basics of the MCBJ method used to make a single molecular junction.
2.1 Transition voltage spectroscopy and the Simmons model
2.1.1 Transition voltage spectroscopy
In 1963, John G. Simmons constructed a generalized formula for the electric tunnel effect between similar electrodes separated by a thin insulating film [25]. In his paper, Simmons described the general case of electron tunneling through a potential barrier of arbitrary shape. In 2006, Beebe et al. developed a new analytical technique for analysing molec- ular and short-range vacuum tunneling named transition voltage spectroscopy (TVS), based on the Simmons model [16].
The main idea of TVS is to find the minimum in a graph of ln I/V2 against 1/V . This signifies the point at which two different mechanisms of charge transport competes, which is called Vtrans or alternatively Vmin as it is the minimum point in the plot. I will use the notation Vtransfor the transition voltage for clarity. The reasoning behind the exotic choice of axes in the TVS plot, or more commonly referred to as the Fowler-Nordheim plot, will be discussed below.
4
In a simplified manner, the transition voltage can be seen as the transition from tunneling seen at a low applied bias voltage to electron field emission at high applied bias voltage as described in a paper by Fowler and Nordheim in 1928 [26]. The reason for the transition in this model is the extreme electrical field occurring at a higher bias voltage.
Since the voltage drop occurs over a distance of ∼ 1 nm, the electrical field is in the range of 109V/m. Successfully applying a high enough bias voltage to obtain a resonant tunneling condition through the molecular orbital has proved close to impossible since the junction breaks down due to Joule heating [24]. Beebe et al. suggested that the transition voltage, which occurs at a lower voltage, gives an estimate of the difference between the electrode Fermi level (EF) and the frontier (conducting) molecular orbital (ε0). This energy difference is important for understanding the electronic structure of the molecule, especially for gating applications. However, the one-to-one ratio proposed between transition voltage and |EF − ε0| has since been improved and a more general interpretation was given by Chen et al., where an asymmetry factor was included, leading to the ratio |EF − ε0|/eVtrans ranging from 0.86 to 2.0 [23].
Lately, it has been suggested by Markussen et al. that one can find even higher transition values by changing the exponent to ln I/Vα, where 1.5 ≤ α ≤ 2 for practical purposes, with EF − ε0 deviating by approximately 0.1 eV at α = 1.5 [22]. In addition to this, they provided a linear model for determining the connection between transition voltage and EF− ε0for two cases of α, shown in Equations2.1and2.2. Here, 0 ≤ η ≤ 0.5 is the junction asymmetry factor, where η = 0 is a completely symmetric junction and η = 0.5 is a completely asymmetric junction.
|EF − ε0|/eVtrans = 2.3η + 0.85 α = 2 (2.1)
|EF − ε0|/eVtrans = 3.7η + 1.5 α = 1.5 (2.2)
2.1.2 Simmons approximation
In this section, I will discuss Simmons’ arguments in order to provide a necessary foun- dation for understanding TVS. Simmons’ simplified approximation assumes that the junction consists of a rectangular barrier that gradually becomes triangular at higher bias voltage. In Figure 2.1, a band diagram of Simmons model for zero bias and very high bias can be found. It should be noted that the high bias approximation used by Beebe et al. is the case where the voltage is so high that the Fermi level of the right electrode in the Figure is actually lower than the conduction band of the left electrode, leading to no tunneling from electrode 2 to electrode 1 due to no available states. This case is shown in the right side of Figure 2.1.
Figure 2.1: Band diagram of the Simmons model shown for electron tunneling (not to scale). Here, EF denotes the electrode Fermi level, φ denotes the barrier height, Ψ denotes the work function of the metal and s denotes the apparent junction distance as seen by the tunneling hole or electron. The shaded regions in the electrodes represent respective conduction bands, below which are no available states. On the left side, the case for zero bias voltage is depicted where the barrier is rectangular. On the right
side, the case for very high bias voltage is shown where the barrier is triangular.
Formulas derived in Simmons’ paper were expressed in current density; hence, multiply- ing by area seen by the charge carrier tunneling through the junction gives expressions in ampere. The general formula for current through any potential barrier derived by Simmons can be seen in Equation 2.3:
I = I0n φe−B
√φ− (φ + eV )e−B
√φ+eVo
(2.3) where
I0= Ae/2πh(βs)2 B = (4πβs/h)√
2me
φ is mean tunneling barrier height in eV V is applied bias voltage to junction in V
A is area seen by the tunneling charge carrier in m2 β is a correction factor, usually close to 1
s is the barrier length in m
Equation 2.3 can be simplified in four cases of low, intermediate, high and very high applied bias voltage to the junction. For practical purposes, Beebe et al. chose to use the low bias region for explaining the behavior up until the transition voltage and the very high bias region for behavior above the transition voltage [16]. They chose not to include image effects on the junction barrier shape; however, even using the full model including this effect cannot give a good explanation for experimental results as shown by Trouwborst et al. [27]. Thus, for the sake of understanding TVS, I will explain only the low and the highest bias regions in the simplified model.
2.1.2.1 Low bias region
The low bias region is where the bias voltage is approximately close to zero. As shown on the left side of Figure2.1, the barrier is rectangular in this region. Using the approx- imation that the barrier does not change shape with low bias voltage application and a series expansion ignoring higher powers of voltage gives the expression for the current derived by Simmons seen in Equation2.4.
I =
√2mφAe2
h2s Vlowe−(4πs/h)
√2mφ (2.4)
Reconstructing Equation 2.4to Fowler-Nordheim form gives Equation 2.5.
ln(I/Vlow2 ) = ln( 1
Vlow) + ln(
√2mφAe2
h2s ) − (4πs/h)p
2mφ (2.5)
Equation2.5is on the form y = ln(x)+c in a plot of ln(I/V2) against 1/V , which means that it will show a slow logarithmic increasing form from high to low bias voltage.
2.1.2.2 High bias region
The situation is similar to that of field emission as described by Fowler and Nordheim in this region, leading to a straight line in the plot of ln(I/V2) against 1/V . This will be shown in this section.
A high voltage means that the tunneling barrier is triangular, shown on the right side of Figure2.1. Here, tunneling barrier length decreases with increased bias. This means that the molecule will seem shorter for the tunneling charge carrier so tunneling probability increases correspondingly. Thus, the barrier length is inversely proportional to applied bias voltage, s = φ/(eV ). The average barrier height φ is constant and equal to half of the molecular barrier height due to the triangular shape. Simmons used these parameters in Equation 2.3 and calculated the correction factor β = 23/24. He then reasoned that
an extremely high bias will lead to tunneling from only one side since positive electrode Fermi level is below the negative electrode conduction band and obtained the result shown in Equation2.6.
I = 2.2Ae3
8πhφs2Vhighest2 e−8πs
√
2mφ3/2/(2.96heVhighest) (2.6)
Reconstructing Equation 2.6to Fowler-Nordheim form gives Equation 2.7.
ln(I/Vhighest2 ) = −8πs√
2mφ3/2 2.96he
1
Vhighest + ln 2.2Ae3 8πhφs2
(2.7)
This means that if one plots ln(I/V2) against 1/V , it is on the form y = −bx + k, where b is a positive constant. This leads to a simple straight line with negative slope. It is worth noting that this approximation is only valid at extremely high bias voltages, so there will be more error the closer the bias is to the transition voltage.
2.2 Coherent transport model
2.2.1 Landauer model
In the paper written by Landauer et al. in 1985, a model including finite temperature of a one-dimensional single-channel elastically scattering conductor was derived (Equa- tion 2.8) [28]. This has subsequently led to application within molecular electronics as the model is based on the Fermi-Dirac distribution function and general transmission function for the electrons scattering through the junction. The general transmission function T(E) gives room for adapting the junction barrier to molecular junction behav- ior, while the Fermi-Dirac distribution is necessary to describe the nature of electron or hole tunneling transport.
I = (2e/h) Z
T (E)[f (E − µ1) − f (E − µ2)]dE (2.8)
Here, f denotes the Fermi-Dirac function f = 1 + e(E−µ1,2)/kBT−1
. The chemical potentials of the electrodes µ1 and µ2 are in the case of applied bias voltage given by µ1 = eV /2 and µ2 = −eV /2 due to the electrodes shifting symmetrically about the Fermi level as is shown in the energy diagram of the Landauer model in Figure2.2. Inserting µ1,2= ±eV /2 in the general Equation 2.8leads to an equation for current depending on bias voltage as can be seen in Equation 2.9.
I(V ) = (2e/h) Z
T (E)[f (E − eV /2) − f (E + eV /2)]dE (2.9)
Figure 2.2: Energy diagram of the Landauer model. εHOM O,LU M Odenotes the energy of the molecular HOMO, LUMO orbitals respectively. V is the applied bias between electrode 1 and 2. ΓL and ΓR represent the broadening of the molecular orbital close to electrode 1 and 2 respectively. The left side shows a symmetric molecular junction at zero bias, and the right side shows a symmetric molecular junction at finite bias.
If we look at a simplified picture where T ≈ 0, the Fermi-Dirac functions in Equation2.9 are two Heaviside step functions1of different signs offset by eV. This leads to a limitation on which charge carrier energies are able to flow, contained within E = EF±eV /2. Thus, it can be seen that the right side of the integrand in Equation2.9determines the overall window of which energies can contribute to charge transfer, depending on the magnitude of the supplied bias voltage. In the case of zero temperature, the limits of integration would be strictly confined within EF ± eV /2 due to the step function nature of the Fermi-Dirac functions and the right side of the integrand would be replaced with unity.
If finite temperature is taken into account, these simplifications cannot be made. In this case, right side of the integrand expression determines the probability of charge carrier energies. The difference from zero temperature is that at finite temperature, the Fermi-Dirac functions are smooth and not step functions. Thus, the available window of charge carrier energies effectively widens.
1A Heaviside step function is a discontinuous function that is zero for negative arguments and one for positive arguments.
The transmission function T(E) in Equation 2.9plays a similar role in determining the current flow through the junction as the Fermi-Dirac functions. The first main difference is that the transmission function peak is modeled as centered around the charge carrier conducting molecular orbital energy ε0 instead of the Fermi level. The second difference is the underlying reason for different transmissive energy levels. In the case of the Fermi- Dirac function, temperature determines the allowed energy states of the charge carriers whereas in the case of the transmission function, it is the molecular level broadening due to the molecule-electrode covalent bond. The transmission function used by Matsuhita et al. and Youngsang Kim et al. can be seen in Equation 2.10[29,30].
T (E, V ) = 4ΓLΓR
[E − ε0(V )]2+ [ΓL+ ΓR]2 (2.10) Here, ε0(V ) is the apparent HOMO or LUMO2 energy with respect to electrode Fermi level EF depending on applied bias, which will soon be discussed. Defining ε0(V ) as the difference between the frontier molecular orbital and the electrode Fermi level leads to the expression E − ε0(V ) in Equation 2.10 in fact being E − (EF + ε0(V )), where the electrode Fermi level EF has been set to zero for convenience in calculations since it is a reference energy level. The broadening function around the molecular orbital energy ε0(V ) is denoted by ΓL,R for bonding to the left and the right electrode in Figure 2.2 respectively. The presence of ΓL,Rincludes the junction asymmetry into the transmission function, where a perfectly symmetric junction corresponds to ΓL= ΓRshown in Figure 2.2since the orbitals are equally broadened on electrode 1 and 2.
The voltage dependence in ε0(V ) was used by Matsuhita et al. because of the asymmetric I–V behavior observed for different bias polarities on some molecular junctions. The simple model used is shown in Equation 2.11.
ε0(V ) = ε0+eV 2
ΓL− ΓR
ΓL+ ΓR (2.11)
If the junction is perfectly symmetric, ΓL= ΓR and the second term in Equation 2.11 becomes zero. In this case, the apparent charge carrying molecular orbital energy with respect to the Fermi level is equal to ε0 and is not dependent on bias voltage. On the other hand, if the junction is asymmetric as shown in Figure 2.3, the apparent ε0(V ) energy will follow the potential of the stronger bonded electrode. This means that if the molecule is HOMO conducting (ε0 < 0) and it is more strongly bound to the left electrode than the right (ΓL−ΓR> 0), a positive bias polarity will give a smaller |ε0(V )|
and a higher current. This is clearly shown in Figure2.3b), where the bias voltage makes the Fermi level of the right electrode enter the molecular frontier orbital. A negative
2HOMO and LUMO denotes highest occupied molecular orbital and lowest unoccupied molecular orbital, respectively.
bias polarity gives a larger energy gap between the electrode Fermi level and the frontier molecular orbital as is shown in a), which in turn leads to a lower conductance. If the molecule was LUMO conducting, or the right side was more strongly bonded, the case would be the opposite.
Figure 2.3: Energy diagram of bias application to an unevenly coupled molecule.
The left electrode is more strongly bonded than the right electrode in both cases. The energy level of the frontier molecular orbital, in this case the HOMO, follows the bias polarity of the electrode it is most strongly coupled to, a) for negative bias voltage and
b) for positive bias voltage.
Maximum transmission for Equation 2.10, T (E, V ) = 1, is obtained under two condi- tions:
i) the energy of transferable charge carriers is equal to |EF − ε0| ii) a perfectly symmetric junction ΓL= ΓR
The first requirement leads to E −ε0(V ) = 0. In other words, this means that the frontier molecular orbital is perfectly aligned with the electrode Fermi level. T (E, V ) = 1 is called resonant tunneling, for which the corresponding conductance can be found: the integral in Equation 2.9 is approximately equal to the potential energy eV for a large bias voltage, leading to a conductance of G = I/V = 2e2/h = 1 G0. 1 G0 is the conductance quantum of a conducting single atomic chain. In principle, obtaining 1 G0 for a molecular junction is difficult. However, recently, 0.7 G0 was reported by Getty et al. through a ferrocene-based molecular wire and up to 1 G0 was reported by Kiguchi et al. through a Pt-benzene-Pt junction [31,32]. The requirement for a relatively resonant tunneling is reachable if eV /2 ≥ ε0. Applying this bias to a molecular junction is difficult because of the large difference usually seen between the electrode Fermi level and the frontier molecular orbital. This leads to the molecular junction breaking down due to Joule heating before resonant tunneling is achieved. If the frontier molecular orbital ε0
is close to the Fermi level of the electrodes, it is easier to achieve resonant tunneling and a higher conductance.
Different groups have used different transmission functions to describe the molecular orbital characteristics, where the Lorentzian shape seems to be the most common ap- proach. The contents of T (E) varies, but the general idea is that it contains the molecular energy level ε0, and molecular orbital broadening Γ. Huisman et al. used a transmission function closely related to the one used by Matsuhita et al.,
T (E, Γ) = η(1 − η)Γ2 Γ2/4 + (E − ε)2
where η is a measure of junction asymmetry. In this case, η = 0.5 is a perfectly symmetric junction. This equation was further developed into a transmission function dependent on molecular length in order to explain experimental results showing that zero-bias conductance of alkanes exponentially decreases with increased length:
T (E, d) = 1
1 4η(1−η)+
E−ε EF−ε
2
eβd
where β is the tunneling decay constant and d is the molecular length. In comparison, Markussen et al. used a simpler transmission function
T (E; ε0, Γ) = f
(E − ε0)2+ Γ2/4
where f was added as a constant factor to include other effects such as several molecules connected in parallel, asymmetric coupling and so on. The main benefit of the Landauer model is that it is easily adjustable to advances in the field and can be effectively used to explain experimental results.
2.2.2 I–V characteristics of a molecular junction expressed by a single level tunneling model
The Landauer model can be used for fitting to experimental results. In this work, this has been done in line with methods used by Matsuhita et al., where only one of the molecular orbitals are assumed to conduct charge carriers at any one time [29]. Firstly, the I–V curve for temperature T = 0 was derived. Then, a correction term was added for modelling the I–V curve at a finite temperature. They modeled voltage drop over the junction mainly occurring at electrode-molecule contacts and a higher voltage drop over the weakest molecule-electrode bond in an asymmetric junction by using Equation2.11.
Matsuhita et al. solved Equation2.9using Sommerfeld expansion and the transmission
function 2.10 with ε0(V ) as given in Equation 2.11. The final result obtained can be seen in Equation2.12.
I(V ) =8e h
ΓLΓR ΓL+ ΓR
"
tan−1
ΓR
ΓL+ΓReV − ε0 ΓL+ ΓR
!
+ tan−1
ΓL
ΓL+ΓReV + ε0 ΓL+ ΓR
!#
−8π2
3he(kT )2ΓLΓR
ΓR
ΓL+ΓReV − ε0
ΓR
ΓL+ΓReV − ε02
+ (ΓL+ ΓR)2
2 (2.12)
+
ΓL
ΓL+ΓReV + ε0
ΓL
ΓL+ΓReV + ε0
2
+ (ΓL+ ΓR)2
2
The top part of Equation2.12is the zero temperature variant of the solution, while the second part of the equation is the correction term. This equation may be used for fitting to experimental data in order to find the parameters ΓL,R and ε0.
There is no precise connection between the Landauer model and the transition voltage.
The qualitative connection is that the transition voltage in the graph of ln I/V2 against 1/V occurs when the Fermi level of one electrode enters the broadened molecular or- bital. When the applied bias shifts the electrode Fermi level, it enters the transmissive energies allowed described by the Lorentzian transmission function T (E). Although be- ing finite for all energies, it is vanishingly small for energies far from its peak. When the Fermi level enters the distinguishable part of the Lorentzian transmission function, the I–V characteristics of the junction gradually changes, giving the transition voltage characteristic [21].
2.3 Mechanically controllable break junction
2.3.1 Instrumental and experimental principle
In this work, MCBJ is selected for making a single or few molecular junction. A schematic diagram of MCBJ is shown in Figure 2.4 a). The driving rod is used for breaking the junction by pushing the rod upwards using a micrometer screw and a piezoelectric actuator. The ceramic stoppers holds the substrate down while the driving rod is pushed. This leads to the thin silicon substrate being bent and the gold electrodes placed on the substrate are subsequently slowly retracted from each other. When the substrate is bent enough, the connection is broken. This leaves a very short distance be- tween the electrodes, giving possibility for a molecule to bridge the gap through bonding
to the tip gold atoms. It is possible to break the junction down at ˚Angstr¨om precision by bending the silicon substrate that the junction is deposited on. This precise distance control is the main benefit of the break junction method.
Figure 2.4 b) shows a schematic of the displacement ratio (r) between the junction elongation (dl) and the vertical pushing distance (dz), which is approximately given by r = dl/dz = 6tu/L2. Here, t is the substrate thickness, u is the approximate distance between the electrodes created by thermal evaporation and L is the distance between the ceramic stoppers [33, 34]. In this experiment, values are approximately t = 100 µm, u = 4 µm, L = 15 mm. The smallest dz length division applicable through the piezoelectric actuator is about 1µm (1% of the maximum stroke length). This gives the smallest dl possible in this set-up of dl ≈ 6×100×10−6×4×10−6×10−6/ 15 × 10−32
≈ 1 × 10−11m. This corresponds to a precision of 0.1 ˚A, which is accurate enough for observing molecular junctions.
Figure 2.4: a) Set-up of mechanically controlled break junction. b) Schematic of the small difference dl obtained when the driving rod is pushed up a distance dz.
2.3.2 Preparation of the electrodes
In order to perform MCBJ measurements, there is a need to produce two electrodes joined with a small contact area suspended over a substrate. The junction contact was formed by electroplating gold onto the base electrodes prepared by photolithography.
The detailed experimental procedure for preparing the electrodes and a graphical exam- ple can be found in Appendix B.
In Figure 2.5, a schematic drawing of a junction before and after electrodeposition can be seen. Photolithography and deposition by evaporation was performed to create two gold electrodes with a spacing of about 4µm. Since gold cannot be deposited onto silicon easily, chrome was first deposited as an adhesive layer. In order to accumulate the deposition process around the gap region, insulating layers were formed on the gold electrodes except the at the gap and wire contact pad regions by Cr deposition and oxidation with UV/O3 treatment. The substrate was then coated with a water-repellent layer using hexamethyldisilizane (HMDS) and the gold wire contacts were fastened to the contact pads using silver paste. The junction was finished by electroplating using a potassium gold cyanide solution and a gold counter electrode.
2.3.3 Preparing a molecular junction
Conductance of an atomic gold junction is quantized in units of G0 = 2e2/h ≈ 77.5µS, which is the conductance of a single gold atom chain. This corresponds to a resistor of approximately 12.9 kW. When the junction is broken, the electrical conductance through the gold junction subsequently falls. At around 10 G0 the quantization starts to be visible, shown in Figure 2.6.
When breaking the junction after adding a solution of conducting molecules, a new plateau can be seen below 1 G0. Figure 2.7 shows an example of a conductance-time transient during the breaking of the gold junction with BDT molecules absorbed on it.
A conductance plateau was observed at 0.01 G0, which was attributed to the formation of a Au-BDT-Au junction. The conductance of the molecule is dependent on the gold- molecule-gold contact geometry, concluded by Ulrich et al., leading to the conductance of a molecule not having a definite value [11]. The molecular junction breaks under further elongation of the gap distance. Huang et al. compared STM stretching results to a thermodynamic bond-breaking model and found that it is the Au-Au bond near the molecular junction that breaks rather than the molecule-Au bond [13].
Figure 2.5: a) shows a schematic drawing of a junction before electroplating. Notice the difference in scales, where the ∼ 1 mm gap region is considerably contracted com- pared to the ∼ 4µm gap created by photolithography. b) shows a schematic drawing
of a junction after electrodeposition.
Figure 2.6: Conductance quantization seen from 10 G0 at constant electrode with- drawal rate, with an illustration of a single gold atom junction.
Figure 2.7: From gold atom conductance plateaus to BDT conductance plateaus with an illustration of single BDT molecule junction. The electrodes were retracted at a constant rate. Broken junction indicates that the molecular junction has lost contact
from one of the electrodes.
Development of Experimental System
3.1 Design of a transition voltage spectroscopy measure- ment system
3.1.1 Hardware design
Figure3.1shows a schematic diagram of the hardware connections for transition voltage spectroscopy measurement. The position of the driving rod was controlled by a stepping motor (New Focus Picomotor 8310V) and a stacked piezoelectric actuator. A function generator (Iwatsu SG-4105) was used for controlling the stepping motor. A signal from a digital to analog (DA) converter (NI-PXI 4461), amplified through a Mess-Tek M-2654 Piezo Driver, was used for controlling the piezoelectric actuator. A voltage drop over the junction was generated by the DA converter (NI-PXI 4461). The current was amplified by an I/V amplifier (FEMTO DLPCA-200) and the resulting voltage was recorded by the analog to digital (AD) converter (NI-PXI 4461).
Figure 3.2 shows a photograph of the experimental set-up. In order to reduce 60Hz noise, it was vital to ground the MCBJ stand to the machines used in the set-up. When using the self-made amplifier, it was also important to keep the cable distance between amplifier and junction short in order to increase the amplified signal bandwidth.
18
Figure 3.1: Schematic picture of the experimental set-up.
Figure 3.2: Entire experimental set-up.
3.1.2 Software design
Measurement parameters and data recording was controlled through LabVIEW 2010.
A program was created that could control amplifier gain setting, piezoelectric actuator height, external electric field voltage, applied DC bias voltage for checking junction conductance and triangular voltage wave application. Programs were also made for applying an external electric field voltage sweep at constant bias voltage, bias voltage sweep under constant gate voltage and magnetic field sweep at constant bias voltage.
Figure 3.3demonstrates two bias voltage sweeps taken during a short time period.
Figure 3.3: LabVIEW environment as seen while performing voltage sweeps on a resistor. The top blue line in the left graph is conductance plotted against time, the green center line is applied voltage and the red bottom line is the current response.
The right side graph shows the frequency spectrum of the current from 0 Hz to 2 kHz.
In order to analyse the data collected through LabVIEW, several programs were devel- oped in MATLAB. The main program created was for making the Fowler-Nordheim plot of the TVS data. This program can make a current-voltage and current-time plot of the data, check offset at low voltage values to find bias current, average the current values from positive and negative slope of the triangle sweep, remove the bias found and finally construct the FN-plot. A simple program to plot conductance during break junction was made. The Cloudplot function found on the MATLAB File Exchange was used in a script that can check files used for recording transition voltages, import data from the correct bias sweep and finally plot all data used for recording transition voltages.
Programs were also made to construct a histogram over all transition voltage values
recorded and plotting transition voltage values against respective averaged sweep cur- rent. A code was written to automatically use the curve fitting tool add-on in MATLAB for fitting Equation2.12to the recorded I–V data. AppendixAshows commented code of the created programs.
3.2 Test of experimental set-up
Figure3.4shows a test of the experimental set-up using a 10 MΩ resistor as a junction.
Upon examining the diagram, the voltage-current characteristic is perfectly linear, and the scaling is also correct since I = V /R = V × 10−7A. The difference between the averaged positive slope current and averaged negative slope current in the triangle sweep was found to be approximately 80 pA. This is a capacitive effect, which is described by Equation 3.1.
IC = CdV
dt (3.1)
For the positive voltage slope, we have Itot = IE + IC and for the negative slope we have Itot = IE − IC, where Itot denotes total current through the resistor, IE denotes expected current through the resistor and IC denotes the additive capacitive effect. The reason for the sign change between positive and negative slope is the derivative changing sign in Equation 3.1. If the difference between the positive slope current and negative slope is taken, we find (IE+ IC) − (IE − IC) = 2IC. Hence, half of the difference between positive and negative slope sweeps gives the correct value for capacitive current in the circuit. Since the difference between positive and negative slopes was found to be 80 pA, the capacitive current is IC = 40 pA. This, coupled with dV /dt = 24 V/s gives C = IC/ (dV /dt) = 40 × 10−12/24 ≈ 1.7 pF. The current between the negative and positive slope was automatically averaged in Fowler-Nordheim plot analysis in this work in order to remove the capacitive effect from the data.
Figure 3.4: I–V sweep test using a 10 MW resistor instead of a junction. a) shows voltage and current plotted against time. The left vertical axis corresponds to the blue voltage curve, while the right axis corresponds to the red current curve. The inset in a) shows the linear I–V behavior. b) shows a Fowler-Nordheim plot of the resistor data where red and blue colors indicate positive and negative bias voltages, respectively. No
minimum point is present in the plot.
Current Voltage Characteristics of BDT, HDT and 5T-di-SCN
4.1 Molecules used in experiment
Figure 4.1 shows the structures of the molecules used in this study, HDT, BDT and an oligothiophene (5T). The thiol (SH) group is known to form a -S-Au covalent bond, replacing the bonded hydrogen atom [35]. The group acting as an anchor to the elec- trodes for 5T is the thiocyanate group (SCN). Although the structural details are not well known, Yamada et al. supposed that the R-SCN group reacts with the gold elec- trodes to form R-S-Au and a gold atom with an adsorbed CN, Au-CNads [36].
Figure 4.1: Schematic drawings of the investigated molecules.
Since affinity to gold is much higher for the thiol and thiocyanate groups than for alkyl groups, one can be certain that the molecule-electrode contact is in fact occurring at these locations and not on other parts of the molecule. For 5T, since the alkyl groups are very long, molecular conductance is actually lower than that perceivable in this experimental set-up if that is the molecule-electrode contact geometry. The UV–vis absorption spectra of 5T has been measured by Yamada et al. and it was found to
23
absorb at around 430 nm [36]. The HOMO-LUMO gap in eV can thus, in free solution, be determined to approximately E = hc/eλ ≈ 2.9 eV for the 5T molecule. The EF−εHOM O gap for HDT was calculated to 2.14 eV and the HOMO-LUMO gap was calculated to 7.5 eV by Li et al. using DFT [37]. The EF − εHOM O for BDT was calculated, using DFT, to be 1.16 eV by Baheti and the HOMO-LUMO gap is ∼ 5.1 eV according to Chen and Tao [35,38,39].
4.2 Measurement method
The substrate was placed at the stand and the gold wires were fastened to the signal input and output. A drop of the molecular solution with a concentration of about 1 mM was added to the sample surface using a pipette. Argon gas was fed to the metal case containing the sample at a rate of 0.5 L/min to remove nitrogen, oxygen and possible contaminants in the air, which also can produce conductance peaks.
After letting the solvent dry for 10–15 minutes, the driving rod was pushed up from below the sample to bend it. The I–V amplifier gain was adapted to avoid overloading.
The junction was first completely broken and then reattached until the conductance was at a molecular level (0.01 G0 to 0.0001 G0). The bias voltage was then swept using a single triangle wave from 0 V with a controllable amplitude. The sweep frequency was 4 Hz and the data was collected at a sample rate of 100 kS/s. The most stable junctions were obtained when the molecule solution on the substrate had been left untouched for about 15 minutes so the solvent had evaporated, instead of measuring directly after adding the solution.
4.3 BDT
4.3.1 Transition voltage spectroscopy
Figure 4.2shows an example of unacceptable data for BDT. A general trend for all the molecules was instability of the junction, which led to transition voltages being hard to distinguish. Data that had unclear transition voltages due to jumps in conductance close to the transition point were discarded. As the positive and negative slopes were averaged, data with different current behavior between positive and negative slopes in the same bias voltage polarity were discarded. If the minimum point in the Fowler-Nordheim plot was broad, the approximate center point was recorded as data. A moving average filter was used to smooth all the data with a window of nine samples in order to decrease high frequency noise. This was performed to make the transition voltage values clearer.
Figure 4.2: Not ideal data for BDT. a) shows an I–V plot with clear jumps in current.
In the negative bias region, the curves between positive and negative sweeps cross. b) shows an FN-plot displaying uncertain transition voltages due to jumps close to the transition region. c) shows current and voltage plotted against time with clear jumps
in the current with respect to applied voltage.
Figure 4.3 shows one of the best recorded examples of a BDT molecular junction re- sponse to the applied voltage. At approximately ±0.8 V, the current response changes characteristic and the current increases more at higher voltages. The reason for the offset between positive and negative voltage slopes shown in Figure 4.3 b) is the capacitance C of the junction as described by I = C × dV /dt. Details on capacitance calculation is shown in section 3.2. Averaging the positive and negative slope currents in MATLAB gives a capacitive current IC = 6.9 nA. Since the voltage amplitude of the triangle wave is 1.5 V and the frequency is 4 Hz, dV /dt = 6/0.25 = 24 V/s. Thus, the capacitance can be calculated as C = I/ (dV /dt) = 6.9 × 10−9/24 = 290 pF. Since the capacitive current is additive to the current of interest, averaging the positive and negative slope removes the effect. The large capacitive current is the reason why a higher triangular sweep frequency than 4 Hz was not used in experiments.
Figure 4.3: a) shows current characteristic over time as voltage is swept in a triangular wave on a BDT junction. b) shows a current-voltage plot. The positive sweep gradient is above the negative sweep gradient, showing a clear capacitive effect in the junction.
The dip in the center part of the curve is due to the last values seen in plot a) in combination with the moving average filter.
The corresponding Fowler-Nordheim plot for the data in Figure 4.3 is shown in Figure 4.4. On the left side, voltage values from 0.05 V to 1.5 V are shown. This indicates that voltage values below 0.05 V are unnecessary for TVS analysis as the transition voltage occurs at a much higher voltage. There is logarithmic growth from the transition voltage to higher 1/V values and a nearly linear relationship between ln I/V2 and 1/V for low 1/V values. This is expected and explained in the Simmons model in Chapter 2.
In general, the vertical offset between negative and positive sweep shown in Figure 4.4 was found in both directions and was also absent for some junctions. The transition voltage values were found by zooming in on the minima in the FN-plot and writing down the values in an Excel file. Transition voltages could not be found when over 100
Figure 4.4: The left side shows a typical Fowler-Nordheim plot for benzenedithiol.
The blue and red lines correspond to negative and positive bias voltages, respectively.
The right side shows a zoomed picture of the transition voltages.
measurements were performed on mesitylene, the solvent used for BDT, in molecular conductance region.
In order to provide an overview of the transition voltages recorded, a 2D histogram program called Cloudplot is used. All the data used to calculate the average transition voltage for BDT is shown in Figure 4.5 a). It is shown that the transition voltages does not change with the amplitude of the current response, where current amplitude is visible as a vertical shift in the graph of ln I/V2 against 1/V . The average current for each negative and positive sweep used as data was plotted against the corresponding transition voltage in Figure4.5b) for providing a summary of the results. The transition voltage is clearly not dependent on the molecular conductance.
In Figure 4.6, histograms of the data taken for two different measurement occasions and a summary of both occasions are shown. Different junctions were used between the sessions. Average transition voltages of the separate sessions were spaced by 0.06 V, which is within one σ. It can also be clearly seen that the transition voltage spectroscopy yields a Gaussian distribution at a large amount of data.
The transition voltage value found for an average of 392 BDT measurements in this work was 0.83 V ± 0.11 V. Song et al. reported a value of 1.14 V ± 0.04 V in their 2009 paper and about 1.11 V ± 0.07 V (extracted from a diagram) in their 2011 paper, using an electromigrated breaking process [19,40]. The values found by Song’s group deviate by 0.3 V with the results found in this work. Since none of the transition voltages are in the range of vacuum tunneling junctions > 1.4 V as shown by Trouwborst et al., the only conclusion that can be drawn is that different values were found between this experimental set-up and that used by Song et al. [27].
Figure 4.5: Summary of BDT TVS. a) shows a 2D histogram, Cloudplot, for all BDT TVS measurements used for determining 392 transition voltage values. Redder colours indicate higher density of data. b) shows averaged sweep current against transition voltage for all 392 BDT measurements. The vertical axis is logarithmic for better
overview as the averaged currents are found over several orders of magnitude.
Figure 4.6: Histogram of all BDT transition voltage results. The data for the top left diagram was recorded on the 28th of May 2013. The data on the top right of the diagram was recorded on the 24th of June 2013. In the bottom summarizing figure, the
results have been automatically fitted with a normal distribution curve.
The broad range of conductance for BDT seen in Figure 4.5 is consistent with that of other groups [11, 12, 30,41,42]. Ulrich et al. have attributed this to the geometrical junction to junction variations, which has become a frequent explanation in various ar- ticles [11]. The fact that the transition voltage of BDT is not dependent on conductance shown in Figure 4.5 b), means that the electrode contact geometry does not affect the molecular orbital position. This in turn might indicate that the same type of I–V be- havior can be expected for different contact geometries. If the molecule is to be used as a transistor, the prevalent factor affecting molecular conductance only affects the magnitude of the on/off state and not the gate bias voltage switching property. How- ever, the challenge of creating junctions with similar transition voltage in order to create transistors with the same gate voltage switching still remains an unsolved problem for BDT.
4.3.2 Single level tunneling model
Equation 2.12 was used to fit a line to the experimental I–V data obtained using the curve fitting tool in MATLAB. The parameters extracted from fitting were the coupling strengths to the left and right electrodes, ΓL,R, and the difference between the electrode Fermi level and the frontier molecular orbital, ε0. The averaged transition voltages found were firstly used as an initial parameter for ε0, and it was then updated to the averaged ε0 found in the first fitting. Γ = 0.03 was used according to the value found by Matsuhita et al. for BDT [29]. Initial parameters for Γ were then updated to the averaged Γ values for faster fitting. The fitting of one data file finished when either 400 iterations had been made (case of divergence) or the fit had come as close to the result as possible within a strict function tolerance of 10−15. It was observed that the fit could converge to most of the I–V behavior seen for the molecules. An example of a good fit and a bad fit is shown in Figure4.7.
To avoid incorporating extreme values into results due to bad fits, the goodness of fit (R2) parameter was used to automatically remove fitting that did not converge to the experimental data. All fits with an R2 value lower than 0.974 were removed from the results. The few results that had a positive value for EF − ε0 were also removed since this is not the case for the molecules studied. These results were likely due to divergence of the fitting.
The 465 data files recorded for the BDT molecule were used without selection for fitting analysis. Many of these files included bad data, such as amplifier overload due to acci- dental junction reattachment during bias application. Of the 465 data files, 295 had an R2 value above 0.974 in fitting. Eleven of these fits showed a positive ε0 value and one
Figure 4.7: Fitting behavior for different curve types seen for BDT. a) shows the curve adapting well to the experimental data. b) shows the curve fit having a different behavior than experimental data. This means that the I–V behavior cannot be fitted
with the equation and given parameters.
of the 295 values did not reflect the molecular junction behavior (ε0 = −39.57). The fit was able to converge to data with considerable noise, but sometimes it diverged from smooth data. The total time taken for fitting was about 15 minutes. From the same data set, a total of 392 transition voltage values were found from an ideal maximum of 930 values (two possible values per sweep). Transition voltages had to be found man- ually as the minima were not clear and had to be estimated, which took about half a day to perform. A slight disturbance in data observed close to the transition voltage was observed to drastically change its value. Conclusively, data analysis by fitting could give more reliable results in a much shorter time compared to TVS.
Figure 4.8 a) shows a plot of averaged current against molecular frontier level found by fitting. This is shown in order to check if ε0 and conductance are connected. Since the points in the figure are randomly scattered, no correlation is found between ε0 and conductance. This is analogous to the case for transition voltages as shown in Figure 4.5. Figure 4.8 b) shows an ε0 histogram of the 283 values found from BDT data. ε0
does not show a normal distribution.
The averaged ε0 from 91 fits in the first experiment is −0.9134 eV ± 0.0977 eV. The av- eraged ε0from 192 fits in the second experiment using another substrate is −0.9001 eV±
0.0857 eV. This clearly shows that the result is reproducible. The total average found is −0.9045 eV ± 0.0896 eV. The averaged ΓL,R found are 0.0137 eV ± 0.0652 eV and 0.0076 eV±0.0059 eV respectively. The large standard deviation in ΓL,Ris due to asym- metric fits. It is shown in Figure 4.9 that ΓL ≈ ΓR ≈ 0.005 eV. Figure 4.9 b) shows a measure of the junction asymmetry by the absolute difference of the two bonding
Figure 4.8: a) shows averaged current against ε0 for BDT. The vertical axis is in logarithmic scale in order to present the data clearly. A few values are not included due to choice of axis limits. b) shows a histogram of BDT ε0 for all data recorded.
strengths. The large amount of data close to zero clearly shows that the junction was mostly symmetric.
Figure 4.9: a) shows a histogram of ΓL and ΓRfor BDT, where ΓL has been plotted on the negative energy scale for comparison. b) shows a histogram of |ΓL− ΓR| as a measure of junction asymmetry. A few outlying data points have been excluded due to
choice of axes for both histograms.
Figure 4.10 shows averaged sweep current between 0.095 V and 0.105 V bias voltage plotted against ΓL,R. The reason for continuous bond strengths may be that the param- eter Γ does not only incorporate bond strength. For example, Huisman et al. reasoned that Γ should contain a junction length dependence [20].
In order to provide an explanation for the connection between ΓL,R and current seen in Figure 4.10, it is natural to look at the transmission function 4.1used in the fitting model. The expression after dividing by 4ΓLΓR can be seen in Equation 4.2. If we
Figure 4.10: ΓL,Rplotted against averaged current between 0.095 V and 0.105 V bias for BDT. A few values are not seen in the figure due to choice of axis limits.
look at a perfectly symmetrical junction for simplicity’s sake, the expression reduces to Equation4.3. This shows that if E − ε0(V ) is not zero, a higher Γ leads to a decreasing denominator and thus a higher transmission.
T (E, V ) = 4ΓLΓR
[E − ε0(V )]2+ [ΓL+ ΓR]2 (4.1)
= 1
[E−ε0(V )]2
4ΓLΓR +[Γ4ΓL+ΓR]2
LΓR
(4.2)
= {ΓL= ΓR} = 1
[E−ε0(V )]2 4Γ2 + 1
(4.3)
T = 1
ε20
4ΓLΓR +[Γ4ΓL+ΓR]2
LΓR
(4.4)
Figure 4.11 shows the averaged current around 0.1 V bias plotted against an appar- ent transmission value. The apparent transmission values are calculated according to Equation 4.4. The relationship obtained is clearly linear. The data recorded at higher conductance is more uncertain, which is why the linear fit is weighted with the inverse of the x scale values in order to produce a valid fit.
Figure4.11has another implication; if the current at 100 mV bias of a molecular junction is known, it is possible to estimate ε0 or average Γ given that the junction is approx- imately symmetric and one of the values are known. For example, if a molecule’s I–V characteristic is first investigated by the single level tunneling model to find the average
