chapter-5 · the below figure shows a general approach of quantum mechanics in developing various...

CHAPTER-5 Computational Modeling and Molecular Simulation Approach

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CHAPTER-5

Computational Modeling and Molecular Simulation Approach


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Molecular computation focuses on the computational power of molecules and

attempts to realize information processing [162]. Many-sophisticated probabilistic

modeling approaches have been brought in to computational chemistry such as

Heuristic approach, Integral calculus, Langaragian interpolation, Newton Raphson

method, Linear programming and Monte-Carlo (stochastic approach).

The design of molecular electronic devices and the processes to make them on an

industrial scale will require a through theoretical understanding of the molecular and

higher level processes involved. Hence the development of modeling techniques for

molecular electronics devices is a need from basic point of view and from an applied

nanotechnology point of view.

Modeling molecular electronics devices requires computational methods at all length

scales-electronic structure methods for calculating electron transport through organic

molecules bonded in inorganic surfaces[163], molecular simulation methods for

determining the structure of self-assembled films of organic molecules on inorganic

surfaces[163], mesoscale methods to understand and predict the formation of

mesoscale patterns on surfaces and macroscopic scale methods for simulating the

behavior of molecular electronic circuit elements.

The underlying physical laws necessary for the mathematical theory of a large part of

physics and the whole of chemistry are thus completely known, and the difficulty is only

that the exact application of these laws leads to equations much too complicated to be

soluble.


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5.1 Molecular Modeling

Molecular modeling is generally done with structure determination and selection of

calculation methods in computational chemistry. In practice, Computation, Theory and

Modeling are interchangeable and can be regarded as simulations.

The molecular model can be represent as

Fig. 5 .1(a) Molecular Computation and Modeling Strategy

Quantum mechanics takes care of bond formation, many force field parameters and

charges of interest [164]. The molecular mechanics generally takes care of molecular

geometry and mechanical principle lies in forming structure. The molecular dynamics

takes care of free energy in the surface and computing simulation in various time factors.


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The molecular model can be designed as below.

Fig.5.1 (b) Molecular Models, Simulation and Experiment

Simulation does not mean only programming. Experimentalists can and do perform

good simulations. Simulations tend to produce lots of data and not just one simple

answer [165]. The physics of a system need to understand in detail in order to design a

simulation. Simulations can be serial or parallel. Today, most people use PCs for serial

jobs and supercomputers for parallel applications. The approach taken here is a

sequential approach of molecular simulation.

The main ingredients of a molecular simulation are the molecular model and the

simulation algorithm. The molecular model is comprised of atoms, or atom groups, that

interact with potentials (or “force-fields”) described by empirical mathematical

expressions [166].

Assuming that the simulation algorithm is executed correctly, the accuracy of the

force-field is one of the important properties that determine how well the predicted

properties agree with experimental ones [166].A typical simulation box will include

anywhere from 102–106 atoms/molecules and employ periodic boundary conditions to


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emulate a bulk system. The main power of molecular simulation lies in the fact that the

observed macroscopic properties, or trends therein, can often be explained by the

molecular-level behavior [167].

Molecular simulation algorithms can be deterministic or probabilistic. Next, two main

molecular simulation techniques: molecular dynamics (MD) and Monte Carlo (MC) will

be discussed in more detail [168].

5.2 Molecular Simulation

A molecular simulation is a computer experiment based on a molecular model.

Elements of a molecular simulation generally consist of objects that represent molecules

coupled with potentials that describe the interactions (e.g. collisions) between the

molecules [170]. Molecular simulations provide methods of isolating preferred

configurations or states (which might be impossible in the material world), and further,

measuring the physical conditions of these states [170]. Therefore, any programming

tool intended as a molecular simulation package must incorporate objects that

sufficiently represent the aforementioned participants of a molecular simulation. This

includes, molecule objects that describe all the properties of molecules and atoms

(diameter, mass, shape), potential objects that describe these molecules’ interactions,

environment objects that spell out the space occupied by and the phase containing

these molecules and finally display objects for drawing graphical pictures of the system

or displaying results from the objects[169].

5.2.1 How do perform Simulation The use of molecular simulation techniques has been increasing steadily during the

last years. In parallel, new simulation and analysis techniques are constantly being

developed. Nevertheless, it often takes a long time until a new technique becomes

available to the scientific community in an easy-to-use implementation, if it ever

happens. In addition, only a minority of computational scientists has the possibility to

implement or modify simulation methods. There are both technical and legal causes for

this situation. The simulation packages that are commonly used for molecular

simulations are large and complex, making it impractical for anyone but the original

developers to extend or modify the code.


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The below figure shows a general approach of quantum mechanics in developing various system.

Fig.5.2.1 (a) Quantum Mechanics Application [172]

If we analyze the tree, some of the data required for simulations at one level can only

be obtained from the results of simulations on the level below. The descent continues

right down to the quantum mechanical atomic/molecular level, where the many-electron

Schrodinger equation at last provides a concise and accurate universal law of nature

[172].

To understand the role of quantum mechanics in determining the electronic structure

and properties of real materials, the accurate and reliable methods is start from the

lowest box in Figure 5.2.1(a) to the levels above.

Mechanical calculations start from first principles and require only the atomic numbers

of the atoms involved as input. It would provide a firm base on which the higher levels of

the simulation tree can rest.

The equation that can derive from Schrodinger equation is,


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It's a linear second-order complex partial differential equation of a well-known type,

the only significant computational problem being that the sums over i and j include terms

for every electron in the system[171][172]. The many-electron Schrodinger equation is a

partial differential equation in roughly 1023 variables. Most scientific and engineering

problems involve partial differential equations in 3 or at most 6 variables, so dealing with

1023 is quite a challenge.

Instead of considering all the electrons together, the idea has been to look at the

electrons one by one, replacing the complicated punctuating forces due to the others by

an average force known as a mean field. This trick reduces the 1023 dimensional many-

electron Schrodinger equation to a much simpler three-dimensional equation for each

electron. The mean field depends on the electron density, which can not be t known until

the one-electron equation has been solved to find all the one-electron quantum states,

so it's necessary to use simulation of iterative procedure to home in on a consistent

solution.

As discussed earlier in chapter 3 the solids generally use density functional theory. The

density functional theory involves an unknown quantity called the exchange-correlation

energy functional which has to be approximated. The simplest approximation which is

generally used is the local density approximation (LDA). Unfortunately, the LDA is as

such not so or reliable. Some time it clicked for simple metals and semiconductors, but

not for transition metal oxides, superconductors, and hydrogen-bonded materials [172].

There’s no simple systematic way to improve the standard density functional approach

and one is forced to go back to the full many-electron Schrodinger equation instead.

This work has taken a primary level approach towards molecular simulation carried

out in molecular devices. This approach has not taken any software simulation tool,

rather put an idea of making a software where molecular simulation can happen with

many science theory and techniques for different molecules depend on types donor and

acceptor chosen and trying to recreate a physical environment and the interactions

between the molecules as well as the electron transport in a quantum mechanical way.

The overall objective in this section is to develop a software package for simulation of

non-equilibrium quantum transport of electrons in molecular structures.


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This work has been taken into consider the various science theories and methods

such as Schrödinger equation, Hartree-Fock method, DFT (Density Function Theory)

and NEGF (Non-equilibrium Green’s function) for simulation study. Later part it has

taken molecular dynamic approach through quantum Monte Carlo simulation.

Locating critical points on the potential energy surface(PES) is the major task of

molecular simulations[173] .The PES is the collection of energy values at the given

arrangement of atoms .Quantum mechanics works with nuclei and electrons and

quantum chemical methods where as Classical mechanics (molecular mechanics) works

with atoms bonded or non bonded ,usually connections between atoms predefined. Here

the work has carried out with both approaches.

Fig.5.2.1 (b) Generation of Molecular Surface

The vdw(vanderwall) surface of a molecule corresponds to the outward-facing

surfaces of the vanderwall spheres of atoms[174]. The molecular surface is generated

by rolling a spherical probe on the vanderwall surface. The molecular surface is

constructed from contact and re-entrant surface elements. The centre of probe traces

out accessible surface. The hardware and software issue depend on the choice of

molecular model, force field and sample size[173]. These depend on the property

someone is interest in, required accuracy of prediction and the computing power to

generate the ensemble.


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Fig. 5.2.1(c) Choice of molecular Model, Force field and Sample size

Referring to Fig.4.6.(b) of chapter-4 ,where the molecular surface has considered.

This molecular surface has created using molecular rectifying diode described in

section4.6.The simulation happens with different molecule combination that has been

taken by taking different donor and acceptor substitutes.

The donating group can be (X) = (-NH2, -OH, -CH3, -CH2CH3, etc.)

The different types of Acceptor residue that we take as (Y) = (-NO2, -CH, -CHO, - NC

etc.)

The different types of Insulator that we take as (R) = (-CH2-,-CH2-CH2-, etc.)

The different combinations that can be taken considered are as follows

a) Fixing one X (say –NH2), we may have varieties of Y ((-NO2, -CH, -CHO, - NC) and varieties of (R) = (-CH2-,-CH2-CH2-)

b) Fixing one Y (say –NO2), we may have varieties of (X) = (-NH2, -OH, -CH3, -CH2CH3) and (R) = (-CH2-,-CH2-CH2-)

c) Fixing one R (say –CH2), we may have varieties of Y ((-NO2, -CH, -CHO,- NC) and (X) = (-NH2, -OH, -CH3, -CH2CH3)


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These are the molecules that have designed out of these combinations.

Fig.5.2.1 (d) Molecules with Constant X (Donor) for varying Y (Acceptor) and R

(Insulator)


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Fig.5.2.1 (e) Molecules with Constant Y (Acceptor) for varying X (Donor) and R

(Insulator)


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Fig.5.2.1 (f) Molecules with Constant R (Insulator) for varying X (Donor) and Y

(Acceptor)


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The simulation properties get studied with surface properties and electron transport

properties of the molecule of choice. The molecular wire is full of electrons since each of

the atoms in the wire brings at least one electron with it. These electrons are spread on

the molecular surface from molecular orbitas.Each molecular orbital in other way is

made up of different atomic orbitals.

The electronic properties of all these molecules of interest depend on the electron

affinity, ionization potential and bonding energy.These electronic properties responsible

for the conductivity inside the molecular wire.

In each of the combination shown in Fig.5.2.1 (d) to Fig.5.2.1 (f), the left and right

regions are mostly will be identical atoms of molecules. The middle region contains

different atoms and/or disorder. The model is called the nearest neighbor tight-binding

approximation by physicists and among chemists it is known as the Huckel

approximation.

The circuit can be formed with any of these molecular structures acting as inputs.

Depending on the molecular structure there molecular properties get changed, so also

the conductivity properties. Out of these 24 structures formed, if minimum two needed to

form a molecular gate at least 276 gates can be formed. But out of these all may not be

suitable for conductivity.

Monolayer structures of different derivatives formed due to molecular self-organization

are determined. By comparison of the different molecular species it is possible to identify

the chemical groups that are responsible for the anchoring to the substrate, the

intermolecular bonding, and the molecular orientation.

The basic equation of mechanics describing the quantum situation is the Schrodinger

equation. It is the basis for an electron tunneling through a potential barrier [83]. Such a

barrier exists between two slabs of metal separated and electron flow in the surface of

metal. Classical physics would say that the electron had insufficient energy to cross the

barrier, but application of the Schrodinger equation reveals that there is some probability

that the electron would cross the barrier. In tunneling, none of the electrons has energy

larger than the barrier. It is the wave nature of the electrons that makes tunneling

possible. This recall the relation of the wave function of the Schrodinger equation and

probability of particle location.


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The Schrödinger equation, as described earlier represent by HΨ=EΨ. However, it can

be solved exactly only for one-electron systems (e.g., a hydrogen atom) and numerically

for any a system having more electrons.It can apply in nuclei and many electron

scenarios as

Fig.5.2.1 (f) Nuclei-Electron Features in a Molecule

H = Tn + Te + Vnn + Vee + Vne ------------------------------------- Eq (5.2.1)

Where,


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To solve the Schrödinger equation approximately, assumptions are made to simplify the equation by (i) Born-Oppenheimer approximation and (ii) Hartree-Fock approximation [83].

The Born-Oppenheimer approximation allows separate treatment of nuclei and

electrons. In this method Nuclei are much heavier than electrons (ma / me ≥ 1836) and

move much slower [83]. Effectively, electrons adjust themselves instantaneously to

nuclear configurations. Electron and nuclear motions are uncoupled, thus the energies of

the two are separable. For a given nuclear configuration, one calculates electronic

energy. As nuclei move continuously, the points of electronic energy joint to form a

potential energy surface on which nuclei move.

The Hartree-Fock approximation is the independent electron approximation allows

each electron to be considered as being affected by the sum (field) of all other electrons

[83]. In a Many-electron wave function all electrons are independent, each in its own

orbital and form a Hartree product.

A Fock operator F is introduced for a given electron in the i-th orbital:

Fi Φi = Єi Φi

Φi is the i-th molecular orbital, and ei is the corresponding orbital energy. The total energy is not the sum of orbital energies. If you sum them up, you count the electron-electron interactions twice.

The Fock operator is given as

∑ −+=N

jjjii )( KJhF

Where

hi=Core-Hamiltonian operator= Kinetic energy

term and nuclear attraction for the given electron. Jj=Coulomb Operator=Columbic energy term for

the given electron due to another electron Kj = Exchange operator = Exchange energy due

to another electron (A pure quantum mechanical

term due to the Pauli principle, no classical

interpretation)


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The self consistent shows each electron “feels” all the other electrons as a whole

(field of charge), .i.e., an electron moves in a mean-field generated by all the other

electrons [83]. The Fock equation for an electron in the i-th orbital contains information of

all the other electrons, i.e., the Fock equations for all electrons are coupled with each

other. All equations must be solved together (iteratively until self-consistency is

obtained). — Self-consistent field (SCF) method[83].

As per variational principle based on the LCAO approximation, each one-electron

molecular orbital is approximated as a linear combination of atomic orbitals.

Ψ = c1Φ1+c2Φ2+…

The energy calculated from any approximated wave function is higher than the true

energy. The better the wave function, the lower the energy. One can vary the

coefficients to minimize the calculated energy. At the energy minimum, dE = 0, and one

has the best approximation of the true energy

Examples of molecular simulations are Molecular Dynamics (MD) and Monte Carlo

(MC). Molecular Dynamics is a deterministic method that integrates the equations of

motion to examine the evolution of a system over a period of time [170]. Contrary to this,

Monte Carlo simulations are stochastic processes that do not have an element of time.

Instead they use ensemble averaging to generate a large number of randomly selected

configurations of a system [170]. In both MD and MC simulations, averages are taken

over the ensemble of configurations, and these can be used to “measure” the physical

properties of the model system. The work in this thesis has mostly considered the Monte

Carlo simulation.


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5.2.2 Molecular Dynamics

Molecular dynamics simulation consists of the numerical, step-by-step, solution of the

classical equations of motion [175], which for a simple atomic system may be written

=

=

For this purpose we need to be able to calculate the forces fi acting on the atoms, and

these are usually derived from a potential energy U(rN), where rN = (r1; r2; :: : rN) represents the complete set of 3N atomic coordinates. The Molecular dynamics

generally takes care of potential energy from non-bonded interactions, bonding potential

and force field from molecular interactions.

Fig.5.2.2 Geometry of simple Chain Molecule [175]

The molecules are building for molecular systems from site-site potentials. Typically, a

single-molecule quantum-chemical calculation may be used to estimate the electron

density throughout the molecule, which may then be modeled by a distribution of partial

charges, or more accurately by a distribution of electrostatic multipoles [175]. For

molecules we must also consider the intramolecular bonding interactions.


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The simplest molecular model includes terms of the following kind

+ (

+ 1/2

The bonds will typically involve the separation rij = ri-rj between adjacent pairs of

atoms in a molecular framework. The bend angles θijk are between successive bond

vectors such as ri - rj and rj - rk, and therefore involve three atom coordinates:

We can simulate large molecular and nano systems, but challenges still exist as all

simulations present a challenge to balance the detail of a simulation with the length of

time it takes to run. Thus the alternate method is to go for random selection of molecules

in a stochastic assembly.

Molecular dynamics simulation is a deterministic method based on computing the

motion of each atom in the simulation box by integrating Newton’s equation of motion

[166]. Each of the N atoms or molecules in the simulation is treated as a point mass and

given an initial velocity chosen from the Maxwell-Boltzman distribution. The physics of

the model is contained in the force-fields (potentials) acting between the atoms. These

force fields provide the potential energy associated with a given arrangement of the

atoms within a system. The force on each atom is given as the gradient of this potential

energy. A variety of useful microscopic and macroscopic information can be obtained by

using MD simulations such as transport coefficients, phase diagrams and Structural or

conformationalproperties[176].


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MD simulations might become computationally “large” depending on the number of

atoms and number of time steps [166]. To achieve the sub-micron scale, many

thousands or millions of atoms must be simulated. As increased time step size provides

more “real time” per MD step, the time step size should be selected as large as possible

to get more “real time” per CPU time. However, the time step size is limited by the

highest frequency motion of the atoms, which needs to be accurately tracked [166]. For

bonds involved with the system, the time step should be chosen based on the vibrational

motion of the atoms. This limits time steps to the pico second scale, thereby tens or

hundreds of thousands of time steps are necessary to simulate even picoseconds of

“real” time [177].

MD simulation was used as the main tool for predicting diffusivity and infinite-dilution

solubility in the composite material study, and it was used for all simulations in the

MEMS (Micro-Electro-Mechanical Systems) study [166][178].

5.2.3 Stochastic Assembly and Montecarlo Simulation

Demultiplexor created from the random deposition of gold particles is a good example of

how random approaches can yield useful deterministic devices is a [179,180]. Randomly

distributed collection of gold particles can be done between the input address wires and

the molecular-scale nanowires .The set of connections to each nanowire acts as a code

to select that nanowire [179]. If we can arrange the code space and statistics of the

random connections appropriately, we can arrange for most all of the core nanowires to

each have a unique address code. With 50% of the potential connections randomly

connected, a code space with 4log2 (N) address bits will allow one to uniquely address

almost all of the N nanoscale wires [179]. Once constructed, this demultiplexor can be

used to allow a set of lithographic-scale wires to selectively address any of a large

number of molecular-scale wires. The feature of the structure is that the order necessary

to create a useful logical device is discovered after the device is fabricated. Once the

functionality is discovered, then the device is programmed to capitalize on what was

created [179]


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The expression "Monte Carlo method" is actually very general. Monte Carlo (MC)

methods are stochastic techniques--meaning they are based on the use of random

numbers and probability statistics to investigate problems. The use of MC methods to

model physical problems allows us to examine more complex systems than we

otherwise can. Solving equations which describe the interactions between two atoms is

fairly simple; solving the same equations for hundreds or thousands of atoms is

impossible. With MC methods, a large system can be sampled in a number of random

configurations, and that data can be used to describe the system as a whole

Computer-generated numbers aren't really random, since computers are deterministic

[181]. But, given a number to start with--generally called a random number.

To start with, the computer must have a source of uniformly distributed psuedo-

random numbers. A much used algorithm for generating such numbers is the so-called

Von-Neumann “middle-square digits”[182]. Here, an arbitrary n-digit integer is squared,

creating a 2n-digit product. A new integer is formed by extracting the middle n-digits from

the product. This process is iterated over and over, forming a chain of integers. The

various factors that MC taken care of is types of material, distribution of electrons and

running time. The MC method has used extensively in finding the result of ENIAC

computer.ENIAC in his stored program form have a capacity of 1800 instructions from a

vocabulary of about 60 arithmetical and logical operations.

Monte Carlo is the art of approximating an expectation by the sample mean of a

function of simulated random variables [183].

Consider a random variable X having probability mass function or probability density

unction which is greater than zero on a set of values X. Then the expected value

of a function g of X is

If X is discrete, and


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If X is continuous. Now, if we were to take an n-sample of X's, (x1;x2 : :: ; xn), and we

computed the mean of g(x) over the sample, then we would have the Monte Carlo

estimate

(x)=1/n

We could, alternatively, speak of the random variable

(X)=1/n [Monte Carlo estimator of E (g(X)]

If E (g(X)), exists, then the weak law of large numbers tells us that for any arbitrarily

small ε.

This equation signifies that as n gets large, then there is small probability that (x)

deviates much from E(g(X)). For our purposes, the strong law of large numbers says

much the same thing-the important part being that so long as n is large enough, (x)

arising from a Monte Carlo experiment shall be close to E(g(X)), as desired. One other

thing to note at this point is that (x) is unbiased for E (g(X)):

Going back to our original notation, we have the random variable (x), a Monte

Carlo estimator of E(g(X)). Like all random variables, we may compute its variance (if it

exists) by the standard formulas:

If X is discrete,


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if X is continuous, then everything can be done in terms of integrals over continuous

variables, but it all applies equally well to sums over discrete random

variables[183].There are numerous ways to reduce the variance of Monte Carlo

estimators. The “variance-reduction" techniques is one of them which is very useful.

Monte Carlo (MC) simulation is a probabilistic technique in which atomic configurations

are sampled according to their probabilistic weight. Conventional MC techniques can not

be used to predict dynamic phenomena. In contrast to molecular dynamics, Monte Carlo

seeks to determine only the equilibrium properties of a system. Although the MC method

can be used for all equilibrium property calculations for which MD can be used, it is most

commonly used for the calculation of free energies and phase equilibria.[184]

In MC, an initial state is constructed, often stochastically, that is preferably one of high

probability. Then, a “Markov chain” is started by attempting to move a particle to a new

position or orientation [185]. If we define as probability of going to state j given that you

are in state i, states in the Markov chain must satisfy the important condition of

microscopic reversibility.

Because a Monte Carlo simulation is not required to evolve a system through time

according to Newton’s Laws, it can often be made much more efficient in sampling

molecular configurations than molecular dynamics. A Monte Carlo simulation is allowed

to make highly unphysical moves that would never occur in the natural dynamics [166].

The number of steps in Monte Carlo simulation depends on the complexity of the

intermolecular potentials and the desired accuracy of the results. 104-105 MC steps per

particle are usually sufficient in order to determine thermodynamic properties of a fluid

interacting with a Lennard-Jones potential [166]


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5.2.3.1 Inverse Monte Carlo (IMC)

The intent of introducing the inverse Monte Carlo method to bridge the gap of length

and time scale in molecular computer Simulations. In a molecular system, typically a few

dozens of atoms can be simulated during a few picoseconds. As referred to above

discussion of Monte Carlo method of randomly generated inputs generated from

probability distribution and interaction of molecules in simulation process, the inverse

Monte Carlo reconstructs interaction potential between molecules if the distribution

function known .If the radial distribution function between particles are known, we can

calculate the corresponding interaction potential based on distribution functions[172].

5.2.3.2 Quantum Monte Carlo (QMC)

Many-body quantum theory provides the key ideas for understanding the behavior of

materials at the level of the electrons which bind the atoms together. Describing the

complex behavior of materials at the atomic level requires a sophisticated description of

the correlated motion of the electrons, which can be achieved using ab initio

computational methods [172]. The quantum Monte Carlo (QMC) method has many

attractive features for probing the electronic properties of real systems. It is an explicitly

many-body method which takes electron correlation into account from the outset. It is

capable of giving highly accurate results while at the same time exhibiting a favorable

scaling of computational cost with system size.

The main approaches used to solve the many-electron Schrodinger equation in real

solids are Quantum Monte Carlo (QMC) and the GW method [172]. The two are

complementary, QMC simulations tell you about ground state properties while the GW

method gives information about excitations and we use both. In this article we'll

concentrate on our QMC work since it's slightly easier to explain and particularly straight

forward to implement on massively parallel computers. The generic term quantum Monte

Carlo" covers several different methods, but the ones used here are called Variational QMC and Diffusion QMC.


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5.2.3.2.1 VMC (Variational QMC)

According to quantum mechanics, the probability that a measurement of the positions

of all N electrons in a molecule finds them at position r1; r2……; rN is the many-electron

wave function. The idea behind variational QMC is to use a computer to generate sets of

random positions distributed in exactly the same way as the results of this idealized

measurement, and to average the outcomes of many such computer experiments to

obtain quantum mechanical expectation values. Given the form of the many-electron

wave function so that one can generate the required samples, but the exact many-

electron wave function is an unknown function of an enormously large number of

variables and has to be approximated [86].

The first approximation is to replace the macroscopic piece of solid containing roughly

1023 electrons by a small model system containing no more than a few thousand. The

replacement of an effectively infinite solid by a small model system subject to periodic

boundary conditions is still an approximation, of course, and the associated finite-size

errors caused us significant problems at the beginning. The second approximation is

more problematic and harder to improve. Since we don't know the exact many-electron

wave function (even for our small model system), we have to guess it. At first glance this

looks like a hopeless task, but a surprisingly large fraction of those few quantum many-

body problems that have ever been solved" have in fact been solved by guessing the

wave function. Most of our calculations are done using trial functions of the Slater type,

Where D and D↓ are Slater determinants of spin-up and spin-down single-particle

orbitals obtained from Hartree-Fock or LDA calculations, the function u(ri;rj) correlates

the motion of pairs of electrons, and X(ri) is a one-body function. The Slater

determinants build in the antisymmetry required by the Pauli principle, and the u and X


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functions are adjusted to minimize the total energy (or more precisely the variance of the

total energy) in accordance with the variational principle [172].

The total and cohesive energies are accurate to within 0.2eV per atom, and

approximately five times better than the best LDA calculations. The full range of

molecular crystal for which this trial wave function is accurate is still not known, but so

far it has exceeded all reasonable expectations.

In a nutshell The VMC is type of QMC

• The many-electron wave function is unknown

• Has to be approximated

• Use a small model system with no more than a few thousand electrons

• May seem hopeless to have to actually guess the wave function

• But is surprisingly accurate when it work

The limitation with VMC is that nothing can really be done if the trial wave function isn’t

accurate enough; therefore, there are other methods like DMC.The QMC is used for

Total Energy Calculations and can use Monte Carlo to calculate cohesive energies of

different solids.

5.2.3.2.2 Diffusion QMC (DMC)

The major limitation of the variational QMC method is that when the assumed trial

wave function isn't accurate enough. The direct approach is to add more variational

parameters and resort to brute force optimization, but this limits one to an assumed

functional form which may not be adequate, particularly in strongly correlated systems

where the quantum state is markedly different from the Fermi state found in weakly

correlated molecules [172].

An alternative and much better approach is diffusion QMC, which is based on the

imaginary-time Schrodinger equation,


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Where we have adopted a very condensed notation in which R = (r1; r2; : : : ; rN) is a

3N-dimensional vector containing all the electron positions ,and V (R) is the sum of all

the potential energy terms appearing in the ordinary many electron Schrodinger

equation.

The diffusion QMC method finds the overall ground state of the system, and for the

many-electron Schrodinger equation this happens to be a totally symmetric function of

the particle coordinates. To ensure that the diffusion QMC simulation produces an

antisymmetric state we guess the shapes of the regions of configuration space within

which the wave function is positive and negative and solve the imaginary-time

Schrodinger equation to find the lowest energy state consistent with that guess[172].

Fixed-node diffusion QMC is perhaps best regarded as a variational method in which,

instead of assuming a trial form for the whole wave function, we assume only a trial form

for the nodal surface. We use fairly simple guesses based on mean field wave functions

and our experience so far suggests that these give very good results, accurate to

considerably better than 0.1eV per atom.

This is in accurate choosing the right wave function and can use to calculate cohesive

energies of different solids. This uses a Green’s function to determine the energy

function at different orbital levels of electron energy transmission like ground state and

Eigen state.

Diffusion Monte Carlo (DMC) is a quantum Monte Carlo method that uses a Green's

function to solve the Schrödinger equation.


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5.2.3.2.3 MOLECULAR MONTE CARLO ALGORITHM (MMC)

This uses Monte Carlo method to determine structure and properties of matter. The

problems that can be solved using MMC are (i)Surface Chemistry(ii)Metal-Insulator

Transitions(iii)Point Defects in Semi-Conductors(iv)`Excited States and (v)Simple

Chemical Reactions

The root of quantum montecarlo is the Schrodinger Equation which handles many

electrons in the equation and it generate sets of random positions as result of comparing

electron positions to the many-electron wave function.

MMC simulation works by generating transitions between different states. This

involves: (a) generating a random trial configuration (b) evaluating an ‘acceptance

criterion’ by calculating the change in energy and other properties in the trial

configuration; and (c) comparing the acceptance criterion to a random number and either

accepting or rejecting the trial configuration[186].

Before proceeding to MMC algorithm designed for molecular simulation, the markov

chains of states generally considered.

A Markov chain is a sequence of random values whose probabilities at a time interval

depend upon the value of the number at the previous time. A simple example is the non

returning random walk, where the walkers are restricted to not go back to the location

just previously visited.

The controlling factor in a Markov chain is the transition probability; it is a conditional

probability for the system to go to a particular new state, given the current state of the

system. For many problems, such as simulated annealing, the Markov chain obtains the

much desired importance sampling. This means that we get fairly efficient estimates if

we can determine the proper transition probabilities [186], [187].

A general MMC Algorithm1 is illustrated below:

1 Manas Ranjan Pradhan, “A Monte Carlo Scenario to Molecular Computing” published in the proceeding of International Multiconference on Intelligent Systems & Nanotechnology (IISN-2010) organized by Computational Intelligence Laboratory (CI-Lab),Centre For Excellence In Nanotechnology (CENT),Institute Of Science And Technology, Klawad (ISTK) held on 26th-27th Frebuary,2010,pp-367-370.


[349]

Part 1 Create an initial configuration (config).

Part 2 Generate a Markov Chain for Ncycles:

Loop i =1 ... NCycles

Create a new configuration (configTrial).

Find the transition probability w (config, configTrial).

Generate a uniform random number (R) between 0 and 1.

If (w > R) then Accept move (config = configTrial).

Else Reject move (config is unaltered).

End if

End loop

In part 1 of the algorithm after an initial configuration is generated, the part 2 of

algorithm works by repeatedly either accepting or rejecting new configurations. The

algorithm takes markov chain to realize that not all states will make a significant

contribution to the properties of the system. Attention goes to those states that make the

most significant contributions in order to accurately determine the properties of the

system in a finite time. This is achieved via a Markov chain, which is a sequence of trails

for which the outcomes of successive trails depend only on the immediate predecessor.

In a Markov chain, a new state will only be accepted if it is more’ favorable than the

existing state. At this stage there are two important facets of the above algorithm :(i) the

evaluation of configurations involves assumptions concerning the nature and extent of

intermolecular interactions; and (ii) the acceptance criteria is ensemble-dependent [172].

In general, the potential energy (Epot) of N interacting particles can be evaluated as

Where the first term represents the effect of an external field and the remaining terms

represent particle interactions, i.e., u2 is the potential between pairs of particles and u3 is

the potential between particle triplets etc.


[350]

The calculation of energy can be achieved by using the following

Nested double loop:

loop i ¬ 1 ... N - 1

loop j ¬ i + 1 ... N

Evaluate rij.

Evaluate u (rij).

Accumulate energy.

end j loop end i loop

The Model studying the dynamics of driven polyphenylene chain through a molecular

wire has attracted considerable attention from experimental or theoretical perspectives in

recent years. Understanding this process is very helpful for movement of substitute

molecules in the molecular wire. Consider a polyphenylene chain consisting of N similar

molecules of size a, which are driven through a chain of length L.

At each instant, a molecule is picked up at random and attempts to move in any

direction, and the move is accepted with probability p = min[1, exp(-∆U/kT); where ∆U is

the energy change of the chain and kT is thermal energy. In our model, the energy of

polyphenylene chain can be expressed as

U=U Electric+ UC-H Bond + U H-bond-Bond

U Electric is the electric potential energy due to a constant electric field in the z-direction,

is the sum of bond energy between consecutive

benzene molecules I,i+1(of bond length bi)

An important issue in this model is to relate a MC step to duration in real time. To

estimate this MC step, we consider the movement and attachment of substituent

molecule through benzene chain. The velocity of a molecule is approximately

Vs=qE/6πηa; where q is the net electric charge per molecule and ‘a’ is the molecule

size. In our simulations, the driving electric force is of order 10_12 J/m; the molecular

force η is about 10-3Ns/m2; and a ≈ 10-9 m.Therefore, tunneling velocity is about 0.1 m/s.

Since the tunneling time for one electron to cross a barrier say suppose 10-8 m in our


[351]

simulation is about 100MC steps, we estimate that 1MC step is equivalent to 10-9 s: This

need to check with experimental results. For a chain of 100 molecules driven by an

electric field E ≈ 107 V=m; the average blockade time in our simulations is of order

1000MC steps (or 1 µs). We can set for N=5, N=10,N=15…etc.

Monte Carlo simulation is a method for iteratively evaluating a deterministic model

using sets of random numbers as inputs. This method is often used when the model is

complex, nonlinear, or involves more than just a couple uncertain parameters. It is a

simulation can typically involve over-10,000 evaluations of the model. By using random

inputs, we are essentially turning the deterministic model into a stochastic model.

Monte Carlo simulation is categorized as a sampling method because the inputs are

randomly generated from probability distributions to simulate the process of sampling

from an actual population. Choosing a distribution for the inputs that most closely

matches data we already have, or best represents our current state of knowledge. The

data generated from the simulation can be represented as probability distributions (or

histograms) or converted to error bars, reliability predictions, tolerance zones, and

confidence intervals.


[352]

5.2.3.2.4 The Metropolis Algorithm

The Metropolis algorithm generates a random walk of points distributed according to a

required probability distribution. From an initial ``position'' in phase or configuration

space, a proposed ``move'' is generated and the move either accepted or rejected

according to the Metropolis algorithm. By taking a sufficient number of trial steps all of

phase space is explored and the Metropolis algorithm ensures that the points are

distributed according to the required probability distribution [89]

Fig.5.2.3.2.4 The Metropolis algorithm


[353]

5.3 Use of Density Function Theory in Molecular Simulation

As discussed in chapter 3 Density functional theory (DFT) is among the most popular

and versatile methods available in condensed-matter physics, computational physics,

and computational chemistry.

Three types of DFT calculations exist ( 1)Local density approximation (LDA) which is

fastest method, gives less accurate geometry but provides good band

structures.(2)Gradient corrected-gives more accurate geometries.(3)Hybrids (which are

combination of DFT and HF methods)-give more accurate geometries[135].

In order to understand the physics behind mean-field approximations such as the LDA,

it helps to think about one particular electron moving through the sea of nuclei and other

electrons making up a solid. This electron is attracted to the positively charged nuclei,

which are so massive that they can be treated as immobile, and repelled by the other

electrons, which are negatively charged. As a rough approximation, it seems sensible to

replace the fluctuating forces due to the other electrons by the static electrical (Coulomb)

force due to the average electronic charge density. This simple mean field

approximation, known as the Hartree approximation, helps keep the electron away from

regions where there are lots of other electrons on average, which is a good start, but

misses something important.

As the electron moves around, the others stay out of its way; you can think of the

labelled electron carrying round a little exclusion zone", usually known as the exchange-

correlation (XC) hole, within which other electrons rarely venture[172]. The electron

density near this electron is therefore less than the average density, and the Hartree

approximation doesn't take this into account. Other mean field approaches such as

Hartree-Fock theory attempt to build in the effects of the XC hole in an approximate way,

but these approximations aren't particularly accurate.

Density functional theory is based on a remarkable theorem, which states that it is in

principle possible to devise an exact mean-field theory. In other words, the mean field

can be chosen in such a way that the energies and electron densities obtained by

solving the one-electron equations come out exactly right. Practical applications of

density functional theory have had to rely on approximations such as the LDA only


[354]

because no available method has been able to calculate the shapes of XC holes in real

materials.

Quantum Monte Carlo methods can calculate accurate XC holes. However; density

functional theory is an extremely successful approach for the description of ground state

properties of metals, semiconductors, and insulators. The success of density functional

theory (DFT) not only encompasses standard bulk materials but also complex materials

such as polyphenylene, proteins and carbon nanotubes[188].

DFT methods take less computational time than HF calculations and are considered more accurate.

5.4 Use of NEGF in Molecular Simulation

Nano-scale and molecular-scale systems are naturally described by the discrete level

models, for example eigen-states of quantum dots, molecular orbitals, or atomic orbitals.

As it is needed to calculate the current through the nanosystem, it is assumed the

contacts are equilibrium, and there is the voltage V applied between the left and right

contacts. The calculation of the current in a general case is more convenient to perform

using the full power of the non-equilibrium Green function method. In a non-interacting

system, the NEGF can be evaluated by solving Schrödinger equation. In the case of

interacting systems, the other approach, known as the method of tunneling (or transfer)

Hamiltonian (TH),plays an important role, and is widely used to describe tunneling in

superconductors, The main advantage of this method is that it is easily combined with

powerful methods of many-body theory[189].

Canonical transformations from the tight-binding (atomic orbitals) representation to the

eigenstate (molecular orbitals) representation play an important role. From knowledge of

this function one can calculate time-dependent expectation values such as currents and

densities, electron addition and removal energies and the total energy of the system.

The approximations within the nonequilibrium Green function method can be chosen

such that macroscopic conservation laws as those of particle number, momentum and

angular momentum are automatically satisfied. Dissipative processes and memory

effects in transport that occur due to electron-electron interactions and coupling of

electronic to nuclear vibrations can be clearly diagrammatically analyzed[189]


[355]

When one does actual calculations the Green functions are often expressed in a basis, i.e. one writes

Where 'i represents a suitable chosen basis such as Hartree-Fock molecular

orbitals. The coefficients Gij are in fact the Green functions with respect the annihilation

and creation operators with respect to this basis.

A complete theoretical analysis of rectification is complex and will be described

elsewhere, but we can address the most important experimental findings through a

qualitative analysis of the system.

Formulation of the current-voltage dependence of a molecular junction (e is the

charge of an electron, h the Planck’s constant, V the applied voltage, Ef the Fermi

energy of the electrode, ∆0 the spectral density of the electrodes, G the element of the

Green’s function for the system as defined in above equation, and η the energy variable.

The Green function for the system contains the molecular Hamiltonian for the system i.e.

H is the molecular Hamiltonian. The model also assumes that the electron is not

localized on any portion of the molecular bridge for an appreciable time during the

electron transport process; that is, the mechanism of electron transport does not involve

a hopping step to a trap state [191].


[356]

5.5 Analysis on DFT and NEGF The molecular surface undergoes various voltage range test in the region of insulator,

so that multiple donor and acceptor sites can be incorporated. The contacting electrode

in both side of donor and acceptor can be gold. Green’s function and density functional

theories are used to study electron transport characteristics through single molecules

addressed by two metallic contacts. Each contact is modeled with one nanoscopic end

connected to the molecule and one macroscopic end connected to an external potential

difference. The method can be applied to any molecular system for which ab initio

calculations can be performed [190]. It allows us to determine the molecular orbitals

participating in the electron-transfer process, the current−voltage characteristics of the

junction, the density of states, and the transmission function, among other properties,

providing a fundamental tool for the development of molecular electronics

Current−voltage characteristics are in excellent agreement with a break junction

experiment and with other ab initio calculations, yielding new insights regarding electron

transport through single molecules[190].

This work has been used density functional theory (DFT) calculations on molecular

junctions consisting of a single molecule between two gold electrodes. The molecules

consist of an alkene bridge connecting acceptor-nitro group, donor amine end groups in

various combinations. The molecular geometries are optimized and DFT calculates

wave functions and eigenstates of the junction .The electron transport properties for the

junction are calculated by non-equilibrium Green's function (NEGF) formalism. The

current–voltage characteristics for the various molecules in the position of X and Y are

then compared with rectification is observed for these molecules, particularly for the

donor–bridge–acceptor case where the bridge is fixed as alkene. However, at smaller

bias rectification is in the opposite direction and is attributed to the lowest unoccupied

orbital associated with the acceptor group [161],[154].

.The nonequilibrium Green’s function (NEGF) approach in combination with density

functional theory (DFT) was used to investigate the electron transport properties of a

single diode molecule that consists of two weakly coupled electron donating π-system

and electron withdrawing π-system, resembling the conventional p–n junctions. The

rectification ratio for different sets of X, Y then calculated. The mechanism of the

rectification behavior was analyzed in terms of the evolution of molecular energy levels,


[357]

the change of spatial distribution of molecular orbitals, and the electron transmission

spectra. The asymmetric evolutions of the energy levels and the alignment of the states

localized on the electro donating and electron withdrawing segments with the applied

voltage were found to be essential in generating this rectification behavior for the

molecular system [155].Referring to the molecule of our interest in chapter 4 ,The

nonequilibrium Green’s function technique and density functional theory were used to

perform quantum-mechanical calculations of electronic transport of single molecular

wire .To inspect the substituents effect on the electronic transport properties, both the

electron-donating (–NH2) and electron-withdrawing (–NO2) groups were asymmetrically

introduced into the conjugated molecular wire, resembling the semiconductor p–n

junction. The results demonstrated the rectification behavior of the molecular wire. The

asymmetric evolutions of the energy levels and spatial distributions of the frontier

molecular orbitals with the applied voltage are found to be essential in generating this

current–voltage asymmetry [156]. Results of this theoretical study are compatible with

the assumptions that electron transport occurs through the lowest unoccupied molecular

orbital, that the conduction barrier is determined by the molecule chemical potential, and

that the molecule becomes charged as the external potential increases. We can explain

the nonlinear character of the current−voltage characteristic of the molecule and its

temperature dependence [157].

5.6 Conductivity in Molecule and Charge Transfer

As we have seen in fig.5.2.1(d) to 5.2.1(f), so many molecular structures can be

formed with different molecule position. These shows that these molecules can help in

designing so many molecular gates. As an example out of these 24 structures, at least

276 gates can be formed. But all these gates may not suitable for conductivity properties

or current flow in the molecular circuit. Therefore it is needed to study the conductivity

properties of various molecules by taking various structures in a more scientific method.


[358]

5.6.1 Conduction in Nanoscale

Consider an ideal one-dimensional conducting wire of length L which tends to infinity.

An electron transfer from left to right is described by wave equation

φk = (exp(ikx) and corresponding energy єk = (h2/2m)k2

The current carried by this mode

and the current carried by the electrons having the energies εk Є [E,E+∆E] and

propagating to right is

Suppose now the wire connected in the left to a reservoirs of electrons having a

chemical potential µ1 and emitting electrons to the right (K>0) connected to right

reservoir of electrons having chemical potential µ2= µ1-eV and emitting electrons to the

left (K<0) [192].

The total current I is

= =e2/V


[359]

Fig.5.6.1 (a) Condutance associate with Electronic mode

The conductance associate with an electronic mode is exactly e2/h.This is basis of

landaeur approach of quantum transport [192].

Considering a conductor of finite cross section, with band width of w, the electrons are

confined by a abrupt wall.A mode propagating to right is confined by two indices, a wave

vector k in the direction of x and an integer n transverse in y direction.

The energy and wave vector can be given as

If now the contour placed in between two reservoirs having the chemical potential µ1

and µ2= (µ1-eV) with eV 0, the total current leads


[360]

=

Fig.5.6.1(b) Eigen Mode Contribution to Molecular Wire

Where Nmax corresponds to last occupied sub band i.e. the chemical potential lies

between

Every occupied eigenmode contributes to the conductance with e2/h.If spin degeneracy

occurs it would be 2e2/h.

If one varies the chemical potential, one expects to see a step in the conductance, each

time a new sub band is occupied. The conductance counts the number of transmitted

modes.


[361]

5.6.1.1 The Landaeur Formula

The landauer fourmal which is considered to be the most suitable method to study the

charge transfer and conductance in a molecule can be derived as follows.

For a conservatives system The time independent Hamiltonian is given by

The probability current defined as

So that

It can be derived

Using the fact that

If the spatial dependency of the wave function is real function, then This is

the case for instance for vanishing wave ~exp(-kx) or stationary wave ~cos (kx).On the

other hand for a progressive wave like exp (ikx)[192].

For a localized potential є>U, an incoming wave from left exp (ikgx) is at same time

reflect as to left exp (-ikgx) and transmitted to right as exp (ikdx).Since the current is

conserved as jd=jg, we should consider probability current amplitude rather than


[362]

probability amplitude. So we can consider unit incoming current from which we will

deduce the reflection r and transmitted t amplitude.

Rewriting continuity for the probability current amplitude and its derivative, one gets

And defining eta (η) =Kd/Kg, the expression for transmitted and reflected amplitudes read

as

Again for energy є>U, one can consider an incoming wave from the right with unit

probability current amplitude which is reflected to the right with an amplitude r’ and

transmitted to the left with an amplitude t’.

one finds then

t’=t, r’=-r

This allows us to indroduce a diffusion matrix S which connects the probability current

amplitudes incoming in the region responsible for diffusion (potential step) with the out


[363]

coming probability amplitude or in other words which connects the incoming waves with

the out coming transmitted or reflected waves

An important property of S is unitarity which comes from the current conservation

=

If S†S = 1

This implies the relation

This implies important conclusion

which represents transmission and reflection probabilities .One can directly measure

these quantities in a transport experiment.


[364]

Before going further, consider a case of vanishing wave in right for 0<=є<U, defining

є=kd/kg, where u-ε=h2kd2/2m

One can find

The wave is completely reflected r 2=R=1, one can notice that the dephasing delta,

tan(δ/2)=Kd/Kg, which correspond to the delay that wave took during “visiting” virtually to

the region in right.

5.6.1.1.1 Conductance at Zero temperature

Consider the same example with zero temperature. Only the energies greater than U will

interference since in the other case the wave is completely reflected. It is supposed that

the electrons emitted from the left come from a reservoir with a chemical potential

μg>U.This means that incoming wave from the left with energy 0≤Є≤ μg, are each one

occupied with one electron. This results in chemical potential difference of μg- μd=eV.

To compute the total current, it is enough to consider the for instance the case of x>0.

The current from left is

and from right

Therefore the total current is I=Ig-Id


[365]

For V 0, one defines the conductance G=I/V.

is the landaeur formula. The conductance is the transmission. The same result will also

predict for x<0.

5.6.1.2 Quantum Current

The probability current J(x, t) is given by

--------------Eq(5.6.1.2.1) is defined such that

----------------Eq(5.6.1.2.2) describes the change in the probability Pab (t) of finding a particle in the region a < x < b

at time t.

where the probability can be find as

---------------------Eq(5.6.1.2.3) The e wave function Ψ(x, t) describing the particle is normalized and can derived as

---------------------Eq(5.6.1.2.4)


[366]

Going from the first to the second line one uses the Schrödinger equation

, and its complex conjugate (the potential terms cancel), and from the second to the third line one uses partial integration. Setting b = a + dx with dx infinitesimal, allows one to write Eq. 1.2 as

----------------------Eq(5.6.1.2.5) With ρ(x,t)=| Ψ(x,t) |2, the probability density. You might recognize Eq. 1.4 as a continuity

equation, which describes the relation between a density and a current.

Probability currents may seem rather abstract, but they are easily related to something

more familiar. Suppose the particle has a charge q, then the expected charge found in

the region a < x < b at time t is

Defining the electrical current as I(x, t) = qJ(x, t), Eq(5.6.1.2.2) can be rewritten as

This makes the rate of change of charge is given by the difference between the

current flowing in from one side minus the current flowing out from the other side. Even if

the wave function cannot be normalized, but the probability current according to

Eq.5.6.1.2.1 is still a well-defined quantity. Free particles often enter in scattering

problems, where we are interested in quantities like reflection and transmission

coefficients. Since the latter can be directly defined in terms of probability currents, we

can get away with using non-normalizable wave functions.


[367]

In a stationary state

Suppose now that Ψ(x, t) describes a stationary state, i.e.

Then one finds from Eq(5.6.1.2.3)

and from Eq(5.6.1.2.1) and Eq(5.6.1.2.2)

The probability current is constant, i.e. independent of position and time. For example,

consider a free particle with the wave function

From Eq. (5.6.1.2.3) we calculate

Since Pab is the probability of finding the particle in the interval between x = a and x= b,

i.e.an interval of length b − a, we can interpret |A|2 as the probability density per unit

length. It is also called the particle density.

The probability current is easily calculated from its definition, Eq(5.6.1.2.1)

According to de Broglie’s relation p = hk is the momentum of the particle and

is then the velocity of the particle. The electrical current is given by I=qJ=qvρ which is the usual definition of an electrical current, namely charge×velocity×density.


[368]

For the wave function of Ψ(x)=Aeikx ,both velocity and density are constant, so the wave

function describes a uniform current. Suppose q > 0; then if k > 0 the current flows to the

right, if k < 0, the current flows to the left. From now on we assume that k > 0.

Now let’s go to the more complicated wave function

‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ Eq(5.6.1.2.6)

with A, B constants. The associated probability current is

which is interpreted as a right going current minus a left going current. In a scattering

problem one would interpret the first term on the right hand side of Eq(5.6.1.2.6) as the

incident wave and the second term as the reflected wave. Then interpreted as the

difference between incident and reflected currents

Figure 5.6.1.2 One-dimensional scattering problem.

In the left region the potential is a constant V (x) = VL, in the middle region the potential

V (x) can be anything, and in the right region the potential is a constant V (x) = VR. The

middle region is called the scattering region. The left and right regions are called the left

and right leads. In the left lead we have an incoming wave AeikL

x and a reflected wave

Be−ikL

x and in the right lead we have a transmitted wave FeikR

x.


[369]

The reflection coefficient R is defined as the ratio between reflected and incident

currents

The solution in the left region is given by Eq (5.6.1.2.6) with k replaced by kL

The solution in the right region is given by the transmitted wave

, x in right region,

with

One can calculate the transmitted current as

The transmission coefficient T is defined as the ratio between transmitted and incident

currents

--------------------------------Eq(5.6.1.2.7) From the fact that the current has to be independent of position everywhere, J(x,t)=J , it

follows that


[370]

This relation expresses the conservation of current, or: “current in = current out”

(reflected plus transmitted). No matter how weird the potential in the middle region is, the

current going into it has to be equal to the total current coming out of it[192]. No particles

magically appear or disappear in the middle region.

Now it can be seen that

i.e. the reflection and transmission coefficients add up to 1. Since these coefficients

denote the probabilities that a particle is reflected or transmitted, this simply states that

particles are either reflected or transmitted [192]. 5.6.1.3 Quantum Conductance 5.6.1.3.1 Tunnel Junction The device shown in Fig. 5.6.1.3.1 (a) is called a tunnel junction. The left and right

regions consist of metals and the middle region consists of an insulator material, usually

a metal-oxide [192].Such devices can be made in a very controlled way with the middle

region having a thickness of a few nano meters. One is interested in electrical currents,

i.e. the transport of electrons through such junctions, or more generally in the current-

voltage characteristics of such a device. On this small, nanometer length scale electrons

have to be considered as waves and quantum tunneling is important. Nano-electronics is

the general name of the field where one designs and studies special devices that make

use of this electron wave behavior.

Starting with the simplest possible one-dimensional model of a tunnel junction. The

atoms of a material attract electrons by their nuclear Coulomb potential. The electrons in

low lying energy levels are localized around the atomic nuclei and form the atomic cores.

If the atoms are closely packed and the material is sufficiently simple, all these atomic

potentials add up to a total potential that is relatively constant in space. The constant

potential depends on the sort of atoms a material is composed of, so it is different for

every material. The potential in the tunnel junction of Fig.5.6.1.3.1 (a) along the transport

direction can then be represented by a square barrier, as shown in Fig. 5.6.1.3.1 (b).


[371]

Figure 5.6.1.3.1 (a) Schematic representation of a Tunnel Junction

The yellow balls represent atoms of a metal; the blue balls represent atoms of an

insulator. The left and right regions stretch macroscopically far into the left and right,

respectively. The electron waves in the metal are reflected or transmitted by the insulator

in the middle region shown in Fig. 5.6.1.3.1(a)

Figure 5.6.1.3.1 (b) Simple approximation of the potential along the transport

direction of a tunnel Junction


[372]

In the metal (left and right regions) the potential is constant, V (x) = V1.In the insulator

the potential is also constant, V (x) = V0, where V0 > V1. The incoming, reflected and

transmitted waves are given by Aeikx Be−ikx and Feikx.

Figure 5.6.1.3.1 (c) Potential when bias voltage applied to Left and Right leads The potential when a bias voltage U is applied between the left and right leads. This

changes the potential of the right region by ∆V = −eU to V1 − ∆V with respect to the left

region. The voltage drop is indicated schematically. If the bias voltage is small, i.e.∆V ¿

V0 − V1, then we can still use the transmission coefficient T calculated for the unbiased

square barrier (given by the dashed line).

Now as discussed above the Launder formula can be deduced as below According to Eq(5.6.1.2.7) the transmitted electrical current is given by

Using the definition I = qJ (the charge q of an electron is −e). The incoming current Iin is given by

To find the velocity v and density ρ of the incoming electrons,we need to find the

incoming current created in a device .In an experimental setup this is done by applying a

voltage difference U between the left and right regions. The left and right regions are

metals, which can be connected to the two ends of a battery, for instance.


[373]

This results in a potential drop ∆V = −eU between left and right regions, as shown in Fig. Figure 5.6.1.3.1 (c). We supposes that the temperature is zero and the metals are non-magnetic, so we

have spin degeneracy. Then vρ is given by the simple expression

If the potential drop ∆V in Fig. Figure 5.6.1.3.1 (c) is small compared to the barrier

height V0 − V1, we can use the unbiased square barrier potential from Fig. Figure 5.6.1.3.1 (b) to calculate the transmission coefficient T. This is the so-called linear

response regime. Then below Equations give the transmitted electrical current, also

called the tunneling current .

The conductance G as current divided by voltage can be shown as

-------------------------------------------Eq(5.6.1.3.1) Since T is just a dimensionless number between 0 and 1, (e2/π h) has the dimension of

conductance. It is the fundamental quantum of conductance; its value is (e2/πh ≈ 7.75 ×

10−5 Ω−1.

Eq(5.6.1.3.1) is called the Landauer formula; it plays a central role in nano-electronics.

The simple derivation of Eq(5.6.1.3.1) can be done but here only simple introductory

quantum mechanics approach has done.


[374]

5.6.1.3.2 The Pauli Exclusion Principle and the Fermi energy The left and right regions of a tunnel junction consist of metal wires shown in Figure 5.6.1.2 (a).These wires are supposed to be very, very long compared to the size of the

middle region. In a simple-minded model the potential of a metal wire looks like

Fig.5.6.1.3.2. The potential is approximately constant inside the wire and it has steps at

the beginning and end of the wire to keep the electrons in.

The energy levels of this square well potential are,

----------------------Eq(5.6.1.3.2.1) The spacing between the energy levels, En – E (n−1), scales as (1/L2 ) with the length L

of the wire. If L is large, the spacing becomes very small, so from a distance the energy

level spectrum almost looks like a continuum, as illustrated by Fig. Fig.5.6.1.3.2.

The wave functions are given by

-------------Eq(5.6.1.3.2.2) These are not exactly what we need, because they correspond to standing waves,

whereas we need traveling waves to describe currents. For the incoming current we only

need the exp (iknx) part. Setting A = 1/(i√2L), the corresponding electron density

according to Eq.1.10 is


[375]

Fig.5.6.1.3.2: Schematic drawing of the potential and the energy levels of a long

wire.

The points (−L/2) and (L/2) mark the beginning and the end of the wire. The spacing

between the energy levels is so small that the energy spectrum almost looks like a

continuum. EF marks the Fermi energy, i.e. the highest level that is occupied in the

ground state by an electron.

The wire is full of electrons since each of the atoms in the wire brings at least one

electron with it. Filling the energy levels according to the Pauli principle, and having N

electrons in total, the highest occupied level is EN/2.The highest occupied level in the

ground state is called the Fermi energy or EF .


[376]

5.6.1.3.3 Incoming and Tunnel currents Analyzing the tunnel junction and fill its potential profile in Figure 5.6.1.3.1 (b), with

electrons from the left and right wires. This is shown in Fig.5.6.1.3.3 (a). The Fermi

energy EF in the left and right regions is the same. The exclusion principle then tells us

that there can be no flow of current.

Fig.5.6.1.3.3 (a) Tunnel junction where left and right regions are filled with

electrons

With Tunnel junction where left and right regions are filled with electrons, the Fermi

energies EF on the left and right side are identical. The exclusion principle forbids

electrons to trespass from left to right or vice versa.

Any electron on the left side that would try to go to the right side finds an energy level

that is already occupied by an electron, which excludes any other electron from going

there, and vice versa. This confirms what we know from everyday life; in an unbiased

system no current flows.


[377]

Now apply a bias voltage between left and right regions as in Figure 5.6.1.3.1(c). The

result is shown in Fig.5.6.1.3.3 (b). The bias voltage lowers the right region in energy.

Suddenly the electrons on the left side that occupy energy levels En with EF − ∆V <En < EF find empty levels with that energy on the right side. They can tunnel through the

barrier to occupy these levels. Provided the potential drop ∆V is small, we can

approximate the transmission coefficients of all these electrons by T at an energy E = EF

[192]. The incoming current of Iin = −evρ has contributions from all electrons with

energies between (EF − ∆V) and EF.

---------------------Eq(5.6.1.3.3.1) The factor of 2 is there because there are two electrons in each level. This sum in Eq(5.6.1.3.3.1) is rather awkward, but by a trick we can turning it into an integral

-----------------Eq(5.6.1.3.3.2) Where

Looking at Eq(5.6.1.3.2.1), turning the sum into an integral is allowed because L is

very large, so ∆k is tiny. The lower and upper bound of the integral in Eq(5.6.1.3.3.2)

should correspond to the energies (EF − ∆V) and EF , whereas the integral is over dk,

which is again awkward. We can however turn it into an integral over dE, using the

following


[378]

----------------- Eq(5.6.1.3.3.3) Putting Eq(5.6.1.3.3.3) in Eq. Eq(5.6.1.3.3.2) it gives

------- Eq(5.6.1.3.3.4) Collecting Eq(5.6.1.3.3.4), Eq(5.6.1.3.3.2) and electron density ρ =1/2L in Eq (5.6.1.3.3.1), we find for the incoming current

-------------------Eq(5.6.1.3.3.5) This is the required expression for the incoming current.

The tunnel current is then given by

----------------------------- Eq(5.6.1.3.3.6) Where the transmission coefficient T needs to be calculated for the energy E = EF. Note

that with ∆V =−eU, where U is the potential difference (in Volts), this corresponds to

.

Then Landauer formula Eq(5.6.1.3.1) can be derived straightforwardly.


[379]

Fig.5.6.1.3.3 (b) Tunnel junction with an applied bias voltage.

All the levels occupied by electrons in the left region with an energy (EF − ∆V) <En < EF correspond to empty energy levels in the right region. The electrons in these levels

can tunnel from though the barrier from left to right.

The Landauer formula expresses the conductance in terms of a transmission

coefficient. In other words, the problem of finding the conductance becomes a problem

of solving the scattering problem.

To study the kinetics of a polyphenylene chain passing through a ployphenylene chain,

we have simulated its tunneling process 100 times for each set of parameters. Each

chain is randomly deposited above the surface. Due to the applied electric field, the

polyphenylene chain will land on the surface and search for the molecule in donor and

acceptor section to bind.


[380]

Summary

This chapter has been discussed the computational modeling and computational

approach that can be used for molecular computing of electronic devices. This chapter

mainly shows the conductance of molecules with different molecule of choice and how

various simulation methods helps in predicting the conducting feature of molcules.The

molecular simulation and molecular dynamics of molecules has been discussed in brief.

The simulation approach has been discussed with Molecular Monte Carlo algorithm

(MMC).The QMC (Quantum Monte Carlo) has discussed with Inverse Monte Carlo

(IMC), Variational QMC and Diffusion QMC.The Metropolis algorithm has discussed

in a nutshell. The Use of Density Function Theory (DFT) and NEGF (Non-Equilibrium Green's function) for studying the voltage-current properties in Molecular Simulation

has been discussed. At last the conduction of molecule in nanaocsale with quantum

properties has been discussed with Landaeur formula.

chapter-5 · the below figure shows a general approach of quantum mechanics in developing various...

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