Electron Transport in Single Molecule Transistors

Ryan Fillman

April 16th

, 2009

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Contents

1. Introduction…………………………………………………...3

2. Molecular Transistors………………………………………...4 2.1 Configurations

2.2 Fabrication

2.3 Molecules

3. Molecular Conduction………………………………………..8 3.1 Charge Transport Overview

3.2 Simple Two Terminal Energy Level Analysis

3.3 Coulomb Blockade Theory Overview

3.4 Molecule Charging Effects

3.5 The Co tunneling Theory

3.6 The Kondo Theory

3.7 The Non-Equilibrium Green Function (NEGF)

4. Example Results using NEGF……………………………...19 4.1 Quantum Point Contact (QPC)

4.2 Phenyl Dithiol (PD)

5. Conclusion…………………………………………………...21

6. References…………………………………………………...22

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Introduction

The realization of individual molecules as transistors being the ultimate in

miniaturization is a primary goal of the field of molecular electronics [1]. Besides size as one of

the benefits, molecular transistors also offer tune-ability and the ability to self-assemble due to

different electrostatic properties of the molecules allowing them to recognize each other. The

study of electron transport in single molecules is very important in order to gain a full

understanding how these may be used as transistors sometime in the near future. Recent

advances in the manipulation and control of single molecules currently allow scientists to

connect a molecule between two electrodes and measure its electron transport properties. Single

molecule transistors have been created and are improving in quality, speed, and control.

Creation of these transistors is very difficult and much of it is left to statistical probability. Gap

formation and the insertion of the molecule between the gap are two major problems with the

creation of single molecule transistors today. Characteristics of these molecular junctions seem

to be dependent on the chemistry, geometry, and nanoscale physics including the Coulomb

blockade. Each molecule reacts differently with electron transport so any analysis of the charge

transport requires detailed information of the molecule and how it is bonded in the experimental

setup.

The electrical conductivity of these molecules is one of the major focus areas. Each

molecule reacts differently depending on the length of the molecule, geometry, the type of

bonding to the electrodes, and a number of other factors. Co Tunneling and the electron

transport of the Kondo Theory help explain the electron transport across the molecule along with

the Coulomb blockade and many theories that are still being developed. There are a number of

theories for calculating the molecular resistance and conductivity of specific molecules but there

is no theory that can be applied to every molecular junction. The simplified methods are easy to

understand while the more realistic calculations require large calculations and quantum physics

matrix mathematics. The Non-Equilibrium Green Function is an example of a realistic

calculation that depends heavily on a full understanding of the molecule and the type of bonding

between the molecule and the metal electrodes. The NEGF is briefly described towards the end

of the report with a few examples of the application and results. Two-terminal devices are the

main focus throughout the paper to gain a better understanding of the interesting quantum

physics molecular conductance.

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Molecular Transistors

According to the International Technology Roadmap for Semiconductors of 2007, the

potential for miniaturization of these molecular transistors is on the 1E-12 m scale while the

potential switching speed is in the 1 THz range. This degree of miniaturization at the extremely

high operating frequency is one of the main drives of molecular transistor research. It is also

estimated to have the highest binary throughput of any of the emerging research devices.

Although there are many promises of molecular transistors, they are still in their most basic

forms and far from their potential.

Configurations

The first types of single molecule transistors were created in what is called the two-

terminal configuration. This structure consists of a source and a drain electrode attached

together by the molecule as seen in Figure 1. In this figure, the pyramidal atoms represent the

molecule attaching to a single atom of the gold electrodes. This is not necessarily the case in

most experiments, but it allows for simplified equations for the calculations of molecular

conductance that will be shown later in the report.

Figure 1: Two Terminal Single Molecule Transistor with a short Phenyl Dithiol (PDT) between source and drain

contacts [5]

Early forms of these transistors included nano-pore structures, crossed wires, mechanical

break junctions, scanning tunneling microscope, and conducting probe atomic force microscope.

The switching mechanism in these were either dependent on a relatively high bias voltage or a

significant mechanical deformation. This type of mechanical deformation makes the study of

these systems extremely difficult and the reproducibility of the switch very poor. Once the

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molecule is deformed, the energy levels of the molecule drastically change. The two terminal

configurations will be analyzed without the consideration of the deformation.

Three-terminal configurations for single molecule transistors allow for the manipulation

of the Fermi energy through a gate voltage. The setup is very similar to the two-terminal

configuration as previously explained. There is a source and a drain electrode that are bridged

by a single molecule. The gate electrode is placed in close proximity to the molecule so that it is

able to couple electromagnetically to the molecule as seen in Figure 2.

Figure 2: Conceptual picture of a 3-terminal molecular transistor with a short Phenyl Dithiol (PDT) between source

and drain contacts. [5]

This allows the molecule to have electron transport through tunnel junctions between the source

and drain electrodes, which will be explained later in this report. The actual switching

mechanism is therefore no longer a type of mechanical deformation. It is dependent on the gate

electrode having an electric field causing a shift in the source and drain Fermi levels. [3]

Fabrication

The fabrication of these single molecule transistors is one of the biggest feats that must

be overcome in order to realistically use these transistors in circuitry. One of the major problems

with creating these single molecules is the creation of a separation between the source and drain

electrodes. This separation must be on the order of one to four nanometers depending on the

molecule being used. Numerous methods have been attempted to create this nanometer scaled

gap but consistency and speed are two factors that never coincide. Some of these processes

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include using electron-beam lithography, an electro migration technique, or a metal oxidation

method.

The electro migration technique involves the scattering of conduction electrons by static

disorder. This results in a momentum transfer from the electrons to the lattice. This momentum

transfer, or “electron wind”, leads to the reconstruction of the metal structure creating a

nanometer-sized gap. A few hundred mV is needed to create a break in an 80-120nm wide

section. This gap that is created is then the perfect size for a molecule, but it is very irregular in

shape and ranges in gap size from 1-4 nm. These features depend on the voltage used and how

the electrodes separate from each other. [3]

Another method that was recently introduced is the use of the oxidation of metal with lift-

off techniques to create this nanometer-sized gap. These gaps are created using a self-aligned

technique that utilizes the lateral oxidation of thin, conductive metal films. The gap size is then

directly proportional to the thickness of the layer of metal that is placed on top of the liftoff

metal. It also enables the fabrication of nanometer-sized electrodes with two different metals.

This allows the tailoring of the end-group chemistry used to insert molecules for each electrode.

In addition, it facilitates the experiments in which a molecule is connected to electrodes having

different physical properties (e.g., magnetic and non-magnetic). [2]

The positioning of the metallic gate electrode is also extremely difficult. Not only is the

precise placement for the electrode critical but the distance from the molecule must also be exact

to affect the electrostatic potential of the molecule. If the separation is too large, the gate

electrode will not be able to manipulate the coulomb blockade. If the separation is too small,

current will flow from the gate to the molecule and the other two electrodes creating a short. For

this reason, most experimental molecular transistors are typically the two-terminal configuration.

Statistical probability is utilized to insert the molecule into the gaps by placing the electrodes in a

molecular solution. There is a very low rate of success with the two-terminal devices, so trying

to place a gate electrode in the exact location that the molecule statistically bonds correctly is

extremely difficult without even considering the distance that the gate electrode is from the

molecule.

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Molecules

The molecules that are used for attachment in the inter-electrode gap are also critical.

Not only is the bonding of the molecule to the electrodes important, but also the length and

overall size of the molecule is critical in relation to the size of the gap. One type of molecule

that is used commonly with gold electrodes is the C60 molecule. They have excellent adsorption

on gold and a high charge transfer which leads to other types of bonding besides the Van der

Waals bonding. Another type of molecule called Transition Metal Coordination Complexes

(TMCCs) has more recently been used. They are two planar, conjugated ligands that are

perpendicular to each other. They form an octahedral crystal field compressed along the inter-

ligand (z) axis. Complexes have been made with M-Co, Cu, Zn, individual ligands, and alkane

chains. The Co(II) complex has a high spin of 3/2 with CV in solution which suggests that the

redox transitions Co(II) ! Co(III) are easily accessible. The Cu(II) complex on the other hand

has a spin of " with CV supporting the redox transitions Cu(II) ! Cu(I). These complexes self-

assemble when bonding to Au in a tetrahy-drofuran (THF) through the loss of the –CN moieties

and formation of Au-S covalent 13 bonds. [3] The binding energies of Au-CN and Au-S are

about 3.9 eV and 1.73 eV. Phenyl Dithiol (PDT) is the main focus of the molecular conductance

calculations later in the report. This is due to the extensive studies of the PDT systems both

experimentally and theoretically for electron transport through molecules. The experimental I-V

curves still do not quite match the theoretical I-V curves, but through the use of NEGF the results

are very similar. When PDT is connected to two gold leads, it results in one of two geometric

configurations. The first is when the sulfur atom sits directly on the top position of a surface

gold atom. The second, used later in this report, is when the sulfur atom sits in the hollow

position of three nearest-neighbor surface gold atoms. The coupling between the organic

molecule and the gold leads is calculated from the DFT numerical calculations as shown later.

The placement of these molecules is extremely difficult. First, molecules are chosen that

will attach chemically to the metal being used for the electrodes. However, the actual bonding of

the molecule to the electrodes in the inter-electrode gap is a purely statistical process. Besides

not being able to control the placement of the molecules, there is no atomic-scale imaging

technique capable of determining the presence of these molecules if they do span the electrode

gap and how they are bonded to the electrodes. This makes analyzing the samples extremely

difficult and a molecule must be determined to be present through electrical tests. Through this

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test, it is possible to analyze the samples statistically to create an estimate of success rate for the

bonding of the molecules between the inter electrode gap. [3]

Molecular Conduction

Charge Transport Overview

Non-resonant transport and resonant tunneling can cause charge transport in these single

molecule transistors. In non-resonant transport, the molecule acts as a potential barrier in the

device. Therefore, the electrons must tunnel through the molecule from the source to the drain

electrodes. As the molecule length increases, the current that can travel through the molecule

decreases exponentially. Resonant tunneling on the other hand is a type of nonlinear conduction.

The surprising part of resonant tunneling is that these single molecule transistors actually have

properties similar to single-electron transistors.

The electron transport in the molecules has an associated Electron Addition Energy (Ea).

This is caused by the increase in energy as the electron tunnels through the molecule from the

source electrode to the drain electrode. This energy level is comprised of the single particle level

spacing (#E) and the charging energy (Ec). The single particle level spacing is the energetic cost

of promoting an electron on the molecule. This electron is promoted from the Highest Occupied

Molecular Orbital (HOMO) to the Lowest Unoccupied Molecular Orbital (LUMO). The

associated charging energy accounts for the coulomb interactions on the molecule. It is defined

as Ec=e2/C. In this equation, C is the capacitance of the electron-electron interactions. The

charging energy will be revisited later in a more thorough analysis.

Another important energy to take into account when studying the electron transport of

single molecule transistors is the Coulomb Blockade. This essentially blocks the electron from

tunneling through the molecule due to the thermal energy of the molecule. When the thermal

energy kBT is much less than the Electron Addition Energy (Ea) changing the charge state of the

electron is energetically forbidden.

This blockade can be overcome in two ways. First, a large source to drain voltage can be

used to allow resonant tunneling. This method is used in the two-electrode single molecule

transistor configuration. This source to drain voltage must create an energy that is large

compared to the electron addition energy in order to allow the resonant tunneling. In the case of

the three-terminal configurations, the gate is used to overcome this Coulomb blockade. By

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applying a voltage to the gate electrode, the electrostatic potential near the molecule is altered.

This is energetically degenerate and changes the charge of the molecule by one electron. Co

tunneling and slow internal relaxation rate of electrons on the molecule are two more ways of

overcoming this coulomb blockade energy. [3][4]

Simple Two Terminal Energy Level Analysis

Typical current-voltage (I-V) and conductance (G-V) curves can be seen in Figure 3.

These curves have been observed in a number of different two-terminal configurations including

many that have been mentioned in the two-terminal molecular transistor overview.

Figure 3: General properties of measured current-voltage (I-V) and conductance (G-V) characteristics for molecular

wires. The solid line is for a symmetrical molecule I-V and the dashed line is for an asymmetrical molecule I-V.

The electrode geometry and bonding also plays an important role in the I-V

characteristics similar to current MOSFETs. A three terminal MOSFET saturates under

increasing bias while the two-terminal diode constantly increases. The I-V characteristics of

molecules are determined with a combination of electrostatics and quantum transport. The first

model that will be considered does not take into account the shift in the energy level due to

charging effects as the molecule loses or gains electrons along with the broadening of the energy

levels due to their finite lifetime due to the contacts the molecule makes with the electrodes.

The main energy levels that need to be considered are the metallic work function (WF),

the electronic affinity (EA), and the ionization potential (IP) of the molecule. Gold contacts have

a work function of about 5.3 eV and the EA and IP of phenyl dithiol are about 2.4 eV and 8.3 eV

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respectively [5]. Figure 4 shows a simplified equilibrium energy level diagram for a metal-

molecule-metal transistor for a weakly coupled molecule.

Figure 4: Equilibrium energy level diagram for a metal-molecule-metal transistor for a weakly coupled molecule.

In this configuration, the molecule will remain neutral without loosing or gaining an

electron as long as both (IP-WF) and (WF-EA) are much larger than kBT. This is due to the

probability of the molecule losing an electron to form a positive ion is equal to e(WF-IP)/kT

and the

probability of the molecule gaining an electron to form a negative ion is equal to e(EA-WF)/kT

. The

molecule will typically stay neutral due to the large energy that is required to gain or loose an

electron. However, when the molecule bonds strongly with the metallic contacts, the strong

hybridization with the delocalized metallic wave functions allow electrons to be added or

removed from the electron with less energy. This causes the energy levels of the molecule to be

shifted by a contact potential as can be seen in Figure 5.

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Figure 5: Equilibrium energy level diagram for two-terminal single molecule transistor where the molecule is

strongly coupled to the contacts. [5]

For simplification of the energy level diagram, the HOMO and LUMO levels associated

with the incremental charge transfer are used instead of the EA and ionization levels. If the

electron charging energy (U) is much greater than the energy broadening (

!

"), then the two-

terminal device is considered to be operating in the Coulomb Blockade (CB) regime. If this is

not the case, then it is in the self-consistent field (SCF) regime with fractional charge transfer.

These topics will be analyzed more thoroughly later in the report. The location of the Fermi

energy relative to the HOMO and LUMO levels is somewhere between the two levels. The

location of the Fermi energy is equal to the number of electrons present in the molecule at any

given time. Therefore, if the charge transferred were equal to +1, the Ef would be on the LUMO

while if it were equal to -1 it would be on the HOMO. Fractional charges occur due to the work

function of the metal contacts causing the Ef to be somewhere between the two unless the

molecule gains or looses an electron. For the phenyl dithiol molecule with gold electrodes, it is

unknown where the Fermi energy is located due to different theoretical calculations placing it

closer to the LUMO while others theorize it is closer to the HOMO.

Current flow through a two-terminal single molecule transistor always deals with a non-

equilibrium situation with different electrochemical potentials

!

µ. For example, if a negative

voltage is applied to the source with respect to the drain, then the drain has an electrochemical

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potential lower than the source meaning that they both have different Fermi levels with each

wanting to bring the device to equilibrium. These non-equilibrium states can be seen in Figure 6.

Figure 6: Schematic energy level diagram of a two-terminal device where the source is positively biased or

negatively biased.

In this situation, the source keeps trying to add electrons to the system while the drain keeps

trying to remove electrons from the system. If this is considered as a simple one level system

where the system’s energy lies between the electrochemical potentials of the two contacts, an

electron can move to the source or drain at a rate

!

"1

!and

"2

! respectively. This is calculated using

a simple kinetic energy equation. If the system energy was equal to either of the two contacts,

the number of electrons occupying the energy level could be calculated with the following

equations where the 2 represents the spin.

!

N1

= 2 f (",u1)

N2

= 2 f (",u2)

If the system was under non-equilibrium conditions, the number of electrons would be

somewhere between the N1 and N2 values. A steady state kinetic equation can solve the net

current between the two junctions where IL is the net current at the left junction and IR is the net

current at the right junction.

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!

IL =e"

1

!(N

1# N)

IR =e"

2

!(N # N

2)

Steady State IL = IR

$N = 2"1f (%,µ

1) + "

2f (%,µ

2)

"1+ "

2

where

I =2e

!

"1"2

"1+ "

2

f (%,µ1) # f (%,µ

2)[ ]

This proves that the current only flows due to the difference in energy levels between the source

and the drain. Using this model, Figure 7 shows the I-V characteristics of this simple two-

terminal device.

Figure 7:I-V characteristics where

!

µ1

= E f "eV

2

,

!

µ2

= E f "eV

2

,

!

E f = "5eV ,

!

"0

= #5.5eV , and

!

"1

= "2

= 0.2eV [5]

When both the source and drain are above the system energy level, the current is equal to zero.

When the drain drops below the energy level, the current increases and when the source drops

below the energy level, the current reverses. The strength of the bonding between the molecule

and the metal contacts determines the current levels. A stronger bond would result in a higher

current flow where a weaker bond would result in a lower current flow.

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The Coulomb Blockade Theory Overview

In the Coulomb blockade theory, the molecule can be modeled in a number of different

ways. First, it can be modeled as a tunnel junction. This means that the molecule acts as an

insulating layer between the two electrodes. The electrons must quantum-mechanically tunnel

from the source electrode to the drain electrode. This tunneling current is dependent on the

cross-sectional area and length of the molecule, ionization potential of the molecule, and the

source-drain bias voltage. Therefore, as the length of the molecule increases, the conductance

decreases exponentially.

The sequential tunneling model is when the molecule is represented as two tunnel

barriers with a discrete energy level in between the barriers. The transparency of these tunnel

barriers depends on the types and quality of the bonding between the molecule and the

electrodes. This model assumes that the rate of electron relaxation within the molecule is faster

than the electron-tunneling rate into or out of the molecule. This model also neglects the higher-

order tunneling such as co tunneling. According to this model, at low temperatures the current

will flow but only if there is a molecular state between the tunnel barriers. This type of

sequential tunneling is observed in all molecular devices.

The Anderson Model describes the delocalized and localized electron-electron interaction

but does not include the exchange interactions between electrons and the vibration modes of the

molecule. However, it does account for the effect of the Coulomb repulsion for adding electrons

to the molecule for transport. This model is used mainly to understand the transport

measurements of single molecule transistors and the magnetic moment. A single molecule

transistor with a magnetic moment can be in the empty orbital regime where $/% < 0 when there

is no Kondo resonance. It can also be in two other states namely the mixed valence regime and

the Kondo regime. The single molecule is in the mixed valence regime when $/% <~ 1 meaning

that both the spin and charge fluctuations are relevant. If the molecule is in the Kondo regime,

then only the spin fluctuation is relevant. [3]

Molecule Charging Effects

A simple model for the charging effects of the molecule is necessary for a more realistic

model of the electron conduction through the two-terminal device. To do this a potential USCF

needs to be included in the calculations due to the change in the number of electrons from the

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equilibrium state. Using the Hubbard model, we can set USCF=U(N-2f0) and allow the system

energy to vary with this potential by setting

!

" = "0

+USCF

. To calculate the potential, we need to

calculate the relationship between USCF and the number of electrons charging the molecule. The

difference between the current and conductance curves between calculations with the charging

effects and without the charging effects can be seen in Figure 8.

Figure 8: I-V and G-V curves where dashed lines do not include charging calculations and solid lines do include

charging calculations

The inclusion of charging broadens the sharp conductance peaks. The energy difference between

the molecular energy level and the Fermi level is directly proportional to the conductance gap

and the charging only affects the plots at higher voltage levels.

The Co tunneling Theory

The co tunneling theory involves the simultaneous tunneling of two or more electrons but

ignores the spin of the electron. This is taken into account with the Kondo effect that is

explained in the next section. The simultaneous tunneling of two or more electrons can create an

enhanced conduction in the blockade region. Elastic co tunneling is when the electron hops from

the source electrode to an energy level on the molecule according to quantum theory. As this

happens, an electron from that level on the molecule then moves to the drain electrode. This

leaves the energy of the tunneling electron and the molecule unchanged and can occur at low

source-drain voltages.

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Inelastic co tunneling is the same as elastic except the electron that leaves the molecule

actually leaves at a lower energy state. This leaves the molecule in an excited state that is in the

form of a vibration mode of the molecule. Energy is still conserved but this type of inelastic co

tunneling can only occur at certain bias voltages. [3][4]

The Kondo Theory

The Kondo Theory basically states that the electrical resistance of a metal will diverge as

the temperature approaches 0K. Applying this to how a molecule reacts is slightly different. If

the spins of the electrons are accounted for, then the molecule should be able to hold no

electrons, one with a magnetic moment, or two electrons. Its important to note that the

incorporation of spin does not change the physics of the Coulomb blockade in single molecule

transistors. Electrons still need the energy equal to the Ea to add the second electron to the

molecule.

As stated before, the Kondo Theory is designed for an isolated magnetic atom in a

metallic environment. By changing the gate voltage, the single molecule transistor can be made

to have a magnetic moment and actually show very similar characteristics to the isolated

magnetic atom. The Anderson Hamiltonian equation describes the conduction electrons of the

electrodes, the localized electrons on the impurity, the coulomb interaction between the

electrons, and the interactions between the conduction electrons and the localized electrons.

When the molecule contains no magnetic moment, the transport characteristics can easily be

explained by the Coulomb blockade. However, through this equation, the interaction between

localized moment and the moments of the conduction electrons can be described. This equation

probes that the single molecule transistors react much like an anti-ferromagnet and prefer to have

their conduction electron spins aligned anti-parallel to that of the localized electron. The Pauli

Exclusion Principle proves that it is energetically favorable for the conduction electrons to have

their spin opposite of the localized electrons. The spin up electron in the impurity state would

not be able to lower its energy by moving to the electrode if the conduction electron at the Fermi

surface was spin up.

The Kondo effect can also explain the interaction between these electrons. The

conduction electron interacts with the local moment and flips. The spin of the electron then also

flips due to this change. After this, the conduction electron can either interact again, which gives

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the conduction electron a finite lifetime or another conduction electron can interact with the

flipped local moment.

As stated before, in bulk materials, decreasing temperatures of the system usually mean

increased resistivity of the system. This is due to the spin-singlet causing an increased scattering

of the conduction electrons. The effect in single molecule transistors is actually the complete

opposite. A decreasing temperature of the system means an increase in the conductance. The

bond creates a new channel for the conduction electron to flow increasing the conductivity of the

molecule. [3][4]

The Non-Equilibrium Green Function (NEGF)

The two-terminal device explained earlier accounted for the three major influences on

molecular conduction. However, most molecules normally have multiple energy levels with

arbitrary broadening and overlap. The previous model can only be used effectively if the energy

levels do not overlap. The Non-Equilibrium Green Function (NEGF) provides a model that can

account for multiple levels with arbitrary broadening and overlap. Green’s function G(E) is

defined in the following equation along with the density of states D(E) and the spectral function

A(E).

!

G(E) = E "# + i$1

+ $2

2

%

& ' (

) *

"1

D(E) =A(E)

2+A(E) = "2Im{G(E)}

This allows us to derive new equations for the number of electrons (N) and the current (I).

!

N =2

2"dE G(E)

2#1f (E,µ

1) + G(E)

2#2f (E,µ

2)[ ]

$%

%

&

I =2e

hdE#

1#2G(E)

2f (E,µ

1) $ f (E,µ

2)[ ]

$%

%

&

The single energy level

!

" is replaced by a Hamiltonian matrix [H] while the broadening

!

"1,2

is

defined as the imaginary parts of

!

"1,2

. Then the spectral function is the anti-Hermitian part of the

Green’s function from which the density of states D(E) can be calculated by taking the trace

allowing the density matrix [

!

" ] to be calculated where the trace of the density matrix results in

the number of electrons (N).

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!

G(E) = (ES "H "1# "

2)"1#

$1,2

= i(1,2# "

1,2

+# )

A(E) = i(G(E) "G+(E))

D(E) =Tr(AS)

2%

& =1

2%f (E,µ

1)G$

1G

+ + f (E,µ2)G$

2G

+[ ]"'

'

( dE

N = Tr(&S)

The current is then given by the following equation.

!

I =2e

hTr("

1G"

2G

+)( f (E,µ

1) # f (E,µ

2))[ ]dE

#$

$

%

Effectively, this procedure is simply replacing the scalar quantities from the simple two-terminal

device with matrices to solve the self-consistent potential matrix USCF. In this model, USCF is a

function of the density matrix

!

" depending on the molecule and electrodes. There are a number

of techniques to solve for the functionals but they are numerically challenging and quantum

chemistry programs such as Gaussian utilize a number of techniques to speed up the calculations.

However, since software only addresses specific molecules in equilibrium, the programs can

only be utilized for portions of the calculations. To do this, the potential of the molecule is

approximated with a flat potential such as follows. [6]

!

USCF =Uappl + F(") =Uappl +U(N # N0)

where

Uappl =U1

+U2

2

Uappl is the approximate solution to the Laplace equation, U is the charging energy, and N0 is the

number of electrons in the molecule at equilibrium. Using a constant potential simplifies the

calculations by either shifting the molecular states up or down. Through this method, the

molecular states can be treated as unchanged and then simply change the chemical potentials of

the contacts by USCF.

!

u1,2

= E f +U1,2"USC

In this equation, U1,2 is the applied potential and the chemical potential of the probe can be set to

!

µp =(µ1

+ µ2)

2. This allows for the number of electrons on the device to be calculated by

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summing over the contacts and the probe. The functions N1,2,p can be calculated once and then

reused in every SCF step for each applied potential.

!

N = N1(µ1) + N

2(µ

2) + Np (µp )

N1,2,p (µ) =

1

2"Tr(F#

1,2,pG+) f (E,µ)dE

$%

%

&

The assumption of a constant potential across the device relies on the molecule making a single

atomic contact with the metal electrode. For large clusters of metal atoms at each electrode, this

assumption will not provide reliable data.

Example Results

Quantum Point Contact (QPC)

Gold films are typical contact points used by molecular researchers to form the contacts

with the molecules. Figure 9 shows the surface density of states (DOS) for a gold contact

surface. This is the number of states that are present for each energy level showing the possible

electron transfer between states. In the Huckel model, each gold atom is denoted with three 5p,

five 5d, and one 6s orbital that all contribute to the total density of states shown in the plot on the

right. The Fermi energy of the gold in this model was calculated to be -9.5eV.

Figure 9: Gold two layered structure on the left and the right shows the contributions of each of

the orbitals to the surface density of states. [6]

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Phenyl Dithiol (PD)

Using the model for a QPC, the Au6 molecule can then be replaced with any other

molecule as long as the bonding type to the gold is known. For phenyl dithiol, it is known that

the sulfur atoms bond to the surface with a gold-sulfur bond of a length of 0.253nm. Figure 10

shows the molecule bonded between the two contacts on the left along with the Density of States

vs. energy plot on the right.

Figure 10: PDT bonded to gold contacts in a two-terminal configuration on the left. The right

plot shows the Density of States vs. Energy with discrete energy levels. [6]

The DOS plot shows that there are a number of states above the LUMO and below the

HOMO levels. The number of states increases at these locations due to the addition or loss of an

electron. Between the HOMO and LUMO levels, there is really only one state, equilibrium, until

you start to get close to these levels due to charging effects from the source and drain on the two-

terminal device.

Fillman 21

Conclusion

Single molecule transistors are an exciting possibility for the miniaturization of self-

assembling circuits. Before we are able to do this we need to gain a better understanding of the

physical and chemical properties at the quantum level. Coulomb blockade, co tunneling, and the

Kondo effect are the beginning of being able to understand this new world of miniaturized

molecular electronics. Although there are a number of theories that allow the calculations of

how these molecules will react, they are limited in their reliability and are typically

mathematically intensive. All theories can predict certain characteristics of the electrical

conductivity of specific molecules, but there is always some experimental data that does not with

the theoretical data. This understanding will be essential to the future implementation of

molecular transistors.

Other than the science involved in the electron transport, new fabrication techniques must

be designed to increase reproducibility and the quality of the inter-electrode gaps and molecule

placement. Gaining better control of the placement and bonding structures of these molecules is

extremely important for the mass-production of molecular transistors with consistent I-V curves

and conductance. As stated before, the International Technology Roadmap for Semiconductors

of 2007 places the potential for miniaturization of these molecular transistors around 1E-12 m

with the potential switching speed in the 1 THz range. Single molecule transistors are estimated

to have the highest binary throughput of any of the emerging research devices. Molecular

transistors have a very bright potential future, but there are still many problems with device

fabrication and achieving their full potential. The vast number of different types and forms of

molecules will also slow down the implementation of molecular transistors due to the extensive

amounts of time it will take to explore all of the different molecular options.

Fillman 22

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2009.

4 Park, Jiwoong. "Electron Transport in Single Molecule Transistors." University of California,

Berkeley (1996). 8 March. 2009.

5 M. Paulsson, F. Zahid, S. Datta . "Resistance of a Molecule" Purdue University (2003). 1 April.

2009.

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