Multi-Scale Modelling of Electron Transport in Molecular...

Multi-Scale Modelling of Electron Transport in Molecular Devices

Hui Cao

Theoretical Chemistry

Royal Institute of Technology

Stockholm 2009

©Hui Cao, 2009 ISBN 978-91-7415-302-6 ISSN 1654-2312 TRITA-BIO Report 2009:10 Printed by Universitetsservice US-AB, Stockholm, Sweden.

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Abstract The main task of this thesis is to develop multi-scale approaches to model electron

transport in molecular device. At the single molecular level, both elastic and inelastic electron-tunneling processes have been treated simultaneously using first principles methods. By comparing with experiments, the mechanism for conductance switching observed in Pd-dithiolated oligoaniline-Pd molecular junctions has been revealed, which are found to be induced by conformation changes of the intercalated dithiolated oligoaniline in the junctions. The possible oxidation/reduction process as proposed by earlier study is ruled out. An effective approach that combines molecular dynamics simulations and first principles calculations has been developed to study statistic behavior of electron transport in electro-chemically gated molecular junctions. It has been applied to simulate conductance of a single perylene tetracarboxylic diimide (PTCDI) molecule sandwiched between two gold electrodes in aqueous solution, revealing the statistical behavior of molecular conductance in solution at different temperatures for the first time. Our calculations show that the observed temperature dependent conductance can be associated with the thermal effect on hydrogen bonding network around the molecule. Under the external gate voltage, an apparent multi-peak behavior in the statistical conductance histograms of a single molecule junction is obtained, which shows that the common practice in the experiments to relate the number of peaks to the number of molecules presented in the junction is not well defined.

4

Preface The work presented in this thesis has been carried out at Department of Theoretical Chemistry, School of Biotechnology, Royal Institute of Technology, Stockholm, Sweden.

List of papers included in the thesis

Paper I. Cao, H.; Jiang, J.; Ma, J.; Luo, Y. Temperature-Dependent Statistical

Behavior of Single Molecular Conductance in Aqueous Solution. J. Am. Chem. Soc.

2008, 130, 6674.

Paper II. Cao, H.; Jiang, J.; Ma, J.; Luo, Y. Identification of Switching Mechanism

in Molecular Junctions by Inelastic Electron Tunneling Spectroscopy. J. Phys. Chem. C,

2008, 112, 11018.

Paper III. Cao, H.; Ma, J.; Luo, Y. Statistical Behavior of Electrochemical Single

Molecular Field Effect Transistor. J. Am. Chem. Soc., submitted (2009)

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Comments on my contribution to the papers included

• I was responsible for all calculations in all papers.

• I participated in the writing and editing of all papers.

6

Acknowledgements

I would like to express my great thanks to my supervisor Prof. Yi Luo for his guidance and inspiration during the research of many interesting subjects in the field of molecular electronics. His great ideas and insight in this frontier scientific area has led me to make a significant difference in my academic work.

I express my sincere thanks to Prof. Jing Ma and Prof. Shuhua Li in China for introducing me to field of molecular dynamics and quantum chemistry. I’m thankful to their guidance and considerable care of my further research, as well as the help in my life.

I would like to thank Dr. Jun Jiang for in-depth discussions on many aspects of molecular electronics. Thanks to Bin Gao for his help in how to exploit the calculation resources and program more efficiently.

Thanks to Prof. Hans Ågren, Dr. Fahmi Himo, and Prof. Faris Gel’mukhanov, who make the research atmosphere more pleasant. Thanks to other researchers in this department for their kindness.

Thanks to my Chinese colleagues and give my best wishes to them for achieving progress in their research field.

Contents

1 Introduction ............................................................................................ 9

2 ..................................................................... 11Elastic Scattering Theory2.1 Introduction ......................................................................................................11 2.2 Molecular Devices........................................................................................... 12 2.3 Electron Transport Properties .......................................................................... 13

2.3.1 Transition Probability ............................................................................ 13 2.3.2 Electric Current...................................................................................... 17 2.3.3 Conductance .......................................................................................... 20

3 .................................................................. 21Inelastic Scattering Theory3.1 General Theory of IETS .................................................................................. 22

3.1.1 Molecular devices.................................................................................. 22 3.1.2 General Theory ...................................................................................... 22

3.2 Applications ..................................................................................................... 26 3.2.1 Identification of the switching mechanism............................................ 26

4 ................................................. 29Solvent Effect on Electron Transport4.1 Introduction ..................................................................................................... 29 4.2 Continuum Model............................................................................................ 30 4.3 Discrete Model ................................................................................................ 31

4.3.1 Theory.................................................................................................... 31 4.3.2 Information From MD Simulations....................................................... 32

4. 4 Applications .................................................................................................... 35 4.4.1 Single Molecular Conductance in Aqueous Solution ............................ 35 4.4.2 Statistical Molecular Conductance in Aqueous Solution Under the External Electrical Field. ................................................................................ 37

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8 CONTENTS

Chapter 1 Introduction

The development of conventional silicon-based microelectronic industry is restricted by the “Moore’s law”, which tells the fact that the number of transistors in a chip doubles every 18 months. When the size of semiconductor devices becomes small enough the quantum effect occurs and the conventional devices do not work efficiently any longer. In this context, the aim of molecular electronics is to construct the molecular circuit on the basis of assembling molecular wires, molecular switches, molecular rectifiers, and molecular transistors together, in order to replace the conventional semiconductor circuit. The perspective of molecular circuit is to satisfy the requirement of high response speed and high density of integration.

Although the concept of molecular electronics was introduced in the early seventies, by a theoretical study of A. Aviram and M. Ratner on the current-voltage response of a molecular rectifier,1 many fundamental aspects of molecular electronics remain obscure. Conventional metal-molecule-metal junctions comprise the molecule wired between metal electrodes. However, the buried interface between molecule and the electrode has not been effectively controlled in experiment and the experimental observations of electron transport are often controversial, even contradictory to each other. Theoretical simulations are thus very important in understanding the electron transport in molecular devices. For example, various possible mechanisms for conductance switching behavior have been proposed, including oxidation/reduction of molecules, rotation of functional groups, rotation of molecule backbones, interactions with neighbor molecules, fluctuation of bonds, and change of molecule-metal hybridization.2-7 However, the lack of a proper characterization tool to determine the exact structure of the molecule in the junction has made it difficult to distinguish different mechanisms. In this case, theoretical simulations of inelastic electron tunneling spectroscopy (IETS)8-10 has been proven to be very useful to identify not

9

10 CHAPTER 1. INTRODUCTION

only the conformation changes of the molecule in the junctions but also the exact bonding distance between the terminal atoms in the molecule and the electrodes.

Many experimental techniques have been developed to investigate the electron transport in molecules or monolayer. In these techniques, breaking junction11-13 has been widely used in studying electron transport properties of a single molecule, which can however introduce two major uncertainties in measurements, namely the structure of metal-molecule contact and solvent-molecule interaction. Therefore, the statistical average method is believed to be the most meaningful approach in studying the electron transport properties of molecular junctions at present. One way to do it is to calculate molecular conductance at all possible contact geometries by artificially moving molecule around the surface of the electrode.14 The shortage of this approach is obvious since it could either miss important configurations or include too many conformations with very low probability in the calculations. In this sense, Monte-Carlo method or Molecular Dynamics (MD) simulations are better choices in getting the samples of different equilibrium conformations.15-17 The research of temperature effect is difficult in the framework of the quantum mechanics, especially when the large number of solvent molecules exist in the molecular electron transport system. But it has been proven to be very convenient to attack this problem by combining the quantum mechanics and MD simulations. Despite the fact that the first principle molecular dynamics simulations can give more correct dynamics behavior of system, the inherent restrict of expensive computational cost determines that it cannot be applied in the large supermolecules system at the moment.

In this thesis, we use the quantum chemical methods, in particular QCME program18, to calculate electron transport properties of molecular devices. MD simulations are used to obtain the equilibrium conformations of transport system as inputs for the QCME calculations. By combining quantum mechanics in the electron scale and MD simulations in the atom/molecule scale, we can efficiently study various behaviors of complex transport system under different temperature and external electric field conditions.

Chapter 2 Elastic Scattering Theory 2.1 Introduction

Traditional electron transport investigations are largely based on the solution of

the Boltzmann transport equation.19 In this approach, the quantum mechanical effect only comes in through the calculation of band structures, which provides the input to the Boltzmann transport equation. As a result, the study of the electron transport can be decoupled from that of the electronic structures. The investigation of electron transport in molecular devices, however, is always directly coupled to the calculations of electronic structures. In fact, the study of electron transport in molecular devices is also more complicated than that in the mesoscopic system, which is highlighted by two quantum mechanical effects reflecting the wave-particle duality of electron, namely, the quantization of electronic charge as observed in the coulomb blockade and the single-electron transistors20 and the preservation of quantum phase coherence which leads to the observation of the conductance quantization in transport through a narrow constriction in the quantum point system. Compared to the mesoscopic transport, the interface between molecule and electrode must be taken explicitly into account because the experimental measurements of transport properties are not from the intercalated molecule itself but from the integral molecular device including the interface, where the atomic arrangement can play an important role in determining the electron transport of molecular devices. In this context, the extended molecule consisting of the molecule and a number of atoms in the electrodes need to be explicitly considered. Due to the fact that the charge and potential perturbation, induced by the adsorption of the molecule, are metallically screened by the electrodes

11

12 CHAPTER 2. ELASTIC SCATTERING THEORY

and extend over only a small region into the electrodes. In practical calculation, it is enough to include only those surface metal atoms closest to the molecule in the extended molecule.

Many approaches have been developed in calculating the electron transport properties of molecular devices over the years. Among them, the jellium model21-23 is an appealing approach, in which the atomic structures of the metal surface are ignored and the electrode are considered only in providing the continuous energy spectrum. In some cases, the jellium model was proven to be very useful in simulating the electron transport phenomena, such as the negative differential conductance effect. However, the jellium model has its inherent shortage in describing the electronic density of states and charge density in the molecule-electrode coupling region because it doesn’t include the detailed information of geometries of the electrode in the region perturbed by the absorbed molecule. For the same reason, it is also not applicable to describing the bonding direction between molecule and metal. Another category of theoretical method is the non-equilibrium Green’s function approach.24,25 In this method, one first gets the initial Fock matrix of the scattering region and uses it as the input to construct the Green’s function. Then, one obtains the electron density from the “lesser” Green’s function and returns it to the electronic structure-calculating program to get the new Fock matrix. This calculation procedure ceases when the quantities from different subroutines are self-consistent. Calculation with this approach is, practically, very time-consuming, especially in obtaining the I-V curves. In this context, we use the non-self-consistent procedure to calculate the electron transport properties with much higher computational speed. This approximation is acceptable when the external bias is very small.

2.2 Molecular Devices

Figure 2.1 (a) shows a typical structure of the molecular device called molecular

junction here, in which the intercalated molecule connects the source electrode and the drain electrode. In ordinary cases, the electrode used in molecular junction is metal. Because of the screening effect in metal only small part of the electrode atoms perturbed by the molecule are needed to be included in the extended molecule. Electrons are driven to pass through the scattering region by the external bias. In elastic scattering model, electron doesn’t change its energy during the scattering process. The molecular orbitals, as shown in Figure 2.1 (b) are considered as the

2.3 TRANSPORT PROPERTIES 13

scattering channel of electron tunneling through the molecular junction.

Figure 2.1 (a) Scheme of a typical molecular deivce; (b) Scheme of the alignment of energy levels of the molecule and the Fermi levels of electrodes.

2.3 Electron Transport Properties26-28

2.3.1 Transition Probability

We briefly introduce here the general elastic scattering theory for electron transport in molecular devices. The Schrödinger Equation of the molecular device is described by

| |H η ηηεΨ = Ψ (2.1)

where H is the Hamiltonian of the system, and can be written in a matrix format as

SS SM SD

MS MM MD

DS DM DD

H U UH U H U

U U H

⎛ ⎞⎜ ⎟= ⎜⎜ ⎟⎝ ⎠

⎟ (2.2)

where , SSH MMH , and DDH represent the Hamiltonian matrix of the source


electrode (S), the molecular part (M), and the drain electrode (D), respectively, and U is the interaction between different parts in the molecular junction.

The wavefucntions can also be partitioned on the basis of the subsystems as following,

,

,

,

, , ,

, ,

, ,

, ,

| | | |

| | ( | )

| | ( | )

| | ( | )

N NS

N NM

N ND

S M D

J JS S J J

i i i iJ i J

K KM M K K

i i i iK i K

L LD D L L

i i i iL i L

a a

a a

a a

η

η

η

η η η η

η η |

|

|

J

K

L

η

η η

η η

φ φ

φ φ

φ φ

Ψ = Ψ + Ψ + Ψ

Ψ = = =

Ψ = = =

Ψ = = =

∑ ∑ ∑ ∑

∑ ∑ ∑ ∑

∑ ∑ ∑ ∑

η

η

(2.3)

where and are the wave function and basis function of subsystems S,

D and M, respectively. Here J, K, and L runs over the atomic sites in the molecule.

, ,S D MΨ , ,S D Miφ

The interaction at energy level can be written in the atomic site representation as

, ,

' '', , '

' '', , '

, ,

| | | |

| ' | | ' |

| ' | | ' |

| | | |

JK KJJ K K J

K L LKK L L K

J J LLJ J L L

JL LJJ L L J

U V J K V K J

V K L V L K

V J J V L L

V J L V L J

η η η η

η

η

η η η

ηη η

ηη η

= +

= +

= +

= +

∑ ∑

∑ ∑

∑ ∑

∑ ∑

η

η

(2.4)

where VAB represents the coupling energy between the layer sites A and B, which can be calculate analytically with quantum chemistry methods using the following expression


0

,

| | | |i i i i

i i

CC OCC

AB A B A BA B

V A H B a a Hν ν ν ν

ν ν

φ φ= =∑ ∑ ∑ (2.5)

where ,| |i iA B AH Fφ φ =

i iB is the interaction energy between two atomic basis functions. Based on elastic-scattering Green’s function theory, the transition operator is defined as

T U UGU= + (2.6)

where G is the Green’s function,

1( ) ( )G z z H −= − (2.7)

For an electron scattering from the initial sites | iξ∑ of reservoirs S to the final sites '| mξ∑ of reservoirs D (with i and j running over the atomic site of the source and the drain electrode, respectively), the transition matrix element at a certain energy level will be

' ''

, ,

| | | |m i mi m i m

T U UGUηξ ξ iξ ξ ξ= +∑ ∑ ξ (2.8)

By substituting Uη of Eq. (2.4) into Eq. (2.8) and ignoring the direct coupling between two reservoirs, we get

' ''

' '' ', , ' , ,, ',

+ +i im m

m m

K K K LK K K J JK Li m K K i m i mL K K J

T V g V V g V V gη η η' ' imK

Vηξ ξ ξ ξξ ξ ξ

ξ ξ≠ ≠

=∑∑ ∑ ∑ ∑ ∑ ξ (2.9)

where 'K Kgη is the carrier-conduction contribution from the scattering channel, which


can be expressed as

'1' | |

' | |

K Kg K Kz H

k Kz

η η η η

η η η η

ηε

= Ψ Ψ−

Ψ Ψ=

−

η

i

(2.10)

where parameter z in the Green’s function is a complex variable, iz E i= + Γ , and

is the energy at which the scattering process is observed. Due to the energy conservation rule, the incoming and outgoing electrons should have the same energy, i.e. belong to the same orbital. Assuming an elastic scattering process, Ei equals the energy of the tunneling electron when it enters the scattering region from the reservoir S, as well as the energy at which the electron is collected at time

iE

+∞ by the reservoir D. 1/ escape rate, which is determined by the Fermi Golden rule iΓ

2 2' ' '

2

( ) | ' | |

+ ( ) | | |

SK K f K

Df K

n E V K

n E V K

η ηξ

η ηξ

π

π

Γ = Ψ

Ψ 2

η

) )

(2.11)

where and are the density of states (DOS) of the source and the drain at the Fermi level E

(Sfn E (D

fn E

f, respectively. Hence we obtain

'

' | |( )K K

i

K Kg

E i

η η η ηη

η ηε

Ψ Ψ=

− + Γ (2.12)

From calculations based on the local density approximation (LDA),29,30 it is

known that the metal atomic orbital is much more localized than the molecular counterparts. So that | |L Kη η η ηΨ Ψ and ' | |K Jη η η ηΨ Ψ should be quite small. Therefore the terms including LKgη and 'K Jgη in Eq. (2.9) can be neglected. Actually, the localized properties of the metal orbitals is reflected by the fact that the


potential of metal-molecule-metal configuration drops mostly at the metal-molecule interface.31 The transition probability can finally be written as

'2

'', , '

| ( ) |im

K K KKi m K K

T V g Vηξξ

η

= ∑ ∑∑ (2.13)

2.3.2 Electric Current

Electric current through the molecular junction under the external bias can be computed by integrating the transition probability over all energy states in the reservoir. It is assumed that the molecule is arranged along the z direction, which is also the direction of the electric current. In the effective mass approximation, energy states in the conduction band of the reservoir can be expressed as the summation,

, where is the conduction band edge and is used as the energy reference. It is assumed that the parabolic dispersion relation holds for the energy states in metal. The electrons in the reservoir are assumed to be aa in equilibrium at a temperature T and Fermi level E

,x y z cE E E E= + + cE

f. When an applied voltage V is introduced, the tunneling current density from the source (S) electrode to the drain (D) electrode can be described according to the Landaur formulism32, as

( ) (

( )'

,

', ,

,

''

2

l lx y z z

l lSD x y z x y z

E E E

l lll z z

ei f E E eV f E

T E E

)Eπ

δ

⎡ ⎤= + − −⎣ ⎦

× −

∑ ∑h

+ (2.14)

where ( )f E is the Fermi distribution function,

( ) /1( )

1f BE E k Tf E

e −=

+ (2.15)


Here is the Boltzmann constant, T the temperature, the transition probability describing

the scattering process from the initial state Bk 'l lT

| l to the final state | 'l . The transition probability is

a function of the quantized injection energies along the z axis, and . lzE 'l

zE

Practically, the electric current density can be discussed in three cases according to the

dimensionality of the electrode, and different working formula for the current density can be

derived. When the electrode is made of an atomic metal wire, it can be treated as an one-dimensional electron reservoir, and the current density through the molecular junction can be simplified as

[ ]

' '1 '

, '

'0

2 ( ) ( ) ( )

2 ( ) ( ) ( ) ( )

l l lD l

l l

S Dl l

ei f E eV f E T E E

e

ll

f E eV f E T n E n E dE

π δ

π ∞

⎡ ⎤= − − −⎣ ⎦

= − −

∑

∫

h

h

(2.16)

where and are the density of states of the source electrode and the drain electrode, respectively.

( )Sn E ( )Dn E

When the metal electrode has the character of a two-dimensional electron system,

for instance, a metal film, and if the energy in the x direction forms a continuous spectrum, the current density should be expressed as

'2 0 0

1 ' 1 1

2 ( ) (

( ) ( ) ( )

l lD x z

S DD x l l D z D z z

ei f E E eV f E

E T n E n E dE

)x zEπ

ρ

∞ ∞⎡ ⎤= + − − +⎣ ⎦

×

∫ ∫h (2.17)

where 1 ( )D xEρ is the density of states per length per electron volt of the source.

When energy distributions in both x and y directions are continuous, the current

density can be evaluated by


*

3 ' 1 13 0( ) ( )

2

ln 1 ln 1f z f z

B B

S DBD l l D z D z

E eV E E Ek T k T

em k Ti T n E n E dE

e e

π∞

+ − −

=

⎡ ⎤⎛ ⎞ ⎛⎢ ⎥⎜ ⎟ ⎜× + − +

⎜ ⎟ ⎜⎢ ⎥⎝ ⎠ ⎝⎣ ⎦

∫hz

⎞⎟⎟⎠

(2.18)

where m* is the electron mass. For one-dimensional electrode, the tunneling current equals to the current density through the molecular junctions

1 1D DI i= (2.19)

In the case of the two-dimensional electrode, the current flowing in the molecular junction can be written as

2 2 2D s DI r i= (2.20)

where r2s is the effective injection length of the transmitting electron and determined by the density of electrons following this relation, 1/ 2

2 2[1 ( )]s dr N π≈ and the density of electrons can be calculated as ( )* 2

2 4 /D fN m Eπ= h

3

. For a three-dimensional electrode system, the total current can be described as

3D DI Ai= (2.21)

where A is the effective injection area of the tunneling electron at the metal electrode, determined by the density of electronic states of the bulk metal. We have assumed that the effective injection 2

3sA rπ≈ , where r3s is defined as the radius of a sphere whose volume equals to that of a conduction electron, , where ( 1/3

3 3 / 4sr Nπ= )3D

( ) ( )3/ 2*3 2 / 3D fN m E 3 2π= h is the density of electronic states of the bulk metal.


2.3.3 Conductance

The differential conductance of the molecular junction when the conduction electron tunneling under the bias can be finally written as

IgV∂

=∂

(2.22)

Chapter 3 Inelastic Scattering Theory

A typical inelastic electron tunneling spectroscopy (IETS) is shown in Figure 3.1. It reflects the contribution of the electronic-vibronic coupling effect to the current-voltage characteristics. Each peak is corresponding to one vibrational mode of the molecule intercalated in the molecular junction.

Figure 3.1 A typical inelastic electron tunneling spectroscopy (IETS) in which each peak corresponds to a vibrational mode of the molecule inside a junction.

IETS has become an effective tool in investigating the chemical bonding between buried molecule and the electrode and in identifying the mechanism of conductance switching, etc. Hence, the investigation of IETS has great significance both in

21

22 CHAPTER 3. INELASTIC SCATTERING THEORY

experiments and in theoretical study.

3.1 General Theory of IETS

3.1.1 Molecular devices

The conformation of the molecular junction discussed in the inelastic electron scattering theory is exactly the same as that in the elastic electron scattering theory as shown in Figure 3.2 (a). The difference lies in that the vibronic structures (shown in Figure 3.2 (b)) of the molecule are involved here.

Figure 3.2 (a) Scheme of a molecular device; (b) Scheme of a molecular device containing vibrational energy levels of the molecule and the Fermi levels of electrodes.

3.1.2 General Theory28,33

Based on the adiabatic Born-Oppenheimer approximation, the purely electronic

Hamiltonian of the molecular junction can be described parametrically as the function of the vibrational normal modes Q, and the one-electron Hamiltonian is finally partitioned as

3.1 GENERAL THEORY 23

( ) ( , ) ( )H Q H Q e H Qν= + (3.1)

where and are the electronic and the vibrational Hamiltonian, respectively. The Schrödinger equation now becomes

( , )H Q e ( )H Qν

( , ) ( ) | ( , ) | ( )

( , ) | ( , ) | ( ) ( ) | ( , ) | ( )

( ) | ( , ) | ( )a aa

H Q e H Q Q e Q

H Q e Q e Q H Q Q e Q

n Q e Q

ν η ν

η ν ν η ν

ν η νηε ω

⎡ ⎤+ Ψ Ψ⎣ ⎦

= Ψ Ψ + Ψ Ψ

= + Ψ Ψ∑ h

(3.2)

where ηε represents the energy of the eigenstate, η, of the pure electronic Hamiltonian, aω and aωh are the vibrational frequency and energy of vibrational normal mode , respectively, and aQ anν the quantum number for the mode in aQ| ( )QνΨ .

The nuclear motion dependent wavefunction can be expanded along each vibrational normal mode by using the Taylor expansion as

00, 0 , 0| ( , ) | ( ) | ... | ( )Q a Q

a a

Q e Q Q QQ

ηη ν η ν

= =

∂ΨΨ Ψ = Ψ + + Ψ

∂∑ (3.3)

where | ( )QνΨ is the vibration wavefunction, 0| ηΨ the intrinsic electronic wavefunction at the equilibrium position, 0Q= . In the adiabatic approximation, we can use the first derivative like ( ) / aQ Q∂Ψ ∂ to represent the vibrational motion part in the above expansion.33

In the atomic site representative, the wavefunction relating to the vibrational mode

can be partitioned into three parts, corresponding to source electrode, the drain electrode, and the molecule respectively.


( ), , ,

,0, 0

, 00, 0 , 0

,

| ( , ) | ( ) | ( , ) | ( , ) | ( , ) | ( )

| ( , ) | ( , ) |

| ( , ) | ... (3.4)

| (

N N

N

S M D

J JS

QJ J

KM

Q a QK a a

D

Q e Q Q e Q e Q e Q

Q e J Q e J

KQ e K QQ

Q

η ν η η η ν

η η η

ηη η

η

=

= =

Ψ Ψ = Ψ + Ψ + Ψ Ψ

Ψ = ≈

∂Ψ = + +

∂

Ψ

∑ ∑

∑ ∑

0, 0, ) | ( , ) |N NL L

QL L

e L Q e Lη η== ≈∑ ∑

In analogous to the discussion in Chapter 2, the electron transmission probability

amplitude can now be calculated following

' ' ' ' | 0 ' |, '

' ,

', , ''

, ''

, ', , ''' ' | 0 ' | 0 '

, ' ', , ''

( ) ( )

1 ( ) | ' ( , ) | | ( )( , )

| ( , ) | ( )

( ) ( )

J L J K Q KL Q OK K

J K Q KL Q KKK K

T V Q V Q

Q K Q e Qz H Q e

K Q e Q

V Q V Q g

η

ν η η ν

ν ν ν η

η ν η ν

η ν ν ν

ν ν ν

= =

= =

=

× Ψ Ψ−

× Ψ Ψ

=

∑

∑

∑ ∑

''

(3.5)

where , ', , '''KKgη ν ν ν is given by

, ', , '' ' , '''

' , ''

''

1( ) | ' ( , ) | | ( )( , )

( , ) | ( , ) | ( )

( ) | ' ( , ) | ( )

( , ) | ( , ) | ( ) /( )

KK

a aa

g Q K Q e Qz H Q e

Q e K Q e Q

Q K Q e Q

Q e K Q e Q z n

η ν ν ν ν η η ν

η

η η ν

ν η η ν

η η ν νη ηε ω

= Ψ Ψ−

× Ψ Ψ

= Ψ Ψ

× Ψ Ψ − −∑ h

(3.5)

3.1 GENERAL THEORY 25

After the Taylor expansion, one can get

, ', , '' ' '' ' ''0 0' 0''

'' ''0 00 0

'1 ' | |

| |

KK a aa aa a a a

a

a aa aa a

Kg K Qz n Q Q

KK Q QQ Q

η ηη ν ν ν η ν ν ν ν η

νη η

η ηη ν ν ν ν η

ε ω⎡ ⎤∂Ψ ∂

= +⎢ ⎥− − ∂ ∂⎣ ⎦

⎡ ⎤∂Ψ ∂× + Ψ⎢ ⎥∂ ∂⎣ ⎦

∑ ∑∑

∑ ∑

h0Q Ψ

(3.6)

With the assumption that the nuclear motion is harmonic, we get

' '' ' | | '' 0 | |12a a a

a

Q Q Qν ν ν νω

= = =h

(3.7)

Thus we get

, ', , ''' ''

', , ''

0 00

0 00 0

12

' ' | |

| |

KKa a a a

a

a a

a a

gn

KKQ Q

KKQ Q

η ν ν νν

ν ν ν η

η ηη η

η ηη η

ε ω ω= ×

−

⎡ ⎤∂Ψ ∂× +⎢ ⎥∂ ∂⎣ ⎦⎡ ⎤∂Ψ ∂

× +⎢ ⎥∂ ∂⎣ ⎦

∑ ∑ ∑h

h

0Ψ

Ψ

(3.8)

From the above equation, one can see that the introduction of the vibrational mode into the Hamiltonian and the wavefunction of the system will contribute the inelastic scattering effect to the electron tunneling current in the non-resonant region. Therefore, the total current is the sum of elastic and inelastic contribution

el inelI I I= + (3.9)

Due to the fact that the inelastic contribution to the total current is rather small, the


IETS is described by the second derivative of the current, or the part normalized by the differential conductance

2 /d I dV 2

)2 2( / ) /( /d I dV dI dV

3.2 Applications

3.2.1 Identification of the switching mechanism34

Recently, Cai et al. observed a switching behavior between two bistable conductance states in the in-wire junctions of dithiolated N-methyl-oligoaniline dimer.35 For this bistable switching, a possible mechanism related to the charging effect had been proposed, which was later challenged by the mechanism of the change of molecular confirmation between two stable conjugated structures of the oligoaniline dimer.36 One can thus hope that a comparison between theoretical and experimental IETS spectra should lead to a definitive conclusion on the switching mechanism.

Figure 3.3. Structures of Pd-dithiolated oligoaniline dimer-Pd junctions with three different conjugated structures: (A) α(PN-NP) (both N-CH3 bonds are coplanar with the outer phenyl rings), (B) β(NPN) (both N-CH3 bonds are coplanar with the inner phenyl ring), and (C) γ(PN-PN) (one N-CH3 bond is coplanar with the outer phenyl ring and another is coplanar with the inner phenyl ring). It is noted that the oligoaniline dimer has three different isomers with distinct conjugations, whose structures are illustrated in Figure 3.3. We have named the three

3.2 APPLICATION 27

isomers as α(PN-NP), β(NPN), and γ(PN-PN) conjugations. Inelastic electron tunneling properties for all three conjugations have been calculated using the QCME program. Geometries and electronic structures of isolated diothiolated oligoaniline dimer in the gas phase have been optimized using the Gaussian03 program package37 at the hybrid B3LYP functional38 level with the 6-31G(d) basis set and the LanL2DZ pseudo potential basis set being applied to nonmetal elements and Pd, respectively. It is assumed that the S atoms are placed on the top of the center of three Pd atoms in a Pd (111) plane.

0.08 0.10 0.12 0.14 0.16 0.18 0.20

Exp. IETS(upper) Theo. IETS(lower) El-Ph Coupling (vertical)

d2 I/dV2 (a

rb. u

nits

)

Voltage (V)

12 4

8 910

1112,13 14

6

(A)

0.08 0.10 0.12 0.14 0.16 0.18 0.20

Exp. IETS(upper) Theo. IETS(lower) El-Ph Coupling vertical)

d2 I/dV2 (a

rb. u

nits

)

Voltage (V)

1 2

34 5 6

7

8 9 10

11

1213

14(B) Figure 3.4. Calculated IETS spectra for molecular junctions of (A) α(PN-NP) and (B) conjugation (lower curves) together with the experimental IETS spectra (upper curves).

Our calculations have found that the calculated IETS spectra for α(PN-NP) and β(NPN) conjugations are indeed in good agreement with the experimental spectra of low and high conductance states, respectively. Figure 3.4A presents the calculated IETS spectrum for the junction of α(PN-NP) conjugation with an electrode gap distance of 19.90 Å, together with the experimental spectrum of the low conductance state at a temperature of 10 K for comparison. The calculated IETS spectrum of β(NPN) conjugation resembles the experimental spectrum of the high conductance state very well, as nicely demonstrated in Figure 3.4B. We have also calculated the IETS spectrum of the positively charged (+1) molecule of α(PN-NP) conjugation to examine the possible oxidation effect. The calculated spectrum for the oxidation state differs significantly from that of the experimental spectrum of the high current state.

We have also adopted a model, similar to what was suggested by Ke et al.,39 by


putting one additional Pd atom on top of each triangle Pd cluster, which is directly connected to the molecule. It has further confirmed that IETS spectra are indeed sensitive to the change of the bonding configurations at the molecule-electrode interface as observed in our previous study.

Chapter 4 Solvent Effect on Electron Transport 4.1 Introduction

The environment plays an important role in the determination of the properties of

substances in condensed phases. The chemical and physical phenomena can change greatly when they are observed in solution. Traditionally, solvent refers to the substance that is liquid under the conditions of application, in which other substances can be dissolved in it. Water is the most ordinary solvent in the natural world, and has been extensively studied. Liquid water has a complex structure due to the ability of the molecule to act as both hydrogen-bond donor and acceptor. Hydrogen bond between water molecules is very strong, which is demonstrated by its experimental gas phase dimerization enthalpy of –3.6±0.5 kcal/mol.40 The formation of an extended, dynamic hydrogen-bonded network stems from the hydrogen-bond interactions in liquid water.41 Qualitative knowledge of the solvent effect can be obtained from the empirical approaches based on specific properties of the solute and solvents. The interaction between solvent molecules can be researched by means of the observation of properties such as the level of structure, polarity or softness, electron pair and hydrogen-bond donor/acceptor ability, polarizability, acidity/basicity, and hydrophobicity/hydrophilicity, etc.42-53

Often, the solvent molecules included in the first solvent shell of the solute plays

the key role in determining the properties of the solvent/solute system. Therefore, large amount of works are focus on these part of solvent molecules.54-64 For a given nuclear conformation, the transfer of the solute from the gas phase to the solute changes the

29

30 CHAPTER 4. SOLVENT EFFECT ON ELECTRON TRANSPORT

electron distribution of the solute, and consequently alters its chemical properties. This change includes diverse properties, such as the lengthening in the dipole moment, the change in the molecular electrostatic potential, the variation in the molecular volume, and even the spin density. For example, the dipole moment of water changes from 1.885 D for an isolated water molecule65 to 2.4—2.6 D in the condensed phase,66 which reflects clearly the extent of the electronic polarization effect. The enhancement of the dipole moment upon solvation process has been estimated to be twenty to thirty percent of the gas-phase values for neutral solutes in aqueous solution. In the push-pull π-conjugated molecule, there are several works, reporting the significant solvent-induced charge redistributions.67 The solvent-induced polarization can even not be neglected in less polar solvents such as chloroform, as shown by the dipole moment increases of 8—10% which have been determined for neutral molecules.68

We will briefly discuss two commonly used theoretical methods for the

description of solvent effects in molecular systems, namely the continuum model and the discrete model.

4.2 Continuum Model

The electrostatic interaction between a solute and its surrounding solvent

molecules depends sensitively upon the charge distribution and the polarizability of the solute. The polarization effect is important due to the fact that the solute and solvent relax self-consistently to each other’s presence during the solvation process. The critical physical concept of continuum model is to consider the solute as distributed in continuum solvent, in which the electric field that the solute has polarized in turn exerts on the solute. The mostly applied approach is in the framework of Polarized Continuum Model (PCM), developed by the Pisa group of Tomasi and co-workers.69 Three main concepts are involved in the continuum model, named, cavity formation, dispersion-repulsion, and electrostatic interaction. Enhancement in energy of system during the process of getting out a cavity to accommodate the solute, called the cavity formation energy. Energy decreases due to the interaction between solute in the cavity and the surrounding solvent, called the dispersion-repulsion energy. Energy drops because of the interaction between charge distribution in solute and the polarized charge distribution in solvent, called the electrostatic energy. The energy summation of three parts mentioned above is the free energy of solvation.

4.3. DISCRETE MODEL 31

The original method of PCM is called DPCM. And the developed PCM include IPCM, SCIPCM, CPCM, and IEFPCM.70,71

4.3 Discrete Model

4.3.1 Theory

In this thesis, we focus on the discrete model. The most direct way to simulate the solvent effect is to surround the solute with a large number of solvent molecules, which are represented at the same level of atomic detail as the solute. To fulfill this, one can apply molecular dynamics simulations to get the dynamic nature of the solvated system, which is especially useful in taking into account the temperature effects. In studying the electron transport properties of the solvated molecular junction, the combination of molecular dynamics ensemble statistics approach with the quantum mechanics calculation of the electronic structure is believed to be the most meaningful method.

What is the most important in the MD simulation is the choice of force fields, which are parameterized to describe the molecular interactions. In the traditional force field, the total energy of system is constructed as the summation of the bonded terms, which account for changes in the potential energy resulting from the modification of bond lengths (stretching), angles (bending), and dihedrals (proper and improper torsions), and the nonbonded terms that account for electrostatic and van der Waals interactions between atoms, as following72

pot str bend tor itor VW eleE E E E E E E= + + + + +

( )21 0str

strE K L L= −∑


( )20bend b

bendE K= Θ−Θ∑

( )1 cos2n

tortor n

VE n α= + Φ −⎡ ⎤⎣ ⎦∑∑

( )1 cos 22itor

itortor

VE = − Φ⎡ ⎤⎣ ⎦∑

12 6 12 6, ,

ij ijKI KIVW VW

K I i jKI KI ij ij

A BA BEr r r r

ζ⎡ ⎤⎛ ⎞ ⎛ ⎞⎡ ⎤⎛ ⎞ ⎛ ⎞

= − + −⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎝ ⎠ ⎝ ⎠⎣ ⎦∑ ∑

, ,

i jK Iele ele

K I i jKI ij

Q QQ QEr r

ζ= +∑ ∑

where K1 and Kb are stretching and bending force constants, L0 andΘ0 are equilibrium lengths and angles, Φ represents the dihedral (proper or improper) angles, α is the phase angle, n is the periodicity of the Fourier term, Vn is the proper torsional barrier for the nth Fourier term, and Vitor is the improper torsional barrier. A and B stand for van der Waals parameters, Q are charges, rij are interatomic distances, and ζ is the scaling factor between 1 and 4.

After the calculation of potential energy, Boltzmann samplings are carried out using Newtonian molecular dynamics.73 And these samplings can be used as the input for the next quantum mechanics calculations to get the corresponding electronic structure.

4.3.2 Information From MD Simulations

4.3. DISCRETE MODEL 33

4.3.2.1 Average Structure Information

Dynamics trajectories analysis provides the time-averaged configuration of the solute when the degrees of freedom of solute are not restricted in the MD simulations. Samplings of the dynamics trajectories from the nonequilibrium processes such as protein folding and unfolding can tell us the time evolution of the protein system.74

4.3.2.2 Solute Conformational Flexibility

In solution the macromolecules are flexible and their special functional role is in turn dependent on these structural fluctuations. Insight into concepts such as “preorganization”, “rigidity”, and “entropy trapping” can be obtained from the computation of the entropy difference between two stable states of the studied macromolecules. Identification of the most important movements in the macromolecules can also be done from the principal component analysis.75-77

4.3.2.3 Solvent Structure

Ordinarily, the solvent structure can be researched by the analysis of radial distribution functions (eq 4.8) and the spatial distribution functions (eq 4.9).78-84 In the case of small spherical solutes, we use the radial distribution functions while for the macromolecules the spatial distribution functions are often applied in practice.

2

( , )( )

4y

y

N r r drg r

r drπρ

+=

( , , )( , , ) y

i j k y

N i j kdf i j k

l l l ρ=

where Ny represents the number of solvent molecules included in the spherical layer located between the distances r and r+dr from the solute, ρy the density of the pure solvent, and i, j, k the grid element of dimensions li, lj, and lk.


From the analysis of the distribution functions, we can get the detailed information of the solvent structure around the solute, especially the solvent distribution in the first solvation shell. For solvent molecules seen by the solute, they are of the characteristic of the short-range order and the long-range disorder. These solvent molecules in the first solvation shell play a key role in changing the charge distributions in the solute when the solvent and/or the solute are polar molecules. We can also estimate the solvation free energies from the analysis of distribution functions. Integration of the spatial distribution functions can provide the preferential solvation free energy of a particular site on the macromolecules (eq 4.10).

, ,

ln( ( , , ))state

soli j k

G RT df i j k∆ = − ∑

where the sum runs over all the grid elements that define the state of interest. 4.3.2.4 Energy Analysis

The total energy of a solvated system has remarkable fluctuation during the trajectory. In practice, long MD simulations can reduce most of the statistical noise generated from the solvent-solvent interactions, and MD-averaged values for solute-solute and solute-solvent interactions to the moderate degree. One can then define a “pseudo-energy” function by postprocessing and averaging the MD trajectrories, as shown in eq 4.11, where the solvation free energy can be calculated from the entire trajectory using the continuum models. This approach has been exploited to rationalize the differences in the stability of different conformations of the solute of macromolecule.85 Shortcomings of this approach lies in (1) the nonnegligible noise in the averages and (2) the difficulty in combing discrete solute-solute interactions with the continuum solvation calculations.

force fieldsolute solute solE E G−

−= + ∆

4.4. APPLICATIONS 35

4. 4 Applications

4.4.1 Single Molecular Conductance in Aqueous Solution15

The experimentally determined single molecular conductance is often obtained

from a statistical analysis of a large number of measurements. However, all theoretical analyses have ignored the intrinsic contribution from thermal motion of the wired molecule, as well as the surrounding solvent molecules, to the statistical behavior of the conductance. Very recently, Li et al. observed that electron transport in perylene tetracarboxylic diimides (PTCDI), covalently bound to two gold electrodes via different linker group, depends on the temperature in the aqueous electrolyte but is independent of temperature in a nonpolar solvent. A two-step thermally activated electron transfer process involving reduction-oxidation of the wired molecule was suggested, but it failed to explain the gate-controlled experiments.86

One hypothesis is that the strength and distribution of the hydrogen bond network is temperature-dependent, which could in turn result in the temperature-dependent electron transport. To verify this hypothesis, we have combined electron transport calculations with molecular dynamics simulations fro the PTCDI molecular junction in water solution. The schematic drawing of the hydrogen bond network is given in Figure 4.1. It is noted that the PTCDI molecule possesses four oxygen atoms that are ready to form hydrogen bonds with the surrounding water molecules.

Figure 4.1 Scheme of the studied molecular junction in hydrogen bond network.


Molecular dynamics simulations have been carried out for the system consisting of the Au-PTCDI-Au molecular junction and 800 water molecules in a 35.00 Å × 20.00 Å × 32.55 Å cell using the pcff force field in the Cerius2 package87 at 298 and 308 K, respectively. In the scattering region of the molecular junction, terminal sulfur atoms are placed at the hollow site of three gold atoms in the fcc (111) plane, and the S-Au distance is set to 2.85 Å. Electronic structure of each junction-water supermolecular cluster is calculated using density functional theory at the B3LYP level with LanL2DZ basis set as implemented in the Gaussian03 package.

The envelopes of the histograms are plotted in Figure 4.2. It can be clearly seen that the conductance at 308 K is about 1.7 times larger than that at 298 K. This finding is in good agreement with the experimental observation. An important message coming out from our calculations is that even within the one-step tunneling model (elastic scattering region), with the inclusion of thermal motion of water molecules, one can also lead to the conclusion that the conductance of a molecular junction becomes larger with the increase of the temperature.

0 5 10 15 200

5

10

15

20

25

30 298 K 308 K

Cou

nts

Conductance (10-3nS)1

2 3 4 5 6 7 8 90.0

0.5

1.0

298K308K

ρ

R (angstrom)

Figure 4.2. Statistical distributions of conductance and radial distribution functions (inset) of water surrounding oxygen atom in PTCDI at 298K and 308K.

The inset of Figure 4.2 shows the radial distribution functions (RDF) of oxygen-oxygen distance between water molecules and the PTCDI molecule at 298 and 308 K, respectively. Apparently water molecules accounted by this distribution are the ones that have strongest interaction with the PTCDI molecule through hydrogen bonding networks. It is probably not surprised to see that the conductance distributions are closely associated with the RDFs of the first solvation shell. The maximum of the RDF at 298K is found to be at 3.69 Å, which is about 0.18Å shorter


than that at 308K. It implies that the water molecules are closely packed around the oxygen atoms of PTCDI molecule at lower temperature, which results in stronger charge localization on oxygen atoms and reduced conductance of the molecule. At higher temperature, the hydrogen bond network is much looser than that at lower temperature, which explains the observation that the conductance distribution of molecule at 308K is broader than that at 298K. By analysis of RDFs, we can also estimate number of water molecules within the first solvation shell that can strongly affect the PTCDI molecule. At both temperatures, the width of the first solvation shell is around 5.3 Å, in which 14 water molecules can be found.

Another important result of the present work is the actual shape of the conductance histogram. In almost all experiments, a Gaussian distribution has been adopted to describe experimental histograms, which is a matter of convenience rather than correctness. Our study presents the first evidence to demonstrate that the use of a Gaussian distribution is absolutely not justified. As clearly shown in Figure 4.2, there is a threshold for the distribution of the conductance. In the case of PTCDI molecular junction in water solution, one cannot find any configurations that result in conductance below 1 (10-3nS). Such a threshold value reflects a simple fact that the distance between the PTCDI and the water molecules can not be infinitely small, i.e. there is also a threshold for the intermolecular distance, which according to the RDFs, should be around 2.5 Å for the O-O distance between water and PTCDI molecules. It is noted that normally the RDF can also be associated with the profile of interaction energy between the wired molecule and the water molecules. In another word, an experimental conductance histogram could thus be used to extrapolate the interaction energy profile between the molecule and the surroundings.

4.4.2 Statistical Molecular Conductance in Aqueous Solution Under

the External Electrical Field.86,88

Under the external electric field, the polar solvent molecules were expected to

rearrange according to the direction of electric field and this change in structure could in turn affect the statistical behavior. For an extensive theoretical simulation, it is important to investigate the influence of solvent, temperature, and electric field on the statistical behavior at the same time in order to give a comprehensive picture in understanding the electron transport in single molecular junctions.


We have carried out molecular dynamics simulations with inclusion of different electric fields, vertical to the π electron conjugation plane, on the PTCDI single molecular junction in water at the temperature of 298 and 308 K, respectively. Figure 4.4 shows the snapshot of the molecular dynamics simulations under the electric field of zero and 3.5e9 V/m, respectively. As expected, the surrounding water molecules change from unordered to the ordered arrangement under the external electric field. The original short-range ordered and long-range unordered microscopic distribution behavior of water molecule is ruined. The dipole moment of the whole system, which is anti-paralleled to the external electric field, increases from zero to about 1870 Debye. The drastic response of water molecules to the external electric field indicates again that the motion of polar solvent molecules is easier to be influenced by the environmental factors, and consequently, the interaction between polar solvent molecules and the polar group, such as the oxygen atoms in PTCDI molecule changes greatly. Figure 4.4 Rearrangement of water molecules around PTCDI after the gate electric field imposed (C). Structure of PTCDI (A) and the scheme of the electrochemical field effect transistor (B) are also depicted.

The statistical behavior of conductance of PTCDI molecular junction with a small bias of 0.1 V under different gate electric field is depicted in Figure 4.5. The statistical behavior of conductance doesn’t obey the Gaussian distribution as we found before. With the increase of the electric field, there is a tendency of the occurrence of multi-peaks in the statistical histograms. In the case of electric field of 3.5e9 V/m at 308 K, one can find that the statistical histogram can be fit with three curves with the


peaks at 3.6, 9.1, and 15.4 (10-3 nS), respectively (Figure 4.5 B). Our results have clearly shown that even with a single molecule, the effect of surrounding can also introduce multi-peaks distribution. It thus calls for the re-examination of existing experimental results and caution for the future studies.

The increase of gate electric field has direct consequence to the molecular conductance. It has shifted the entire distribution of conductance up to the higher values, as clearly shown in Figure 4.5 A-B. There is an indirect contribution from the increased electric field: ordering of surrounding water molecules. The degree of molecular ordering can be partially reflected by the distribution of total dipole moment of the simulated system as given in Figures 4.5 C-2D. It can be seen that larger gate electric field leads to much narrow distribution with bigger dipole moment. The multi-peaks behavior of dipole moment distribution also reflects the fact that the water molecules are not ordered uniformly and the dynamic fluctuation is presented. One could associate the fluctuation of molecular ordering with multi-peaks statistical behavior of molecular conductance.

(D) (B)

(C) (A)

Figure 4.5. Statistic distribution of conductance for PTCDI molecular junction in aqueous solution at 308 K under the external gate electric fields of (A) 2.5e9 V/m, (B) 2.5e9 V/m. Statistic distribution of dipole moment of water molecules under electric field of (C) 2.5e9 V/m, (D) 3.5e9 V/m.


Two of major experimental observations86 can also be well explained by the statistic model used here. Our calculations reproduce the experiment that under the field of 5.0e9 V/m, the molecular conductance has three orders of magnitude increase in comparison with those under the electric fields of 2.5 e9 V/m and 3.5e9 V/m. The sudden change of the conductance is resulted from big changes of molecular electronic structures induced by the strong polarization of external field that push many conducting orbitals into the transport window89. It was also shown in the experiment that under the gate electric field, the conductance is less sensitive on the temperature with increased gate voltage88. This observation is well reproduced by our simulations. This behavior can also be related to ordering of water molecules. Under larger field, the water molecules are well ordered and the thermal fluctuation can thus be largely depressed. This viewpoint is supported by the difference between the calculated maximal total dipole moment of surrounding water molecules at different temperatures, which is found to be 6.2, 2.5 and 1.1 Debye under electric fields of 2.0e9, 3.5e9 and 5.0e9V/m, respectively.

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