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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.
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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.
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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
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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.
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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
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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
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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]
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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].
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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,
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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.
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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
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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
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=
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.
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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
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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
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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.
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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
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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)
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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.
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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.
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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.
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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
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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).
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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
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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.
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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.
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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
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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 .
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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.
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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
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----------------- 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.
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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.
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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.