Multi-Scale Modelling of Electron Transport in Molecular Devices

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Multi-Scale Modelling of Electron Transport in Molecular Devices

Hui Cao

Theoretical Chemistry Royal Institute of Technology

Stockholm 2009

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©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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3

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.

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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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5

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.

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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.

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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

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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 junction^11-13 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.

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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

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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 model^21-23 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

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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 Properties

^26-28

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 Ψ =

^η

ε

_η

Ψ

^η (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 U

H U H U

U U H

⎛ ⎞

⎜ ⎟

= ⎜ ⎜ ⎟

⎝ ⎠

⎟

(2.2)

where H_SS , H_MM , and H_DD represent the Hamiltonian matrix of the source

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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 N

S

N N

M

N N

D

S M D

J J

S S J J

i i i i

J i J

K K

M M K K

i i i i

K i K

L L

D D L L

i i i i

L i L

a a

η

η η η η

η η

|

| J

K L

η

η η

φ φ

Ψ = Ψ + Ψ + Ψ

Ψ = = =

∑ ∑ ∑ ∑

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 M

Ψ φ_i^{S D M}^{, ,}

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

, ,

' '

', , '

' '

', , '

, ,

| | | |

| ' | | ' |

| | | |

JK KJ

J K K J

K L LK

K L L K

J J LL

J J L L

JL LJ

J 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

(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

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0

,

| | | |

i i i i

i i

CC OCC

AB A B A B

A B

V A H B

^ν ^ν

a a

^ν ^ν

H

ν ν

φ φ

= ∑ = ∑ ∑

(2.5)

where | | _,

i i

A H B FA

φ φ =

iBi 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

∑

|ξ_i of reservoirs S to the final sites

∑

|ξ_m^' 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 m

i m i m

T

_{ξ ξ}^η

= ∑ ξ U ξ + ∑ ξ UGU ^ξ

_i (2.8)

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

' '

'

' ' ' '

, , ' , , , ',

+ +

i i

m m

K K K LK K K J J

K L

i m K K i m L K i m K J

T

_{ξ ξ}^η

V

_ξ

g

^η

V

_ξ

V g V

_ξ ^η _ξ

V

_ξ_m'K

g

^η'

V

_i

ξ ξ

≠ ≠

= ∑ ∑ ∑ ∑ ∑ ∑

^ξ ^(2.9)

where g_{K K}^η_' is the carrier-conduction contribution from the scattering channel, which

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can be expressed as

'

' | 1 |

' | |

g

K K

K K

z H

k K

z

η η η η

ε

η

= Ψ Ψ

−

Ψ Ψ

= −

η

i

(2.10)

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

Ei

+∞

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

2 2

' ' '

2

( ) | ' | |

+ ( ) | | |

S

K K f K

D

f K

n E V K

η η

ξ

η η

ξ

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

S(

n Ef n E^D( _f

f, respectively. Hence we obtain

'

' | |

( )

K K

i

K K

g E i

η η η η

η

η η

ε

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^η |Ψ^η Ψ^η |K^η and K' |^η Ψ^η Ψ^η |J^η should be quite small. Therefore the terms including g^η_LK and g_{K J}^η_' in Eq. (2.9) can be neglected.

Actually, the localized properties of the metal orbitals is reflected by the fact that the

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

'

2 ' '

, , '

| ( ) |

mK K K K i

i m K K

T V

_ξ

g

^η

V

_ξ

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 c

E E= +E +E E_c

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 l x y z z

l l

SD x y z x y z

E E E

l l

ll z z

i e f E E eV f E

T E E

)

π E δ

⎡ ⎤

= ⎣ + − − ⎦

× −

h ∑ ∑ +

(2.14)

where f E is the Fermi distribution function, ( )

( )

1

^/

( )

^{E E}^f ^{k T}^B

1 f E = e

⁻

+

(2.15)

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Here is the Boltzmann constant, T the temperature, the transition probability describing the scattering process from the initial state

kB T_{l l}_'

| l to the final state | 'l . The transition probability is a function of the quantized injection energies along the z axis, E^l_z and E_z^l^'.

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 l

D l

l l

S D

l l

i e f E eV f E T E E

e

l l

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.

S( )

n E n E^D( )

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 l

D x z

S D

D x l l D z D z z

i e f E E eV f E

E T n E n E dE

x

E

z

) π

ρ

∞ ∞

⎡ ⎤

= ⎣ + − − + ⎦

×

h ∫ ∫

(2.17)

where ρ₁_D( )E_x 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

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*

3 3 0 ' 1

( )

1

( )

2 ln 1 ln 1

f z f z

B B

S D

B

D l l D z D z

E eV E E E

k T k T

em k T

i T n E n E dE

e e

π

∞

+ − −

=

⎡ ⎛ ⎞ ⎛ ⎤

⎢ ⎜ ⎟ ⎜ ⎥

× + − +

⎜ ⎟ ⎜

⎢ ⎝ ⎠ ⎝ ⎥

⎣ ⎦

h ∫

^z

⎞ ⎟

⎟ ⎠

(2.18)

where m* is the electron mass.

For one-dimensional electrode, the tunneling current equals to the current density through the molecular junctions

1D 1D

I =i (2.19)

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

2D 2 2s D

I =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, r₂_s ≈[1 (N₂_dπ)]^{1/ 2}and the density of electrons can be calculated as N2_D =

(

4πm E^* _f

)

/h²

3

.

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

3D D

I =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 A≈πr_3s², where r3s is defined as the radius of a sphere whose volume equals to that of a conduction electron, r3_s =

(

3/ 4πN3D

)

^{1/ 3} , where

(

^*

) (

^{3/ 2}

)

3_D 2 _f / 3

N = m E h³π² is the density of electronic states of the bulk metal.

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2.3.3 Conductance

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

g I V

= ∂

∂

(2.22)

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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

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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 Theory

^28,33

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

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3.1 GENERAL THEORY 23

( ) ( , ) ( )

H Q = H Q e + H Q

^ν (3.1) where and are the electronic and the vibrational Hamiltonian, respectively. The Schrödinger equation now becomes

( , )

H Q e H Q^ν( )

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

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

(

_a _a

) | ( , ) | ( )

a

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 ν

ε

η

ω

⎡ + ⎤ Ψ Ψ

⎣ ⎦

= Ψ Ψ + Ψ Ψ

= + ∑ ^h Ψ Ψ

(3.2)

where ε_η represents the energy of the eigenstate, η, of the pure electronic Hamiltonian, ω_a and _hω_a are the vibrational frequency and energy of vibrational normal mode Q_a, respectively, and n^ν_a the quantum number for the mode Q_a in

|Ψ^ν( )Q .

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

0

0, 0 , 0

| ( , ) | ( ) |

_Q _{a Q}

... | ( )

a a

Q e Q Q Q

Q

η ν η η ν

= =

Ψ Ψ = Ψ + ∂Ψ + Ψ

∑ ∂

(3.3)

where |Ψ^ν( )Q is the vibration wavefunction, |Ψ^η₀ the intrinsic electronic wavefunction at the equilibrium position, ^Q=⁰. In the adiabatic approximation, we can use the first derivative like ∂Ψ( ) /Q ∂ to represent the vibrational motion part Q_a 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.

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(

^, ^, ^,

)

,

0, 0

, 0

0, 0 , 0

,

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

| ( , ) | ( , ) |

| ( , ) | ... (3.4)

| (

N N

N

S M D

J J

S

Q

J J

K M

Q a Q

K a a

D

Q e Q Q e Q e Q e Q

Q e J Q e J

Q e K K Q

Q Q

η ν η η η ν

η η η

η

=

= =

Ψ Ψ = Ψ + Ψ + Ψ Ψ

Ψ = ≈

Ψ = + ∂ +

∂ Ψ

∑ ∑

0, 0

, ) ^N | ( , ) ^N |

L L

Q

L L

e =

∑

L Q e^η ≈

∑

L^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 O

K K

J K Q KL Q KK

K K

T V Q V Q

Q K Q e Q

z 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 g^{η ν ν ν}_KK^{, ', , ''}_' is given by

, ', , '' ' , ''

'

' , ''

''

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

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

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

KK

a a

a

g Q K Q e Q

z 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 ν

η

ε

η

ω

= Ψ Ψ

−

× Ψ Ψ

= Ψ Ψ

× Ψ Ψ − − ∑ ^h

(3.5)

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3.1 GENERAL THEORY 25

After the Taylor expansion, one can get

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

' '' 0

'' ''

0 0

'

1 ' | |

| |

KK a a

a a

a a a a

a

a a

a a a a

g K Q K

z n Q Q

K Q K Q

Q Q

η η

η ν ν ν η ν ν ν ν η

ν

η η

η ν ν ν ν η

ε ω

⎡ ∂Ψ ∂ ⎤

= − − ⎢⎣ ∂ + ∂ ⎥⎦

⎡ ∂Ψ ∂ ⎤

×⎢⎣ ∂ + ∂ Ψ ⎥⎦

∑ ∑

∑

∑ ∑

h Q Ψ⁰

(3.6)

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

' ''

' | | '' 0 | |1

a a a

2

a

Q

^{ν ν}

ν Q ν Q

= = = ω h

(3.7)

Thus we get

, ', , ''

' ''

', , ''

0 0

0

0 0

1 2 ' | ' |

| |

KK

a a a a

a

a a

g n

K K

Q Q

K K

Q Q

η ν ν ν

ν ν ν η ν

η η

ε ω ω

= ×

−

⎡ ∂Ψ ∂ ⎤

× ⎢ ⎣ ∂ + ∂ ⎥ ⎦

⎡ ∂Ψ ∂ ⎤

× ⎢ ⎣ ∂ + ∂ ⎥ ⎦

∑ ∑ ∑ ^h ^h

Ψ

0

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 inel

I = I + I

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

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

2 /

d I dV²

2 2 )

(d I dV/ ) /(dI dV/

3.2 Applications

3.2.1 Identification of the switching mechanism

³⁴

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

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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 (arb. units)

Voltage (V)

1 2 4 8 9 10

11

12,13 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 (arb. units)

Voltage (V)

1 2

3 4 5 6 7

8 9 10 11

12 13

(B) 14

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.,³⁹by

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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.

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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

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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.

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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 ele

E = E + E + E + E + E + E

( )

²

1 0

str str

E = ∑ K L L −

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32 CHAPTER 4. SOLVENT EFFECT ON ELECTRON TRANSPORT

(

0

)

²

bend b

bend

E = ∑ K Θ − Θ

( )

1 cos

2

tor n

E = ∑∑ ⎡ ⎣ + n Φ − α ⎤ ⎦ V

( )

1 cos 2

2

itor itor

tor

E = ∑ ⎡ ⎣ − Φ ⎤ ⎦ V

12 6 12 6

, ,

ij ij

KI KI

VW VW

K I KI KI i j ij ij

A B

E = ζ ∑ ^⎡ ⎢ ⎣ ^⎛ ⎜ ⎝ r ^{⎞ ⎛} ⎟ ⎜ ⎠ ⎝ − r ^⎞ ⎟ ⎠ ^⎤ ⎥ ⎦ + ∑ ^⎡ ⎢ ⎢ ⎣ ^⎛ ⎜ ⎜ ⎝ r ^{⎞ ⎛} ⎟ ⎜ ⎟ ⎜ ⎠ ⎝ − r ^⎞ ⎟ ⎟ ⎠ ^⎤ ⎥ ⎥ ⎦

, ,

i j K I

ele ele

K I KI i j ij

Q Q Q Q

E = ζ ∑ r + ∑ r

where K1 and Kb are stretching and bending force constants, L0 andΘ₀areequilibrium 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

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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

( , )

( ) 4

y

N r r dr g r πρ r dr

= +

( , , ) ( , , )

^y

i j k y

N i j k df 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.

Multi-Scale Modelling of Electron Transport in Molecular Devices