time dependent simulations of nano-scale conductors

8/7/2019 Time Dependent Simulations of Nano-scale Conductors

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Time dependent simulations of atomic scale

conductors

Kevin Breslin

May 11, 2005

Abstract

A series of simulations were carried out by theoretically placing

an atomic scale conductor between two electrodes, thus forming a

nanometer-scale capacitor. By changing the gap between the plates

and the atomic conductor and the voltage, static and dynamic density

functional methods were used to calculate the discharge parameters of

these simulations. The static transport calculations were performed by

Dr Stefano Sanvito and Maria Tsoneva, TCD. The first series of results

dealt with finding interpretable I-V curves produced from the outputs

of DINAMO, a computer code used in this project to calculate the

discharge parameters in these simulations. The simulations specific to

this project concern the dynamical discharge of finite nano-capacitors

connected by an atomic wire, and comparing these results with corre-sponding calculations, based on the standard static transport formal-

ism. The second part was the investigation of non-conducting charged

states for nanosystems discovered in the first set of simulations.

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Contents

I Background 5

1 Introduction 5

1.1 abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

II Background 6

2 Background 6

2.1 Density Functional Theory (DFT) . . . . . . . . . . . . . . . 62.2 The Kohn Sham equation . . . . . . . . . . . . . . . . . . . . 82.3 Self Consistency in calculating the charge density . . . . . . . 82.4 The Hartree Fock Approximation . . . . . . . . . . . . . . . . 92.5 Hopping Integrals from using the Hartree Fock Approximation 92.6 The Landaeur Formalism:- Static DFT . . . . . . . . . . . . . 102.7 Exchange and Correlation and How the Local Density Ap-

proximation can be made in this case . . . . . . . . . . . . . . 102.8 The Tight Binding Bond Model . . . . . . . . . . . . . . . . . 12

3 Time Dependent Density Functional Theory 12

3.1 The TD Kohn Sham Scheme . . . . . . . . . . . . . . . . . . 133.2 Time Dependent Tight Binding Bond Model . . . . . . . . . 133.3 Single orbital time-dependent tight binding . . . . . . . . . . 14

4 Applying the density functional in practice 15

4.1 Static and Dynamic Density Functional Theory . . . . . . . . 154.2 Major critisms against Static Density Functional Theory . . . 164.3 Coulomb Blockades . . . . . . . . . . . . . . . . . . . . . . . . 16

III An initial set of simulations for the charging of thecapacitor plate system 18

5 Introduction 18

5.1 Motivation for initial tests . . . . . . . . . . . . . . . . . . . . 185.2 Classical expectations . . . . . . . . . . . . . . . . . . . . . . 18

6 Method 19

6.1 An initial series of simulations . . . . . . . . . . . . . . . . . . 196.2 A comment on the System of units used . . . . . . . . . . . . 19

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7 Programming 20

7.1 The Capacitor program . . . . . . . . . . . . . . . . . . . . . 207.2 The Capacicalc program applied for calculating the capaci-

tance for wire based systems . . . . . . . . . . . . . . . . . . . 20

8 Results 21

8.1 Tables of Results . . . . . . . . . . . . . . . . . . . . . . . . . 218.1.1 The size of the externally applied voltage drop . . . . 218.1.2 The length of the lattice parameter . . . . . . . . . . . 218.1.3 The size of the atomic spacing between the plates . . 21

9 Discussion 22

9.1 The Initial set of experiments . . . . . . . . . . . . . . . . . . 229.1.1 Changing the plate width . . . . . . . . . . . . . . . . 229.1.2 Increasing the size of the drop in the externally applied

potential . . . . . . . . . . . . . . . . . . . . . . . . . 229.1.3 Changing the length of the chain . . . . . . . . . . . . 229.1.4 Changing the size of the spacing of the atoms in the

plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229.2 General Comment on results . . . . . . . . . . . . . . . . . . 23

9.2.1 Paralell Plate Capacitor model . . . . . . . . . . . . . 239.2.2 Independent Dipole model . . . . . . . . . . . . . . . . 239.2.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 23

9.3 A Comment on surface effects . . . . . . . . . . . . . . . . . . 24

10 Conclusion 24

10.1 The Initial set of experiments . . . . . . . . . . . . . . . . . . 24

IV A comparison between Static and Dynamic DFT incapacitor discharge systems 25

11 Motivations and Practical Importance 25

11.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2511.2 The Practical Importance of the subject . . . . . . . . . . . . 25

11.3 Applications and Problems with nanostructures . . . . . . . . 2611.4 The Theoretical Importance of the subject . . . . . . . . . . . 2711.5 How the Simulations were carried out . . . . . . . . . . . . . 27

12 Method 29

12.1 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 2912.2 Changing the capacitors dimensions and other parameters . . 2912.3 Discharge Curves . . . . . . . . . . . . . . . . . . . . . . . . . 3012.4 The purpose of Current-Voltage Graphs . . . . . . . . . . . . 3012.5 The purpose of Charge-Time and On-Site Energy-Time Graphs 30

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13 Programming 33

13.1 Programs Necessary to calculate results . . . . . . . . . . . . 3313.2 Creating capacitor geometries from the Capacitance Program 3313.3 Creating Discharge and I-V plots from the Calculate Program 33

14 Results 35

14.1 S imulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3514.1.1 Current-Voltage graphs . . . . . . . . . . . . . . . . . 3514.1.2 Current Discharge Graphs . . . . . . . . . . . . . . . . 3514.1.3 Current-Voltage Graphs . . . . . . . . . . . . . . . . . 3714.1.4 Voltage Position Graphs . . . . . . . . . . . . . . . . . 3714.1.5 Current Discharge Graphs . . . . . . . . . . . . . . . . 37

14.1.6 Voltage Discharge Graphs . . . . . . . . . . . . . . . . 37

15 Discussion 37

15.1 The Comparison between Static and Dynamic DFT . . . . . 37

16 Conclusion 38

V Analyzing the effects of a Coulomb blockage 39

17 Introduction 39

18 Motivation 39

19 Practical Importance 40

19.1 The Practical Importance Coulomb blockades? . . . . . . . . 4019.2 The existence of Coulomb blockades can be useful . . . . . . 4019.3 The existences of Coulomb blockades are problematic . . . . . 40

20 Method 42

20.1 Creating new graphs . . . . . . . . . . . . . . . . . . . . . . . 42

21 Programming 44

21.1 Amendments to the Calculate program . . . . . . . . . . . . . 44

22 Results 45

22.1 S imulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4522.2 Charge Distributions . . . . . . . . . . . . . . . . . . . . . . . 4522.3 Voltage time Graphs . . . . . . . . . . . . . . . . . . . . . . . 4522.4 Current-Voltage Curves . . . . . . . . . . . . . . . . . . . . . 5122.5 Charge Discharge graphs . . . . . . . . . . . . . . . . . . . . . 5122.6 The on site energy graphs . . . . . . . . . . . . . . . . . . . . 5122.7 On site charge graphs . . . . . . . . . . . . . . . . . . . . . . 51

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23 Discussion 52

23.1 The investigation of Coulomb blockades . . . . . . . . . . . . 52

24 Conclusion 54

VI Summary 5524.1 The Initial set of experiments . . . . . . . . . . . . . . . . . . 5524.2 C onclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

25 Conclusion 55

VII Appendices 57

26 Programs Used 57

26.1 The Capacitance Program . . . . . . . . . . . . . . . . . . . 5726.2 The Calculate Program . . . . . . . . . . . . . . . . . . . . . 6026.3 The Capacicalc Program . . . . . . . . . . . . . . . . . . . . 63

27 Acknowledgments 65

28 Appendix 65

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Time dependent simulations of

atomic scale conductorsMaster of Science Project

Part I

Background

1 Introduction

1.1 abstract

A series of simulations were carried out by theoretically placing an atomicscale conductor between two electrodes, thus forming a nanometer-scale ca-pacitor. By changing the gap between the plates and the atomic conduc-tor and the voltage, static and dynamic density functional methods wereused to calculate the discharge parameters of these simulations. The statictransport calculations were performed by Dr Stefano Sanvito and MariaTsoneva, TCD. The first series of results dealt with finding interpretableI-V curves produced from the outputs of DINAMO, a computer code usedin this project to calculate the discharge parameters in these simulations.The simulations specific to this project concern the dynamical discharge offinite nano-capacitors connected by an atomic wire, and comparing theseresults with corresponding calculations, based on the standard static trans-port formalism. The second part was the investigation of non-conductingcharged states for nanosystems discovered in the first set of simulations.

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Part II

Background

2 Background

2.1 Density Functional Theory (DFT)

The transition from Schrodinger equation based quantum mechanics to Den-sity Functional theory occurs when the wave function tends to be more anal-ogous to the charge density of the system. The DFT model is not just ofimportance in the field of condensed matter physics but it is useful in chem-istry, geophysics, biophysics, metallurgy and material science to analyzethe electrical, thermal, mechanical and magnetic properties of the modeledmaterial. A detailed mathematical proof of the existence of a Density Func-tional for a charge density (r) can be found in Finnis book [3]. The energycan be represented as a functional of the charge density, where the resultingfunctional E[] can by variational theory be represented an Euler-Lagrangeequation

E[]

(r)=

F[]

(r)+ Vext (r) = (1)

where Vext is an external potential mentioned later and F[] is an internalenergy functional (i.e. it represents the kinetic energy of the electrons) andthe potential energy between electrons and is a constant.

For any reasonable charge density (one that is positive, continuous andcan be normalized) (r) there exists an asymmetric wavefunction | describ-ing the N electrons in the charge density (r). Density Functional Theorywas initially developed by Thomas [1927] and Fermi [1927], but the func-tional developed by Hohenberg, Kohn and Sham.

The foundations for Density Functional Theory come from the postulatesof the Hohenberg-Kohn Theory of a density functional, the postulates aresummarized below

1. The Uniqueness Postulate:- Ground state expectation value of anyoperator is a unique functional of ground state charge density

2. The Variational Principle postulate :- The charge density (r) is suchthat it minimizes the Energy functional E[].

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3. The Internal Energy Functional postulate:- The Functional F[], which

represents internal energy acts independent of the external potentialVext

In Density Functional Theory, the charge density completely determinesall features of the electronic behaviour of the system in the ground state. Ittherefore means that the ground state of the interacting atom is a uniquefunctional of the charge density, which can be determined by a form of selfconsistent variational theory. The exact functional in most cases is unknownbut the functional of the ground state [15] can be written as

EG () = T[ ()]+ () N() + () H () +EXC [ ()] (2)The first term is the Kohn Sham kinetic energy Term, it represents the

total kinetic energy of the electrons in the system and is given by

T[(r)] =n

fn

(r)

h2

2m2(r) dr (3)

where is the wavefunction defined by

(r) =n

fn(r)2 dr (4)

The second term is an electrostatic energy associated with the electronic

charge and potential due to nuclei with

N(r) =

(r)

40|r Ri|dr (5)

The third term is called the Hartree Potential and is defined as

H(r) =1

2

(r)

40|r r|dr (6)

This is the functional derivative of the Hartree energy, which representsthe electrostatic self-energy of the charge density.

VH

= (r)H

(r) dr (7)

The fourth term is the exchange correlation energy which does not justrepresent the physical effects of correlation and exchange mechanism, it alsocorrects overall for the overestimation of the Hartree approximation on theinter atomic Coulomb repulsions. Using the Local Density Approximationit can be defined as

EXC [()] =

XC ( (()) () (8)

[3]

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2.2 The Kohn Sham equation

To find the energy eigenvalues of any quantum mechanical system wherethe Hamiltonian is known, the solution of the Schrodinger equation must befound

H(r) = E(r) (9)

The time dependent version of this equation is

H(r, t) = ih(r)

t(10)

The version of the Schrodinger equation used for the wave function of thejth electron is also called the Kohn Sham equation

h2

2m 2j(r) + Veff(r)j(r) = jj(r) (11)

As you can see this is a single electron alternative to the complexand insolvable many body problem. The equation is calculated self con-sistently. The method carried out, essentially applies variational methodtechniques to the functional allowing for a calculated charge density. Thetechnique is explained in greater detail in the next section. [3]

2.3 Self Consistency in calculating the charge density

The self-consistency results from applying variational calculus to the func-tional above. An iterative loop is set up until the charge density estimated

from the loop is equal to the charge density from the previous iteration. Inother words the charge density converges to a finite value. The loop consistsof four parts

1. Enter a charge density, calculating the Hartree Potential from theequation

2VinH(r) = 4in(r) (12)

2. An effective potential is then calculated from adding up the bias volt-age the Hartree Potential and the Exchange correlation Potential

Veff(r) = Vbias(r) + VinH(r) + V

inXC(r) (13)

It is estimated from Local Density Approximations

3. Solve the Kohn Sham version of the Schrodinger equation

h2

2m2j(r) + Veff(r)j(r) = jj(r) (14)

4. From the solution calculates the charge density and reiterate untilconvergences

(r) =Ni=1

Nj=1

i(r)j(r) (15)

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2.4 The Hartree Fock Approximation

The Hartree Fock Approximation [3]relies on the variational principle, (atfirst order this is

E= |H|

, (16)

where Eis the energy functional, | is an N-electron state (which can berepresented by a Slater Determinant) and H is the Hamiltonian given inatomic mass units by

i

1

22i +

1

2

i

j

1

|ri rj |+i

I

ZI|ri RI|

. (17)

The Slater Determinant wavefunction | is given in the form of the tensor

1 (x1) 2 (x1) 3 (x1) . . . n (x1)1 (x2) 2 (x2) 3 (x2) . . . n (x2)1 (x3) 2 (x3) 3 (x3) . . . n (x3)1 (x4) 2 (x4) 3 (x4) . . . n (x4)

......

.... . .

...1 (xn) 2 (xn) n (xn) . . . n (xn)

where n (xn) is the wavefunction of the nth particle in this series of elec-

trons. This is clearly problematic to solve mathematically but by applyingthe single electron DFT approximation can diagonalize the tensor to N num-ber of equal single electron equations. The simplest form of the Hartree FockApproximation is given by

T+ Vext + G

|n = n |n(19)

G describes all the electron-electron interactions representing the HartreePotential and Exchange and Correlation effects.

The Charge Density can be represented by the operator

=n=0

|fn(20)

2.5 Hopping Integrals from using the Hartree Fock Approx-imation

The wavefunctions for most real systems are not orthogonal, local nucleican share a charge distribution. Indeed it this process that occurs metal-lic bonding, and explains other aspects of metallic chemistry or indeed any

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other form of solid state chemistry e.g. solids where covalent (e.g. poly-

mers), ionic (e.g. salts and other ionic crystals) and even Van der Wallsforces apply (between graphite layers) though the last would obviously re-quire abandoning the LD Approximation. In these situations the probabilitydistributions (charge densities) produced are also known as overlap or Hop-ping integrals. They represent the electron distributions between differentatoms while electron distribution on a single atom is represented by a selfenergy integral.

[3] Slater and Koster showed that the number of hopping integrals forsystems of atoms of s,p and d orbitals can be greatly reduced by symmetryarguments and from this an overall electrostatic self energy of the chargedensity can be calculated.

2.6 The Landaeur Formalism:- Static DFT

The Static Theory of conduction has a long history starting with the pioneer-ing work of Landauer. The Landaeur formalism has since been re expressedin numerous alternative, but algebraically equivalent forms. The core phi-losophy of this static approach is the following. We consider a nanostructureconnected to two (defect-free ideal metallic) leads. We imagine also that,very far from the nanojunction, each lead is connected to its own particlereservoir, which injects electrons into that lead with effective electrochemicalpotential, appropriate to that reservoir. If the two reservoir electrochemical

potentials are different (this difference is interpreted as the applied batteryvoltage), then a net current flows.The physical assumptions of this model state that the transport through

the quantum confined region is considered always to be coherent, and theelectrons scatter only elastically. But emission and reflection from the par-ticle reservoirs (electrodes) is always incoherent. The quantum wire in thiscase must be in thermal equilibrium with the connected electrodes.

The formalism is limited physically by a dephasing process which maybe crucial under certain circumstances.

The Calculation of conductance, and therefore current can be made fromtwo observations from Landaeur The potential across a conductor can be

viewed as arising from the self-consistent build-up of carriers, rather thanthe current arising from the applied electric field The conductance of adevice can be calculated from the electron transmission through it. Thereis no justification for this procedure to work for interacting electrons

2.7 Exchange and Correlation and How the Local DensityApproximation can be made in this case

The Exchange Correlation term is a feature of the systems asymmetryand represents the inclusion of the affect of the presence of the exchange-

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correlation hole as well as corrections to the functional. The overall probabil-

ity of finding an electron in space is reduced. This occurs because interactingelectrons with parallel spins repel due to the Pauli Principle. The exchangecorrelation hole has a charge of -e with a radius of its electron hole pair isthe distance the charge density of the hole screens that of the electron tocreate a neutral quasiparticle i.e. a particle like body that behaves like acharge less electron. The exchange and correlation potential takes the form

VXC(r) = XC((r)) +

dXC()

d

=(r)

(21)

This approximation can be made due to the Local Density Approximation.

[2]

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2.8 The Tight Binding Bond Model

In the Tight Binding (TB) approximation, the calculations of the energiesrequire the Hamiltonian to include interactions only with nearest neighboursand include only two-center contributions. The sum of the occupied eigen-values is the (attractive) electronic contribution to the total energy. It hasfour main advantages

TB is the simplest model that takes account of atomic structure.

TB allows mechanical, structure and current induced effects to betreated on the concept of the inter-atomic bond.

If the model can be solved as a computational mesh, the results ofTB transport and its wavefunction can be expanded into discrete pro-portions. The model then can be transformed to any other electronicmodel. The TB model reduces the size of the data involved and istherefore computational efficient.

The Orthogonal WavefunctionThe following approximation is made

|i =

n,n,,

i,n,(t)n, (22)

is the notation of type of orbital centred at the nucleus n (= s,p,d,f,

Gausian etc.)so the approximation is made of the different positions basisstates are orthogonal (i.e they do not overlap).

n,|n,

= n,n, (23)

In the orthogonal positional basis the tight binding Hamiltonian is

|Hn,,n, | =

n,n,,

n,Hn,,n,n,(24)

For Static DFt in a TB orbital basis called ab initio TB, the Kohn-Shamequations solved slef-consitently are

HTBn,,n, | = E, \, (25)

3 Time Dependent Density Functional Theory

Physics Rev. Lett 52 (1984) 997 The external potential Vext that a collectionof electrons interacts with is a unique functional of the time evolving electrondensity for a given many body wavelength.

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3.1 The TD Kohn Sham Scheme

The time evolving density is in this case can be found by solving a set oftime dependent one electron electron Schro dinger equations.

h2

2m2n + Veffn = in (26)

This is not however self consistent, the charge density is calculated from thewave-function that solves numerically the differential equation above. In aperfect (ideal) TDDFT the V[XCwould be required to be a functional of theinstantaneous electon density (r) but the time eveolution of(r) also. TimeDependent Density Functional Theory in a TB oribtal basis then consists of

solving the TD one elctron kohn-Sham equation.

ihi,n,=n,

Hn,,ni,n, (27)

In Time dependent DFT the toatl energy of the system is irrelvant, what isrelevant is the responce of the electron density to an external pertubation(i.e. an applied field) and its the time evolution. Section 3 Journal ofPhysics Condensed matter The corresponding Euler Lagrange equations arenow Kohn Sham one electron Schrodinger equations

ihi = T i +n

vni + vhxci (28)

vhxc =E

(29)

E is a Hartree-exchange-correlation functional e.g. LDA

3.2 Time Dependent Tight Binding Bond Model

The modification [4] is made that the bare nuclei are treated as ions, The Nethat appear explicitly are therefore valence electrons with a valence chargedensity . Let Mn Rn(t) and Rn(t) be the mass, position and velocity ofthe ions. Let Rn(t) be defined by the orbital wavefunction for a state |

r| = |[r Rn(t)]|n, (30)

is a normalized orbital state and represents the orbitals on the atom. thevalence-electron density of an atom n in isolation becomes

n = n[r Rn(t)] =n,

n,

fn,[r Rn(t)]2 (31)

0 = 0[r Rn(t)] =n,

n,

[r Rn(t)]2 (32)

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The energy functional is taken to be

E[] = E[0] +

(r, t) 0r, r Rn(t)V0(r, t)dr +

TB [](33)

In general the dynamical equation of the theory is given by the La-grangian

L = ihNei=1

< i(t)|i(t) > Nei=1

< i(t)|T|i(t)| > E[] (34)

+Nei=1

Mn[Rn(t)]2/2 [Rn(t)]

whereE[] =

NZi=1

(r, t)vn[r Rn(t)], dr + Ehxc[]

= (r, t) =i

|i ((r), t)|2

and the kinetic energy operator is given by T= p2/2me

3.3 Single orbital time-dependent tight binding

In this approach [4] the electrons are descibed by the tight binding timedependent single electron Schrodinger equation

ihin =n

HTBnn in (35)

where HTBnn = H0nn + V

TBnn in time dependent tight binding, with the charge

potential given by

VTBnn = nn

Unqn +

n=n

fnnqn

(36)

with fnn describing the Coulomb interaction

fn

n =

e2

40

R2nn +

e2

40U

2(37)

where Rnn = Rn Rn

and U= 7eVin this series of experiments. Uis a parameter of the model. This model is the simplest form of the localdensity functional theory. Describes albeit in a model way onsite Hartreeand interactions and intersite monopole Hartree interactions. The TDSE issolved for the electrons but the ions are treated as being stationary, thatapply an external bias Vbias in this experiment. The values of these aretaken from a fit applied to Gold (Au) Phil mag. A81 (2001) 1833.

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Figure 1: Diagram of the Charge injection in Time Independent DensityFunctional Theory (TIDFT). In Fig 3 above each electrode is connected to aparticle reservoir, which injects electrons into that electrode with an effectiveelectrochemical potential. These are modelled by Fermi-Dirac functions. Aflow of net current occurs if these electrochemical potentials are different.The calculated current and Current-Voltage curves are determined uniquelyby the assumed injection electron distributions, and an assumed relationbetween one electron potential and density.

4 Applying the density functional in practice

There are two methods of DFT that can be applied, they are Static and Dy-namical DFT. In the static case, the charge density at each individual pointcalculated self consistently is then substituted into the functional above tofind a local energy distribution. The differences in applications of the twotechniques are mentioned below.

4.1 Static and Dynamic Density Functional Theory

A key part of this project is to compare the difference in the results taken

from Static (Time Independent) and Dynamical (Time Dependent) DensityFunctional Theory Methods.

In the static theory of transport, each electrode is connected to a notionalparticle reservoir, which injects electrons into that electrode with an effectiveelectrochemical potential. A flow of net current occurs if these electrochem-ical potentials are different. The Calculated current and Current-Voltagecurves are determined uniquely by the assumed injection electron distribu-tions, and an assumed relation between one electron potential and density.One problem is that if there is an error in the imposed injection energy distri-bution functions (assumed a priori to be Fermi-Dirac), this would generate

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erroneous transport properties and gross disagreements between I-V curves

measured experimentally and computed. But the Static DFT method stillremains rigorous for non-interacting electrons; under question at present forinteracting electrons, at high bias, away from equilibrium.

Alternatively because dynamic DFT deals with the movement of elec-trons through the wire, and in particular the response between electrons,it has been proved to give an exact description of electrons away from theground state, including current-carrying systems.

Reference M Di Ventra TN Todorov J of Physics Condensed matter 16(2004) 8025

It places no a priori assumptions about the effective electronic distribu-tions thus it overcomes the limitations of the static approach, though the use

of transient currents in finite systems imposes its own practical difficultiesin trying to back out information about steady states.

The static transport calculations were performed by Dr Stefano San-vito and Maria Tsoneva, TCD. The results of their approach were used toperform a more detailed comparison between the two models. In partic-ular whether anything new could be brought from the time independentapproach.

4.2 Major critisms against Static Density Functional Theory

1. It uses XC functionals and potentials developed for ground state to

attempt to describe excited current carrying electrons. There is noreason why this approach shouls always work, ther has been knownexamples of IV curves producing gross anomolies with experiment.

Physical Review Letters Vol 84 2000 pg 979

The IV curves assigned resonant features to the energies of Kohn Shamquasibound state computed from a static functional. but one electronKohn Sham energies in static DFT are meaningless. Static Iv curveswould place resonant feautures at spurious energies.

Phy Rev Letter 94 (2005) 146803.

2. A priori assignment of fixed electrochemical potentials

The method of applying the Fermi-Dirac distrubutions of electrons ap-proaching the junctions is okay for non interacting electrons. Howevererrors in assigning these functions to non-equillibrium scenario giveserrors in computed IV curves.

4.3 Coulomb Blockades

When a large amount of charge enters from a large space to a smallerspace, the charges may build up on the chain and repel additional charge

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from flowing through it. The resultant Coulomb Blockades are described as

the increase of Coulomb energy, and the subsequent opening of a Coulombgap that occurs when the electronic charge tunnels into small nano-particleislands. This has the effect of increasing the tunneling resistance. TheCoulomb blockade produces Non Conducting charge states There are twoconditions that are necessary for the Coulomb blockade to occur

To avoid thermal fluctuation, the islands capacitance C must be smallenough e2/2C > kBT. At temperatures of 1K the effects should beobservable below 1015F.

To avoid quantum fluctuations. The junction resistance between theplates must be large compared to the resistance quantum. Quantum

fluctuations would allow electrons to pass through the energy barrierin a process called macroscopic quantum tunneling or co-tunneling.

[7]

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Part III

An initial set of simulations for the

charging of the capacitor plate

system

5 Introduction

The theory and method for these simulations will be discussed in the nextsection. These initial sets of simulations help to get a feel for the basic

physics involved in the study.

5.1 Motivation for initial tests

Before a comparison could be made between the two aspects of DensityFunctional Theory in section 2, a set of calculations were made on a pairsmall 7 7 1 atom plates to see if clear trends of the charging part ofthe experiment could be established. These computer experiments testedthe effect of increasing the size of the interatomic spacing on the plates, thedistance between the plates and the drop in the externally applied poten-tial between them. These simulations show the correlation with classicalexpectations of a macroscopic capacitor system.

5.2 Classical expectations

From classical electromagnetic theory the following expectations can bemade.

1. Placing a conductor (in this case an atomic wire) between the elec-trodes should produce a discharge comparable with the exponential de-cay discharge for common capacitors. This is because weaker conductor-slab contact should increase the resistance and the capacitor time con-stant .

2. Increasing the drop in the externally applied potential should increasethe potential difference between the plates because the onsite shiftsproduced by the voltage bias should produce a direct potential effectbetween the plates. However this does not necessarily apply absolutelyfor nanostructures because the wave-nature of electrons is ignoredmacroscopically, so the actual transport properties of these systemsmay be incompatible with classical, macroscopic expectations.

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6 Method

6.1 An initial series of simulations

Before the discharge simulations were carried out, a series of test experimentswere carried out to see how the model worked before discharge. The threesimulations were

1. Testing the effect of what will happen to the discharge if the appliedbias between the metal plates is increased.

2. Testing the effect of what will happen to the discharge if the width ofthe lattice parameter is increased.

3. Testing the effect of what will happen to the discharge if an atomicchain is placed in the middle of the plates and testing the effect ofincreasing the number of atoms on that chain

The average potential and the average charge on each plate was calculatedwhile varying the parameters:

1. The externally applied voltage bias from 0-5 Volts

2. The length of the lattice parameter between 2 and 3 angstroms

3. The atomic spacing between the plates for 1, 3 and 5 atomic spacing.

The atomic spacing refers to the number of atoms that in theory could beplaced in the vacuum between the plates with a atom to atom spacing of2.5 Angstroms ensuring no overlaps.

The results were then processed by a program called Capacicalc. Theanalysis of these results was made in comparison to the classical expectationsthat were stated in the introduction to this project.

6.2 A comment on the System of units used

The currents were measured in microamps, the voltages were measured inVolts and the charges were measured in unit charge. Using these systems ofunits:

The (on-site) energy is measured in electron volts

The capacitance is measured in electrons per volt

The resistance is measured in megaohms and conductance in microampsper volt

In unit resistance and capacitance these units would have a dischargetime constant in the region of 100 femtoseconds.

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This will be useful information in the second section where the discharge

behaviour is analyzed. All distances in these simulations are measured inangstroms, one angstrom is 1010m.

7 Programming

7.1 The Capacitor program

An output file of the co-ordinates of each individual atom in the capacitorset-up was produced in the capacitor program. In these simulations two(two dimensional) capacitor plates of a 7 7 1 atomic length separatedby a distance of 2.5 angstroms, or otherwise dependent on the experiment

was used. This program is discussed in greater detail in the next part of thereport.

7.2 The Capacicalc program applied for calculating the ca-pacitance for wire based systems

The program used to obtain the results was called Capacicalc (capacitancecalculator). It was used in conjunction with DINAMO to obtain the sum ofthe charge on both plates and the average potential along each plate. ThePotential Difference was then taken as the difference of these potentials.Geometries were set up by a program called Capacitor which drew out two

(two dimensional) capacitor plates of 771 atomic length separated by adistance of 2.5 angstroms, or otherwise dependent on the experiment. Theoutput file of the program is like this

Left Plate =

9.578350553862499

AveragePotential =

-2.117306678454579

Right Plate=

-9.578350561882605

AveragePotential=

1.674979218077951

Potential Difference=

-3.792285896532531

Left Plate refers to the charge accumulated on the left plate and Right platefor the charge on the Right Plate.

(Each charge value is in the units of electron charge and each averagepotential is in the units of volts.) The results obtained from this programare in the next section.

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Spacing (A) Left Plate (e) Right Plate (e) Potential Difference(V)

1 2.36616622891230 -2.36616622907380 -0.2777182749764933 1.63377999935660 -1.63378000002360 -0.2669585795366835 1.63333746026630 -1.63333745906980 -0.203729268368392

Table 3: Experiment 3: The size of the atomic spacing between the plates.

9 Discussion

9.1 The Initial set of experiments

9.1.1 Changing the plate width

The length of the gap between the plates: The smaller the gap between theplates is the larger the potential difference is. Charges reduce at smallerdistances, The charges at each plate are also equal in the same manner aspreviously mentioned.

In the next experiment the distance between the plates should be largeenough to ensure a large build up of charge.

9.1.2 Increasing the size of the drop in the externally applied

potential

From the chart it was observed that increasing the size of the drop in the

externally applied potential increases the potential difference between platesand the charge polarity of one plate in comparison with the other.In the next experiment the Drop in the externally applied potential is

stilled varied, this should allow a detailed investigation of the transportbehaviour in the discharge case.

9.1.3 Changing the length of the chain

When the size of the atomic chain is increased the potential difference ineach simulation is subject to the charge balance between the plates. Longerchains also have the effect of increasing the gap between the plates so theflow of charge is significantly reduced from a one atom chain to a three atom

chain. The charges are not equal for longer chains as there could be a flowof charge along the atomic chain

In the next experiment the size of the plates should be large enough toavoid the chain significantly effecting the charge distribution on the plates

9.1.4 Changing the size of the spacing of the atoms in the plate

The potential difference in each simulation reduces as the size of the spacingincreases. This is due to the reduction in the Coulomb Force which goes

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as 1r

. Also less electrons are freed because this reduces the electric field

between the plates.In the next experiment the spacing is increased by the presence of a

chain and an additional spacing between the chain and the plates.

9.2 General Comment on results

There is a model for detailling these results are two explanations for thebehaviour of the plates in general.

A paralell plate capacitor

An independent dipole model

9.2.1 Paralell Plate Capacitor model

At distances where the plate-plate distances (d) is significantly larger thanthe lattice parameter (a) the system operates Sas a paraelell plate capacitor,where the equation of the potential is the sum of the potential due to theapplied field and the potential due to the onsite energy

P D =

Qed

490a2+

2q

eU

(38)where Q is the excess number of electrons per plate and q is the excess

number of electrons per atom.

9.2.2 Independent Dipole model

At distances where the lattice parameter (a) is significantly larger than theplate to plate distance (d) the system acts as 49 independent dipoles. Thepotential in this case is the potential across one of these dipoles given bythe equation

P D = 2

qe

40d

q

eU

(39)

9.2.3 AnalysisIn the case of 5V, and a 2.5Agap in Table 1 the PD calculated by the firstmethod would be 30.7V, but in the second method it would be 1V; thissituation seems to be somewhere between the two methods. In the case of3.0Aspacing and a 2.5Agap in Table 2 the PD calculated by the first methodwould be 14.8V, but in the second method it would be 0.6V, this situationseams to favour the dipole system better

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9.3 A Comment on surface effects

The effects of surface atoms in comparison to internal atoms may be neededto be reduced. Surface atoms, and in particular corner placed atoms donot have the same number of nearest neighbours and therefore provide thesystem with a degree of anisotropy. The effect of surface atoms is lowestwhen the ratio (3) below is smallest.

Surface Area

V olume(40)

For a box (or parallelepiped) shaped substance the formula becomes

2 (lw + wh + lh)

lwh = 21

l +1

h +1

w

. (41)

Therefore the surface effects are reduced in comparison to the bulk materialwhen the size of each of the dimensions (i.e. the length (l) the width (w)and the height of the structure (h)) is large. In the nanoscale (quantum)systems these dimensions are small, therefore the surface effects on the twoelectrodes cannot be ignored.

10 Conclusion

10.1 The Initial set of experimentsIn these experiments, the best system to produce a stable discharge wastested using smaller scale system, these small scale systems gave an aware-ness of the physical properties of a nano-scale capacitor set up. From theresults it was found that it was favorable in creating a discharge system tohave both the voltage drop in the externally applied potential and the dis-tance between the plates great enough to ensure a large build up of chargebetween the plates. Including thicker plates would also benefit the build upof charge. The size of the plates and the spacing between atoms should bothbe of a reasonable value. This is to be expected as classical effects dominatein larger systems.

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Part IV

A comparison between Static and

Dynamic DFT in capacitor

discharge systems

11 Motivations and Practical Importance

11.1 Motivations

In this series of investigations the initial motivation for the present work wastwofold:

To find interpretable I-V curves from the transient dynamical dischargeof finite nano-capacitors connected by an atomic or a molecular wire.The current-Voltage curves show the resistive and transport behaviourof the system. The procedure used was the single orbital TDDFT to

study the flow of electrons across a nanojunction. To compare these results with corresponding calculations, based on the

standard static transport formalism. With the information obtainedfrom the Current-Voltage graph, the differences can be analyzed andthe observations found can be explored in greater detail. This is totest the effect of relieving some of the restrictions placed by StaticDFT e.g the a priori assumption of the energy distrubutions. See ifthe effects can be discovered by TIDFT lie outside the scope of thestatic approach.

From these initial motivations the results that were found could open up

new areas to be Studied but the functionals themselves do not need to beimproved.

11.2 The Practical Importance of the subject

The Density Functional Model of analyzing the charge density to produceresults can be used as a purely mathematical approach to analyze the be-haviour of small scale or nanoscale systems. The approach is quantum me-chanical so it is a statistical approach. However because of constraints placedon this functional by applying Time independent functional theory (Kohn

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Sham and the tight binding approximation) as well as the other physical

limitations of the set-up, (the temperature, the geometry and voltage pa-rameters of the twin plate system used, surface effects etc.) the data outputcan be represented as a continuous set of results. The effects that are pro-duced at this quantum level, should by the Bohr Correspondence principleexplain the classical electrical properties such as resistance and capacitance.The Density Functional Model is a very useful tool in Electronic engineeringparticularly because it replaces the theoretical Schrodinger wavefunction bya measurable parameter, the electronic charge density. The fact that thischarge density is also confined to be real and its value is limited (by conser-vation of charge) also makes it scientifically useful.

There are two practical reasons for simulating an atomic system and

using it to compare the results through experiment in order to develop thescience.

Improving the model on the experimentThe simulated model can beused along with the experiment to explain anomalies such as impuritiesand other known anomalies that can be found in the nanostructure.

Improving the experiment on the model- If effects from experimenta-tion cannot be explained by known anomalous effects highlighted inthe model then the approximations used under the model may needto be re-examined.

Indeed part of this experiment is to compare two different theoretical modelsbased on Density Functional Theory, so in this experiment the accuracy ofthe approximations used under the model is examined. The simulations havea stand alone value that must also not be ignored, these simulations mustfirst be accurate models of the physical system being studied. If simulationsare accurate they can be used instead of expensive nanoscale apparatusfor research purposes. The research can detail both the potential and thelimitations of how these results can be applied by engineering (e.g. they mayhighlight where new structures can be developed, or reveal where complexbehaviour occurs in the system).

The model applied in these simulations does show remarkable consistency

with the physical reality of a nanostructure, but as the practical need fornanostructures increases so will the need for more accurate representations.

11.3 Applications and Problems with nanostructures

The series of simulations that is being carried out deals with a real field ofcondensed matter research called nanostructures. Typically nanostructuresin the form of fine films or thin wires are of dimensions between 1 and 10nanometers. Quantum wires like the ones in the simulation can be developedby either selective etching or electron confinement techniques.

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2. The electric external field is then removed; this creates a voltage bias.

Fig 1.2

3. The result of action in 2 allows electrons to flow across the two capac-itor plates. Fig 1.3 This flow of charge is the simulated discharge

4. This allows the DINAMO program to carry out the calculation ofcurrent and potential drop during discharge. Fig 1.4. The analysis ofthe results is carried out by a new separated program called Calculate,detailed in section 11

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12 Method

12.1 Measurements

Simulations were carried out using the DYNAMO program, which testedthe effective discharges through the two electrodes with no conductor inthe middle of the plates. Then the effect of varying the conductor (with 4length atomic chains, at 2.5A , at 2.8A and at 3.0A chain to plate gaps)at different external Voltages (Voltages 4V, 3V, 2V, and 1V) was testedto find stable discharge scenarios. They work in the same way as a sim-ple Capacitor-Resistor circuit, where the capacitor is first charged and thenis discharged through a resistor. The difference is that the addition of asmall scale conductor is placed in the middle of the capacitor plates causing

a short circuit. At the macroscopic level the current and potential dropacross the capacitor is an exponential decay. (These parameters decay ex-ponentially with a half-life of CR/log 2). However the effects change afterthe adding of a small scale conductor (in the form of a wire) at the mid-dle of the two conductive plates. Placing the conductor in the middle ofthe electrodes should produce a discharge comparable with the exponentialdecay discharge about common capacitors. Increasing the spacing betweenthe plates and the atomic chain should increase the electrical resistance andtherefore increase the capacitors time constant. However the wave nature ofelectrons is ignored microscopically so the effects on the discharges may beof such a magnitude that they may not be so obvious. The results are cal-culated using both Static and Dynamic versions of the Density FunctionalTheory. These methods are used to compare expected discharges. DynamicDensity Functional Theory deals with the movement of electrons throughthe wire, and in particular the response between electrons as the electronsmove through the wire. Static DFT deals with Static Transport in whichEnergy boundaries are set up to control the change of the charge densities.

12.2 Changing the capacitors dimensions and other param-eters

To see how the changing of the capacitor geometry and other parameters

affected the charge distribution in charging and discharging, the followingchanges were made

1. The voltage between the plates were set at 4.0V, 3.0V, 2.0V, 1.0Vand the effects of the effects of their change in setup on the effectivedischarge were determined.

2. A conductor chain was introduced between the plates and its distancefrom the plates is increased to 2.8A, 3.0Awhile noting the effect on theeffective discharge

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When applying DINAMO in calculating the voltage, the voltage across

the conductor was ignored. They are applied over a time period of 50nanoseconds. In our simulations the ions are treated as being stationarywhile electrons are treated as quantum mechanical particles.

12.3 Discharge Curves

When trying to establish stable discharges, Current-Time and Voltage-TimeCurves would provide a good basis for comparison. A stable discharge oc-curs when these curves appear to copy the macroscopic exponential decaybehaviour that would be expected in what is a microscopic setup. From thegraph (Fig4) the overall shape is roughly those expected for a classical ca-

pacitor discharge Initially (within the first femtosecond) there exists a periodof build up of charge, represented by the peak. In this first femtosecond theself consistency calculations required are being made in the simulation, andso the time before this period is complete is unrepresentative of a physicalsystem.

Notice that the residual Potential Difference graph, there is a period ofcharge with no flow of current which suggests a charged but non conductingstate called a Coulomb Blockade. A fuller analysis is carried out in thesection entitled Expansion:- Investigating Coulomb blockages

12.4 The purpose of Current-Voltage Graphs

One of the main analyses of results will come from drawing up Current-Voltage Curves. The aim of drawing up Current-Voltage curves is to explorethe behaviour of the electron transport across the atomic wire during thedischarge of the capacitor type set up. The currents will be affected byCoulombic repulsions of the atomic chain producing large scale jumps. Thiscan be seen quite easily in the static discharge sense. A differential resistancecan be taken from this graph to see by how much this repulsions limits themagnitudes of the currents.

12.5 The purpose of Charge-Time and On-Site Energy-Time

GraphsThe Charge-Time and On-Site Energy-Time graphs primarily show how thedistribution of charge and the potential difference between the plates changesduring the time of discharge. These graphs allow for the examination of thetransport mechanisms that the electrons moved by to be made. Of interestin this case is the comparison between ballistic and non ballistic transportmechanisms.

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Figure 2: The Simulation diagram: The first step shows the system polarizedby external field E, the second step shows the removal of the external field.This in turn allows the electrons to flow across, shown in the third step. Theforth and final step shows that current and potential drop are calculatedfrom the results found during discharge

Figure 3: The plate setup, the diagram shows the initial polarizations ofthe plate slap one slab have a positive bias and the other a negative. Thepotential bias is applied in centre of plates so the chain atoms at each endbecome oppositely polarized to the atoms on the other side

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Figure 4: The Charge, Voltage and Current Discharge Graph. The overallshape is roughly as expected for a classical capacitor discharge. Notice in the

second graph charge and the potential difference V, does not exponentiallydecay towards 0 Volts but instead remains at a residual constant potentialdifference. There is also an excess of charge (q) between the plates whenthere is no flow of current (I): This suggests a charged but non conductingstate- Coulomb Blockade

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13 Programming

13.1 Programs Necessary to calculate results

There are two programs used with DINAMO to give the results

1. The Capacitor program creates a capacitor plate system with a wireacross it. It also sets the voltage bias, and can be adapted to changethe geometry of this capacitor plate system.

2. The Calculate program creates I-V curves for each atom on the chainand makes discharge curves. It calculates the average of the on-siteenergies in order to do so. It also was adapted to count the on site

energies of the individual atoms on the chain.

13.2 Creating capacitor geometries from the CapacitanceProgram

Individual atomic coordinates were set along each plate (x-vertical, y-horizontal)and along the wire, were along the wire, with the centre of the wire lengthat the zero of the x-y co-ordinates and the one dimensional wire alongthe z-axis. A 17 17 4 atom system with two plates seperated by a 4length atomic chain and a wire located seperated from the two plates by2.8 Angstroms is created by the insertion into the Capacitance Programof the following.

AtomicSeparation=2.50_dp

LengthinAtoms=17

HeightinAtoms=17

WidthinAtoms=04

AtomicChain=04

NumberofAtoms=LengthinAtoms*HeightinAtoms*WidthinAtoms*2

+AtomicChain

s=AtomicSeparation

Separation=2.8_dp/s

Vgap=0.50_dp

13.3 Creating Discharge and I-V plots from the CalculateProgram

The Calculate program reads the output of the on site energies and currentsto produce the following graphs

1. Current-Voltage graphs:- I-V graphs were drawn for each of theatomic currents along the chain and the currents between the atomson the ends of the chain and the nearest atom on the nearest plate.

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2. An average Current-Voltage graphs. These were created for the

average current of the atomic currents along the chain and the averageof currents at the endpoints.

3. A graph showing the how the potential difference between the platesvaries with time

4. An average Current-Voltage graphs for the average charge on thechain and the currents on the endpoints.

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-3 -2 -1V / V

0

50

100

150

200

250

i/A

V0

= 4.0 V

V0

= 3.0 V

V0

= 2.0 V

V0

= 1.0 V

d = 2.5

Figure 5: Current-Voltage Graph at a plate to chain gap of 2.5 The overallshape of this graph fits almost exactly to the outline of the static densityfunctional theory graph. The linearity of the graph suggests that in thisballistic case the capacitor plates behave almost like a classical omhic resistor

14 Results

14.1 Simulations

To acquire the results needed, the following simulations were created

for Voltage bias of 4, 3, 2, 1 Volts applied between the plates

and the gap between the plate and the chain being of size 2.5, 2.8 and3.0 Angstroms.

for the 17 17 atom sized parallel plate capacitor with a five atomlong atomic chain across its centre.

In each of these simulations the geometry of the plates was maintained.The discharge simulations were set to run for 25 femtoseconds, with mea-

surements made every 0.01 femtosecond.

14.1.1 Current-Voltage graphs

[hpb]

14.1.2 Current Discharge Graphs

The Voltage discharge graphs were taken over a period of 50 femtosecondsfor graphs of

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-3 -2 -1

V / V

0

50

100

i/A

V0

= 4.0 V

V0

= 3.0 V

V0

= 2.0 V

V0

= 1.0 V

d = 2.8

Figure 6: Current-Voltage Graph at a plate to chain gap of 2.8, The presenceof inharmonic effects in this I-V curve petering out to zero

-3 -2 -1

V / V

0

20

40

60

i/A

V0

= 4.0 V

V0

= 3.0 V

V0

= 2.0 V

V0

= 1.0 V

d = 3.0

Figure 7: Current-Voltage Graph at a plate to chain gap of 3.0 The Currentfor a given voltage drops as d increases and the current peters at somenon-zero residual Voltage, this effect may require further study

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14.1.3 Current-Voltage Graphs

These were taken for Voltage potentials of 1.0V, 2.0V, 3.0V and 4.0V. Asthe plate to chain gap is increased the graphs move from being almost linear(at 2.5 Angstroms) to becoming more spread out and nonlinear in behaviourat higher voltages and distances. At higher voltages it is easier for electronsto tunnel as the energy of the electron is closer to the energy of the Coulombbarrier. (The Coulomb Barrier itself is distance squared related and is in-creased by increasing the plate to chain gap). At large voltages a PotentialDiference Voccurs at which the non current carrying charge phenomenonoccurs at increases. This larger Coulomb barrier is obviously reflective ofthe peak-trough behaviour that occurs as the current peters out.

14.1.4 Voltage Position Graphs

The graphs show the effect of the surface effects on the local voltage distri-bution. The sharp reductions are due to the way the geometry of the platesis numbered. The Voltage difference is represented by the sharp differencebetween the peaks. These results do not include the voltage along the chain.

14.1.5 Current Discharge Graphs

As expected the graphs show signs of classical discharge behaviour but onlyto a point. The residual effects that occur in the current discharge graph

are due to the presence of a Coulomb blockage.

14.1.6 Voltage Discharge Graphs

These graphs act very similar to the current discharge curves. But a VoltageDischarge will not be as sensitive to internal events and collisions as a currentdischarge would be.

15 Discussion

15.1 The Comparison between Static and Dynamic DFT

As stated in the introduction a major motivation for these results was thecomparison between Static and Dynamic DFT. In order to compare Staticand Dynamic DFT, the differences between the Current-Voltage Curves werecompared for both situations. The static I-V graph follows the envelope ofthe dynamic I-V graph very closely. This envelope corresponds to the ini-tial stages of the time-dependent I-V, for a series of different initial voltages.But the drop in current, at a residual large voltage in the time-dependentcase but this has no analogue in the static calculations. These results sug-gest a Coulomb blockade, a blockage of the dynamical current, at a finite

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remaining bias, suggests a form of Coulomb blockade: something that might

naively be expected to be a metallic wire does not conduct, this is due to aspecific stable balance of electron-electron interactions in the biased system.Working on from this knowledge, an investigation into the charge densitiesof the capacitor at the Coulomb blockade can be made. In our time de-pendent DFT approach we seem to be able to access such effects where thestatic approach does not.

16 Conclusion

In summary even at the nanoscale limits the classical expectations men-tioned in 1.2 are maintained, this is shown by the reduction in the initial

potential due to increasing the plate to chain spacing. The comparison be-tween Static and Dynamical DFT shows that Dynamical Density FunctionalTheory can acess effects that the Static case cannot. The results measuredfrom the anomalies that occur for the Dynamical DFT results outside of theStatic DFT envelope may be due to a Coulomb blockade.

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Part V

Analyzing the effects of a Coulomb

blockage

17 Introduction

After the comparison between the two forms of density functional theorywas made it was found that there was the appearance of a non conductingcharged state in the graphs which seamed to indicate a Coulomb block-ade. These non conducting states are of particular interest because nonconducting states (in the form of a Coulomb blockade) can be used in nan-otechnology research such as in developing new diodes and other small scaledevices. The Coulomb blockage is a quantum occurrence which is due tosingle charge tunneling.

The second part of the investigation was to investigate the non-conductingcharged states for nanosystems that were discovered in the first set of sim-

ulations. The duration of the capacitor discharge needed to be extendedbecause these effects only began to occur within the initial discharge time of20 femtoseconds. In these investigations the time was extended to 50 fem-toseconds. These investigations required a set of three coherent dischargecurves (Voltage-time, charge-time and current-time graphs) as well as theCurrent-Voltage curves made in the earlier experiment. In these set of ex-periments instead of measuring these results over the two plates, the resultswere measured on one of the last atoms on the chain.

18 Motivation

By comparing the discharge and Current-Voltage graphs it is hoped thatthe following can be made

to see if there is a clear building up of charge, that can be high-lighted as a being Coulomb blockade. This would be represented on acharge-time graph showing a large charge remaining in one place fora relatively long duration of time.

to make a comparison of the Coulomb blockades in the ballistic caseand in the non ballistic case can be made

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to find what is the effect that the initial voltage has on the size of a

Coulomb blockade

19 Practical Importance

19.1 The Practical Importance Coulomb blockades?

The motivation behind investigating the phenomenon of Coulomb blockadesis in the development of Single Electron Tunneling Devices in Electronics

19.2 The existence of Coulomb blockades can be useful

The theory of Coulomb blockade is useful for creating new singleelectron digital circuits and for creating new devices. An example ofthis is creating a NOR logic gate in Electronics. [8]

Another of the developing uses is in the development of magnetic ran-dom access memories (MRAM), this relies on the physical phenomenonof tunneling magneto-resistance. The plates used could be made intothe magnetic electrodes of a Fe/Ge/Co device, producing a ferromag-netic (or an anti-ferromagnetic) interactor. The gap can be filled witha tunnel junction. The wires are placed parallel between the plates.[9]

Theoretically it is hoped that investigations into the Coulomb block-

ade can help to obtain a current standard at the quantum level. Thereis a problem in obtaining such a standard measure of current physi-cally. The current measured becomes dependent on the resistance andvoltage standards obtained from the Quantum Hall and the Quan-tum Josephson effects respectively. This may be problematic wherethe current is not measured by electrostatic means. e.g. by magneticmeans. [11]

Another hope of Single Electronic Devices is that capacitors can bedeveloped to act independently of temperature, pressure and frequen-cies, and remain sensitive simply to the physical effects of charge and

potential difference. [11]

19.3 The existences of Coulomb blockades are problematic

There are three major problems for such Coulomb blockade devices [10],they require that

The temperature of operation must be raised substantially, implyingthat smaller devices are necessary.

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The transconductance of Single Electron Transistors (SET) is peri-

odic in the gate voltage, a more saturating transconductance wouldbe more suitable property in a SET. If the structure is less than thedevice length the response will not be periodic in the gate voltage ifan external voltage is applied to the gate electrode.

And to avoid large and intolerable spread of device characteristics(capacitance, resistance etc.) occurring in the devices due to Randombackground charges it is necessary to average such random effects overthe device length, making the effect of random charges negligible.

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20 Method

In experiment two the aim was to create new Current-Voltage and chargetime graphs to analyses the effect of the Coulomb blockade. These graphswere applied to one of the end atoms on the 5 atom atomic chain placed inthe centre of the capacitor.

The new graphs would reveal the time evolution of the discharge betweenthe plates and the chain. From these initial tests the objective was to findareas where a large build up of charge occurs in the graph and to analyzethe current and voltage behaviour at this time.

It would be expected that if the charge does build up on the plates itwould form a repulsive electrostatic barrier and reflect the current back into

the plates. This would be seen graphically as a negative current. but thefact that the free charge is blocked itself acts in a way to reduce the current,so the currents will tend to zero. Also a build up of such charge will lead toa build up of the electrostatic energy across the plates in the form of a largerelectric field. Any movement of charge away from this position will reducethe electric field and therefore reduce the on site potential of the atom.

In summary

Charge-time graphs show the charge on that plate. A large build upof charge that lasts, acting to reduce the current and change its directis a good sign of a Coulomb blockade.

Current-time graphs show the current between the atom and the plate,these should be reduced to zero in the presence of a Coulomb block-ade. Negative currents indicate backscattering occurring due to thisblockade.

Voltage-time graphs indicate the energy stored in the plates in the formof electrostatic field, the intensity of which should be proportional tothe on site charge.

20.1 Creating new graphs

Using the results from the previous experiment, new graphs were created toanalyze the discharge of charge with time as well as the time evolution ofthe onsite charge and the onsite energies along the chain and between thechain and the plates.

1. A graph showing the average current along the chain with time.

2. A graph showing the average current between the chain and the platewith time.

3. A graph showing the average charge on the chain with time.

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4. A graph showing the average charge on the chain with time.

5. A graph showing how the on site energy varies on the atom on eachplate which is nearest to atomic chain.

6. A graph showing how the on site charges varies on the atom on eachplate which is nearest to atomic chain.

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21 Programming

The programs and the results that were used in the previous set ofsimulations could be used now, but the calculate program needed tobe adapted to include the ability to develop graphs for the on siteenergy and on site charge measured along the chain.

21.1 Amendments to the Calculate program

Instead of omitting results along the chain, which was done in the firstset of experiments, the calculate program was amended to include theloop given below

ii=0

do i=NumberofAtoms-AtomicChain,NumberofAtoms-1

ii=ii+1

read(24,*) n,q,v

vv(ii)=v

qq(ii)=q

time=j*0.01

enddo

open(35,file=onsitechain.dat)

write(35,(10f26.16)) time,vv(1),vv(2),vv(3),vv(4),vv(5)

open(36,file=onsitechargechain.dat)

write(36,(10f26.16)) time,qq(1),qq(2),qq(3),qq(4),qq(5)

enddo

end program capacicalc

It is this loop that allows the calculation of the onsite charge and onsiteenergy graphs. The final version of the Calculate program is given inthe appendices

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Figure 8: The initial Charge Distrubution

22 Results

22.1 Simulations

In this series of experiments the simulations used the previous outputsproduced by DINAMO and were then analysed by

Current-Voltage graphs

Voltage Time graphs

Current time graphs Charge Time graph

Onsite charge and energy graphs for the atom between the lastatom on the plate and the chain.

These graphs were made for plate to chain gap distances of 2.5, 2.8 and3.0 angstroms and for voltages of 1, 2, 3 and 4V as previously stated.The discharge time in each of these simulations was 50fs. They weremeasure on one of the atoms at the endpoint of the atomic chain.

22.2 Charge Distributions

These show the locality of the charge between the capacitor plates.For the ballistic case the charge appears to be unaffected on the platesbut the chain has both positive and negative sides. Coulomb blockadesmay create a resistance due to backscattering.

22.3 Voltage time Graphs

The voltage time graphs show the same trends as the Charge timegraphs revealing that the overall capacitance of the system remains

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constant in this simulation.

0 10 20 30 40 50

t / fs

1

2

3

V/V

V0

= 4.0 V

V0

= 3.0 V

V0

= 2.0 V

V0

= 1.0 V

d = 2.5

Figure 9: Voltage Time Graph at a plate to chain gap of 2.5:- See ChargeTime Graph at a plate to chain gap of 2.5

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0 10 20 30 40 50

t / fs

0

1

2

3

4

V/V

V0

= 4.0 V

V0

= 3.0 V

V0

= 2.0 V

V0

= 1.0 V

d = 2.8

Figure 10: Voltage Time Graph at a plate to chain gap of 2.8:- See ChargeTime Graph at a plate to chain gap of 2.8

0 10 20 30 40 50

t / fs

0

1

2

3

4

V/V

V0

= 4.0 V

V0

= 3.0 V

V0

= 2.0 V

V0

= 1.0 V

d = 3.0

Figure 11: Voltage Time Graph at a plate to chain gap of 3.0::- See ChargeTime Graph at a plate to chain gap of 2.5

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0 10 20 30 40 50

t / fs

0

2

4

6

8

10

12

q/e

V0

= 4.0 V

V0

= 3.0 V

V0

= 2.0 V

V0

= 1.0 V

d = 2.5

Figure 12: Charge Time Graph at a plate to chain gap of 2.5, at this distanceit is clearly visible that the charge builds up along the chain at 10 and at 25femtoseconds, particularly at higher voltages. This is symbolic of a Coulombblockade .And because of constant capacitance it does not vary from thetrend created for the Voltage time graph

0 10 20 30 40 50

t / fs

0

2

4

6

8

10

12

q/e

V0

= 4.0 V

V0

= 3.0 V

V0

= 2.0 V

V0

= 1.0 V

d = 2.8

Figure 13: Charge Time Graph at a plate to chain gap of 2.8, the discharge inthe first non-ballistic case shows that the charge appears to meet a resistanceat 10 femtoseconds to discharging along the plate, instead of meeting theinterference effect due to the backscattering from a Coulomb blockade.

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0 10 20 30 40 50t / fs

0

2

4

6

8

10

12

q/e

V0

= 4.0 V

V0

= 3.0 V

V0

= 2.0 V

V0

= 1.0 V

d = 3.0

Figure 14: Charge Time Graph at a plate to chain gap of 3.0, The chargedischarge graph takes a longer interval to reach the endpoint in this systembut the build up of charge that appeared in the 2.5 angstrom case doesnot appear to occur here, however a Coulomb blockade may have occurredduring the discharge explaining the longer discharge time.

0 10 20 30 40 50

t / fs

0

50

100

150

200

i/A

V0 = 4.0 VV

0= 3.0 V

V0

= 2.0 V

V0

= 1.0 V

d = 2.5

Current Time Graph at a plateto chain gap of 2.5, in this simulation the current decays within the

first 10 femtoseconds and then a series of inharmonic oscillationsoccur around zero amps, this is symbolic of backscattering and

reflections due to the electrostatic repulsion force due the build up ofcharge

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0 10 20 30 40 50

t / fs

0

50

100

i/A

V0

= 4.0 V

V0

= 3.0 V

V0

= 2.0 V

V0

= 1.0 V

d = 2.8

Figure 15: Current Time Graph at a plate to chain gap of 2.8, the initialcurrent is lower at zero microamps the resonance peaks appear to be lessperiodic than in the 2.5 case

0 10 20 30 40 50t / fs

0

20

40

60

i/A

V0

= 4.0 V

V0

= 3.0 V

V0

= 2.0 V

V0

= 1.0 V

d = 3.0

Figure 16: Current Time Graph at a plate to chain gap of 3.0 :- In thiscase because the initial current is small, the relative resonance peaks thatoccur at 0 microamps. The peaks become smaller at a higher voltage biasbut seam to act independently of the gap between the plates (the currentsoscillated between 20A in all cases). This behaviour is expected as higherenergy particles should produce shorter periods of backscattering

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22.4 Current-Voltage Curves

These are used to find the location of the Coulomb blockage. TheVoltage and the current that occurs when the discharge seizes to obeyexponential behaviour is representative of the behaviour investigatedas a Coulomb blockade.

22.5 Charge Discharge graphs

As mentioned previously Charge time graphs and Voltage time graphwere equal in shape suggesting that the discharge of the charge hadno bearing on the capacitance of the system. At a plate to chain gap

of 2.5, at this distance it is clearly visible that the charge builds upalong the chain at 10 and at 25 femtoseconds, particularly at highervoltages. This is symbolic of a Coulomb blockade .And because ofconstant capacitance it does not vary from the trend created for theVoltage time graph.

22.6 The on site energy graphs

These graphs show the on site energy along each individual atom onthe chain with respect to time as the set up is discharging. Thesewere used to examine the conducting behaviour of electrons at eachindividual time, so that it could be used in conjunction with a voltage-discharge graph or a current-discharge graph to explain the reasons fora possible anomaly like the Coulomb Blockade.

22.7 On site charge graphs

These graphs show the on site charge on each individual atom that ison the chain with respect to time as the set up is discharging. These areused to examine the blockage in a capacitance sense rather than the onsite resistance sense. In comparison with the onsite energy graph the

overall behaviour of the system can be examined. The onsite chargegraph shown is for a spacing of 3.1Aand an applied bias of 4V

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Figure 17: Onsite energy graphs The onsite energy graph shown is for aspacing of 3.1Aand an applied bias of 4VThe removal of the electric fieldreduces the on-site energy producing the discharge. The excess of electronson one plate opposite atoms have paralelldistrubtuions The higher the Elec-tric field the lower the charge- charge

23 Discussion

23.1 The investigation of Coulomb blockades

In this investigation it was found that the charge that is free to flow wasdependent on the Voltage bias applied in general 12 electrons flowedat 4.0V, 9 at 3.0V 6 at 2.0V and 3 at 1.0V. From the 2.5 ballisticcases the charge should begin to discharge into the chain around 10

femtoseconds. It was possible then to determine if blockages At theendpoints of the chain prevented this discharge into the chain.

The height of the blockage acts independently of the distance of theplates, the back reflections occur because of the build up of chargealong the chain and at the two end points, because it is more efficientfor the currents to reverse path than to penetrate the blockade thecurrent flow back into the plate.

In the ballistic case the discharge of the capacitor seams to be leasteffected by the interference of backscattering. This is shown in the

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Figure 18: Onsite charge graph: The onsite charge graph shown is for aspacing of 3.1Aand an applied bias of 4V. The freedom of charge flowalong the plate is directly related to the Electric field applied on that atom.The dominant factor should be the initial electric field due to the plates.The largest charge occurs at the atom closet to the plate with the highestphotochemical potential. However the blue and red distrubution show asenario where the electron-electron repulsions along the chain have a factor.

These represent the atoms the second and fourth atoms on the chain countedin the direction of the Electric field.

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fact that the discharge occurring in the first 10 femtoseconds reaches

a minimum value and then backscattering effects occur separately.While in the non-ballistic simulations the effects of a blockage of chargeseams to happen during the discharge into the chain. Also it must benoted that the duration of the blockage is longer in the ballistic casethan in the other cases.

It should also be noted that because the charge and voltage curvesare so similar, it can be said that the capacitance of the system isunaffected by the presence of a conducting chain across the plates orby the variation of the voltage between them.

24 Conclusion

The blockage of charge shown in the charge time graphs is representedby the large build-up of charge on the atom of the outside of theplate. In the absence of a dynamical current in this graph producesbackscattering. it is possible that these scattering events are due toa Coulomb Blockade. it is only because the currents are measuredby TDDFT which relaxes the energy distrubution criteria that theseevents can be measured by the approach at all. In conclusion it can besaid that these events must be due to electron-electron interactions.The total charge and current on the graph behave in a different way.By conservation of charge the total current on a point should be thetime derivative of the total charge on that point, but this picture maybe different if current are taken for the plate as the direction of theflow of charge is not so linear, especially in a non-uniform electric field,because the field is uniform in these capacitor plate then the need fora model that deals with individual currents is not nescessary.

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Part VI

Summary

A reminder of the main conclusions of the Project

24.1 The Initial set of experiments

In these experiments, the best system to produce a stable discharge wastested using smaller scale system, these small scale systems gave an aware-ness of the physical properties of a nano-scale capacitor set up. From theresults it was found that it was favorable in creating a discharge system tohave both the voltage drop in the externally applied potential and the dis-tance between the plates great enough to ensure a large build up of chargebetween the plates. Including thicker plates would also benefit the build upof charge. The size of the plates and the spacing between atoms should bothbe of a reasonable value. This is to be expected as classical effects dominatein larger systems.

24.2 Conclusion

In summary even at the nanoscale limits the classical expectations men-tioned in 1.2 are maintained, this is shown by the reduction in the initialpotential due to increasing the plate to chain spacing. The comparison be-tween Static and Dynamical DFT shows that Dynamical Density FunctionalTheory can acess effects that the Static case cannot. The results measuredfrom the anomalies that occur for the Dynamical DFT results outside of theStatic DFT envelope may be due to a Coulomb blockade.

25 ConclusionThe blockage of charge shown in the charge time graphs is represented bythe large build-up of charge on the atom of the outside of the plate. In theabsence of a dynamical current in this graph produces backscattering. it ispossible that these scattering events are due to a Coulomb Blockade. it isonly because the currents are measured by TDDFT which relaxes the energydistrubution criteria that these events can be measured by the approach atall. In conclusion it can be said that these events must be due to electron-electron interactions. The total charge and current on the graph behave in a

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different way. By conservation of charge the total current on a point should

be the time derivative of the total charge on that point, but this picture maybe different if current are taken for the plate as the direction of the flow ofcharge is not so linear, especially in a non-uniform electric field, because thefield is uniform in these capacitor plate then the need for a model that dealswith individual currents is not nescessary.

Why do the onsite charge and energy graphs behave in such a strangeway?

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Part VII

Appendices

26 Programs Used

These are the codes used to create the programs for setting up geometries

and creating the data for the graphs used. There are three programs in all1. The Capacitance Program

2. The Calculate Program

3. The Capacicalc Program

26.1 The Capacitance Program

The Capacitance program creates the geometries used in the simulations. The

program was adapted in order to create new simulations. This is the case of creating

a simulation for a four wide atomic chain with a 2.8 plate to chain gap

program capacitor

implicit none

integer :: i,j,k,NumberofAtoms

integer,parameter :: dp = 8

real(dp) :: x,y,z

real(dp) :: xmax,ymax,zmax,s,AtomicSeparation,Separation

integer :: LengthinAtoms,WidthinAtoms,HeightinAtoms,AtomicChain

real(dp) :: Vgap,E, NumberofSpecies

character :: SelfconsistantField

SelfconsistantField=T

AtomicSeparation=2.50_dp

LengthinAtoms=17

HeightinAtoms=17

WidthinAtoms=04

AtomicChain=04

NumberofAtoms=LengthinAtoms*HeightinAtoms*WidthinAtoms*2+AtomicChain

s=AtomicSeparation

Separation=2.8_dp/s

Vgap=0.50_dp

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xmax=(real(LengthinAtoms,dp)+1.0_dp)/2.0_dp

ymax=(real(HeightinAtoms,dp)+1.0_dp)/2.0_dp

zmax=(real(WidthinAtoms,dp))+real(AtomicChain-1,dp)/2.0_dp+Separation

print *,NumberOfAtoms ,NumberofAtoms

print *,%block AtomicCoordinates

E=Vgap*1.0_dp

x=xmax*(-s)

do i=0,LengthinAtoms-1

x=x+s

y=ymax*(-s)

do j=0,HeightinAtoms-1

y=y+s

z=zmax*(-s)

do k=0,WidthinAtoms-1

z=z+s

print (3f13.6,a,f13.6),x,y,z, 1 T,E

enddo

enddo

enddo

E=-Vgap*1.0_dp

x=xmax*(-s)

do i=0,LengthinAtoms-1

x=x+s

y=ymax*(-s)

do j=0,HeightinAtoms-1

y=y+s

z=zmax*(s)

do k=0,WidthinAtoms-1

z=z-s

print (3f13.6,a,f13.6),x,y,z, 1 T,E

enddo

enddo

enddo

x=0.0_dp

y=0.0_dp

E=0.0_dp

zm

time dependent simulations of nano-scale conductors

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