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1

Analysis and Simulation of Transport and/or

Transference in Nanostructures from the

Waveguide Approximation

By

Héctor Manuel Uribe Vargas

A Dissertation

Submitted to the Program in Electronics Science, Electronics Department

in partial fulfillment of the requirements for the degree of

MASTER IN ELECTRONICS SCIENCE

at the

National Institute of Astrophysics, Optics and Electronics September 2013

Tonantzintla, Puebla

Advisors:

PhD. Edmundo Antonio Gutiérrez Domínguez, INAOE Msc. Víctor Hugo Vega González, INAOE

© INAOE 2013 All rights reserved

The author hereby grants to INAOE permission to reproduce and to distribute copies of this thesis document in whole or in part.

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Abstract

Downscaling of semiconductor devices has reached the fundamental limits of physics. Most models for device analysis take into account the electron behavior as a particle, even in the ones that account for quantum corrections; nevertheless, since we are now in the mesoscopic regime (<90 nm of channel length) the carriers have been observed to show up a mixed wave-particle behavior, which makes the particle approach less effective.

In this thesis, a new approach is proposed; the electron will be considered with a full wave behavior. Using Maxwell equations and a full wave simulator, comparisons will be made with the solution of the 2D Schrödinger equation with open boundaries. These are the first steps in using a full wave simulator as a powerful tool for semiconductor mesoscopic device simulation.

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Resumen

El escalamiento de dispositivos semiconductores está llegando a los límites fundamentales de la física. La mayoría de los modelos para análisis de dispositivos toman en cuenta el comportamiento de un electrón como una partícula, aún aquellos que utilizan correcciones cuánticas. Sin embargo, actualmente nos encontramos con dispositivos que se encuentran en el régimen mesoscópico (canal <90 nm), lo que ha ocasionado que los portadores tengan un comportamiento más como onda, haciendo la aproximación de partícula menos efectiva.

En este trabajo, un nuevo enfoque es propuesto, donde se considera que el electrón tendrá un comportamiento exclusivamente ondular. Usando las ecuaciones de Maxwell y un simulador de onda completa, se harán comparaciones con la solución de la ecuación de Maxwell en 2D con fronteras abiertas. Estos son los primeros pasos para utilizar un simulador de onda completa como una poderosa herramienta en el análisis de dispositivos semiconductores de escala nanométrica.

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Contents

Abstract

Resumen

1 Introduction

... 1

1.1 Scaling of the MOS transistor ... 1

1.2 Analysis of MOS transistors ... 2

1.2.1 Drift and diffusion model ... 4

1.2.2 Hydro dynamics model... 7

1.2.3 Density gradient ... 9

1.2.4 Quantum transport model ... 10

1.3 Organization of This Document ... 15

2 Analogy between Maxwell and Schrödinger

equations

... 16

2.1 The Helmholtz equation ... 16

2.2 Maxwell Equations ... 17

2.3 The Schrödinger equation ... 19

2.4 Analogy between Maxwell and Schrödinger equations ... 20

2.5 Propagation of electromagnetic waves in waveguides and resonant cavities ... 22

2.5.1 Waveguides ... 28

2.5.2 Modes in a rectangular waveguide ... 30

2.5.3 Modes in a resonant cavity ... 32

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3 Closed and Open Boundary Conditions

... 39

3.1 Closed boundaries ... 39

3.2 Open Boundaries ... 50

4 Conclusions

... 60

4.1 Technological Trends ... 60

4.2 Simulation trends ... 61

4.3 Future work ... 61

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Figures and Tables Contents

1 Introduction

Table 1.1 Models and their approaches ... 4

2 Analogy Between Maxwell and Schrödinger

equations

Fig. 2.1 Rectangular Waveguide ... 28

Fig. 2.2 Rectangular resonant cavity ... 32

3 Closed and Open boundaries

Fig 3. 1 Cylindrical symmetry between a resonant cavity and the 2D Scrödinger equation ... 39

Fig 3. 2 Rectangular resonant cavity ... 41

Fig 3. 3 First mode of the Electric field in the z-direction viewed from the xy - plane ... 41

Fig 3. 4 Box with infinite high walls ... 42

Fig 3. 5 Wave function of the 2D Schrödinger equation ... 42

Fig 3. 6 Wave function of the 2D Schrödinger equation ... 43

Fig 3. 7 Electric Field in the z-direction ... 43

Fig 3. 8 Subtraction of the wave equation and the electric field ... 44

Fig 3. 9 Second mode of the electric field in the z-direction viewed from the xy - plane ... 44

Fig 3. 10 Wave function of the second sub band of Energy in the 2D Schrödinger equation ... 45

Fig 3. 11 Subtraction of the wave equation and the electric field (Second mode) ... 45

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Fig 3. 12 Third mode of the electric field in the z-direction viewed from the xy -

plane ... 46

Fig 3. 13 Wave function of the third sub band of Energy in the 2D Schrödinger equation ... 46

Fig 3. 14 Subtraction of the wave equation and the electric field (Third mode) ... 47

Fig 3. 15 First mode of the electric field in the z-direction in a cylindrical resonant cavity viewed from the xy-plane ... 48

Fig 3. 16 Wave function of a 2D Schrödinger equation in a circle... 48

Fig 3. 17 Propagation of the second mode of the Electric field in the z-direction viewed from the xy - plane ... 49

Fig 3. 18 Wave function of the second sub band of Energy in the 2D Schrödinger equation ... 49

Fig 3. 19 Scheme of a MOS Transistor ... 50

Fig 3. 20 Scheme of a MOS transistor using Maxwell – Schrödinger analogy ... 50

Fig 3. 21 Resonant cavity with waveguides ports ... 51

Fig 3. 22 Quantum structure with waveguide ports ... 52

Fig 3. 23 Propagation of the electric field in the resonant cavity with waveguide ports in the z- axis ... 52

Fig 3. 24 Wave function of the quantum cavity with waveguide ports ... 53

Fig 3. 25 Superposition of wave function and electric field propagation ... 53

Fig 3. 26 Closed system behaving like an open system ... 54

Fig 3. 27 Resonant cavity ... 55

Fig 3. 28 Resonant cavity with boundaries backed away by a distance L ... 55

Fig 3. 29 Resonant cavity with boundaries backed away by a distance 2L .... 56

Fig 3. 30 Electric field in the z direction in the resonant cavity ... 56

Fig 3. 31 Electric field in the z direction in the resonant cavity with the boundaries backed away by a distance L ... 57

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Fig 3. 32 Electric field in the z direction in the resonant cavity with the boundaries backed away by a distance 2L ... 57 Fig 3. 33 Second mode of propagation for the resonant cavity... 58

Fig 3. 34 Second mode of propagation for the resonant cavity with boundaries backed away L ... 58

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1

1 Introduction

1.1 Scaling of the MOS transistor

The CMOS technology is one of the most important achievements in modern engineering history. Thanks to device down scaling, it has become one of the most profitable industries in the world [1].

In 1960 Gordon Moore made an empirical observation that the number of devices on a chip will be doubled in a span of time of 18 months. This has come to be known as Moore’s law [2]. The “Moore’s Law” is a description of the constant increase in the level of miniaturization. Each time the channel width in a MOS transistor is reduced, we say that a new technology generation has born. Examples of technology generations are 0.18 µm, 0.13µm, 90 nm, 65 nm, 45 nm, 28 nm, and most recently, 22 nm. The numbers refer to the minimum channel width.

Nevertheless, we are getting to the fundamental limits of semiconductor physics. How much further can we go? It is said that the miniaturization of MOSFET transistor is predicted to reach an end by the year 2016. These predictions indicate that the smallest MOSFET that may be fabricated will be around 8 to 7 nm long [3, 10]. In this scale, we face numerous challenges, both practically and theoretical bottlenecks.

In the transport modeling, the downscaling affects the driving field, which yields to a dramatic increase of the channel gradient. As a result, the carrier distribution along the channel can no longer be described by the shifted Maxwellian distribution [10]. In order to properly account for hot-carrier

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and non-local effects, the drift-diffusion and even the energy transport model have to be improved to incorporate the substantial modifications in the distribution function.

Other reason for the semi-classical models to lose gradually their validity is the particle-wave duality nature of carriers. The carrier motion can be described with the classical Newton law only, when the characteristic size of the device is much larger than the corresponding de-Broglie electron wave length. When the device dimensions are getting comparable to the carrier wave length, the carriers can no longer be treated as classical point-like particles, and effects originating from the quantum-mechanical nature of propagation begin to determine transport in ultra-scaled devices [10].

But, before we go further down, we have to review the path that has been followed in the analysis of these devices. First of all, we will overview the drift-diffusion model, then the hydrodynamics model, which will led us to the density gradient model [4, 5], and finally, we talk about the quantum transport modeling.

1.2 Analysis of MOS transistors

Contemporary industrial device simulation relies on the drift-diffusion or on the hydro dynamical equations. These equations treat the carriers like a classical fluid [4, 5, 6, 7, 8]. With ongoing miniaturization of electronic devices, the validity of this description becomes more and more questionable, due to the appearing of quantum effects like the quantization of carrier motion in the potential well of the inversion layer at the silicon/dielectric interface.

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Today’s MOSFETs have gate oxides that are only a few nanometers thick; the channel depth is of the same order of magnitude. This means that quantum mechanical length scales have already been reached. For the lateral dimensions (width W), the length scales are significantly larger [4,9,10].

The rapid progress in technology keeps the simulation of semiconductor devices constantly evolving, with new designs, new materials and processes, and smaller length scales, physical effects that were not important before, start to need careful attention.

Accurate simulation of MOS devices therefore remains very important. From the device simulation point of view, miniaturization poses new problems of different type. The first category is related to statistics. The smaller the device is, then the smaller is the number of microscopic variables that determine its average macroscopic behavior, and the larger are the relative fluctuations. Well-known examples are random dopant fluctuations [8, 9]. Another category of problems is related to ballistic transport [9, 11], because electrons can pass through very short channel distances with only a few scattering events. Due to the strong electric fields, they can reach very high energies. Therefore, extreme non-equilibrium carrier distributions occur, and full band-structure transport description becomes a necessity [4, 9]. In table 1.1 different methods of analysis and the length of the channel are shown for each model.

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Model Improvements

S e m i – C la s s ic a l a p p ro a c h e s

Compact Models Appropriate for Circuit Design

Drift - Diffusion Equations Good for devices down to 0.5 µm, include µ (E) Hydrodynamic Equations Velocity overshoot can be

treated properly

Boltzmann Transport Equation Accurate up to the classical limits Q u a tu m a p p ro a c h e

s Quantum Hydrodynamics

Keep all classical hydrodynamics features +

quantum corrections

Quantum – Kinetic Equation Accurate up to single particle description

Green’s Functions method Includes correlations in both space and time domain

Direct Solution of the n-body Schrödinger equation

Can be solved for small number of particles

Table 1.1 Models and their approaches

1.2.1 Drift and diffusion model

For traditional semiconductor device modeling, the predominant model corresponds to solutions of the so called drift-diffusion equations, which are ‘local’ in terms of the driving forces (electric fields and spatial gradients in the carrier density), i.e. the current at a particular point in space only depends on the instantaneous electric fields and concentration gradient at that point. The complete drift-diffusion model is based on the following set of equations [6, 10, 12]:

Current Equations:

= +

(1.1)

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5 Continuity Equations

=1 ∇ ∙ +

(1.2)

= −1 ∇ ∙ +

Poisson Equation

∇ ∙ ∇ = − − +

(1.3)

where and Up are the net generation-recombination rates. The continuity equations are the conservation laws for the carriers. A numerical scheme,

which solves the continuity equation should keep the total number of particles inside the device being simulated with respect to local positive carrier density (negative density is unphysical), and with respect to monotonicity of the solution (i.e. it should not introduce spurious space oscillations).

Conservative schemes are usually achieved by subdivision of the computational domain into patches (boxes) surrounding the mesh points. The currents are then defined on the boundaries of these elements, thus enforcing conservation (the current exiting one element side is exactly equal to the current entering the neighboring element through the side in common). In the absence of generation-recombination terms, the only contributions to the overall device current arise from the contacts. Remember that, since

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electrons have negative charge, the particle flux is opposite to the current flux.

When the equations are discretized, there are limitations on the choice of mesh size and time step: 1) The mesh size is limited by the Debye length, 2) The time step is limited by the dielectric relaxation time.

A mesh size must be smaller than the Debye length where one has to solve charge variations in space. Carriers diffuse into the lower doped region creating excess carrier distribution which at equilibrium decays in space down to the bulk concentration with approximately exponential behavior. The spatial decay constant is the Debye length:

= !"#&'$(% (1.4)

where N is the doping density.

The dielectric relaxation time, on the other hand, is the characteristic time for charge fluctuations to decay under the influence of the field that they produce. The dielectric relaxation time may be estimated using

)*+ =&(," (1.5)

The drift-diffusion semiconductor equations constitute a coupled nonlinear set. It is not possible, in general, to obtain a solution directly in one

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step, but a nonlinear iteration method is required. The two most popular methods for solving the discretized equations are the Gummel's iteration method [13] and the Newton's method [14].

Finally, the discretization of the continuity equations in conservation form require the determination of the currents on the mid-points of mesh lines connecting neighboring grid nodes. Since the solutions are accessible only on the grid nodes, interpolation schemes are needed to determine the currents. The approach by Scharfetter and Gummel has provided an optimal solution to this problem [15].

1.2.2 Hydro dynamics model

The hydrodynamic model treats the propagation of electrons and/or holes in a semiconductor device as the flow of a charged compressible fluid. The model exhibits hot carrier effects that miss in the standard drift-diffusion model. The hydrodynamic description should be valid for devices with active regions greater than 0.05 microns [8].

First it was hypothesized that the effective carrier injection velocity from the source into the channel would reach the limit of the saturation velocity and remains there as longitudinal electric fields increased beyond the onset value for velocity saturation [6, 8]. However, theoretical work indicated that velocity overshoot can occur even in silicon [16], and indeed it is routinely seen in the high-field region near the drain in simulated devices using energy balance models. While it was understood that velocity overshoot near the drain would not help current drive, experimental work [17, 18] claimed to observe velocity overshoot near the source, which of course would be beneficial and would make the drift-diffusion model invalid.

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In the electronics community, the necessity for the hydrodynamic transport model is normally checked by comparison of simulation results for hydrodynamics and drift- diffusion simulations. Despite the obvious fact that, depending on the equation set, different principal physical effects are taken into account; the influence on the models for the physical parameters is more suitable [6,8].

The main reason for this is that in the case of the hydrodynamic model, information about average carrier energy is available in form of carrier temperature. Many parameters depend on this average carrier energy, e.g., the mobilities and the energy relaxation times. In the case of the drift- diffusion model, the carrier temperatures are assumed to be in equilibrium with the lattice temperature, that is )- = )., hence, all energy dependent parameters have to be modeled in a different way [6,8].

In the drift- diffusion approach, the electron gas is assumed to be in thermal equilibrium with the lattice temperature ) = ). [6, 8]. However, in the presence of a strong electric field, electrons gain energy from the field and the temperature ) of the electron gas is elevated. Since the pressure of the electron gas is proportional to /0) the driving force now becomes the pressure gradient rather than merely the density gradient. This introduces an additional driving force, namely, the temperature gradient besides the electric field and the density gradient. Phenomenologically, one can write

= + D2∇n + D4∇T2 (1.6)

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9 1.2.3 Density gradient

The density gradient (DG) model provides a description of transport in terms of macroscopic quantities, e.g. densities of the particles and their currents. In this aspect, it is similar to the classical hydrodynamic transport and drift diffusion model but in addition it includes contributions that account for certain aspects of the quantum nature of the particles.

First stages of the DG model started from a macroscopic description [21]. They use thermodynamic considerations introducing a lower order dependency on the density gradient into the internal energy per particle and reach an equation of the Schrödinger type for a static one-dimensional system.

The DG model is a generalized drift diffusion model, making quantum corrections to it, by adding gradient - dependent quantum terms, 6& and 6&, which are known as the chemical potentials or “quantum potentials” to the electrostatic potential,

6& =789∇

'

(1.7)

6& =78;∇

'

(1.8)

where

<

=

ħ

=&>9∗+9 ,

<

=

ħ

=&>;∗+; are two coefficients derived from

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10

gradient effect in the electron and holes gases, @∗ and @∗ are the electron and holes effective masses and A and A are fitting parameters.

These new quantum correction parameters are added to the DD model, so it is modified into the DG model.

The derivations of the new model are based on quantum statistical mechanics, i.e. the microscopic transport is described by either the Wigner-Boltzmann equation [22] or its Fourier transformed counterpart, the Quantum - Liouville or von Neumann equation which governs the evolution of the density matrix [23]. Applying the method of moments yields the classical transport models with additional quantum corrections which are then called quantum hydrodynamic (QHD) [24] and quantum drift diffusion (QDD) model. Both models will be briefly explained in the next section.

1.2.4 Quantum transport model

Now that we have described the semi – classical approaches, we will describe the quantum transport, since scaling in the devices have already reached this regime.

Quantum transport in semiconductor devices is suggested by two trends: (1) within the effective-mass approximation, the thermal de Broglie wavelength for electrons in semiconductors is on the order of the gate length of nano-scale MOSFETs, thereby in the limit of wave mechanics; (2) the time of flight for electrons traversing the channel with velocity well in excess of 10CD@/F is in the 10GHF, 10G7F region; a time scale which equals,

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momentum and energy relaxation times in semiconductors which precludes the validity of the Fermi’s golden rule [6, 9,11].

The static quantum effects, such as tunneling through the gate oxide and the energy quantization in the inversion layer of a MOSFET are also significant in nano scale devices. The current generation of MOS devices has oxide thicknesses of roughly 2-3 nm. The most obvious quantum mechanical effect, seen in the very thinnest oxides, is gate leakage via direct tunneling through the oxide. A second effect due to spatial/size quantization in the device channel region is also expected to play significant role in the operation of nano scale devices [6, 9,10]. A recent and very important issue is the non-homogeneous channel conductance properties introduced by the use of axial mechanical strain [20]. This leads to local conductance properties along the channel width and length that requires an atomistic approach [25, 26, 27]]

To understand this issue, one has to consider the operation of a MOSFET device based on two fundamental aspects: (1) the channel charge induced by the gate at the surface of the substrate, and (2) the carrier transport from source to drain along the channel. Quantum effects in the surface potential will have a deep impact on both, the amount of charge which can be induced by the gate electrode through the gate oxide, and the profile of the channel charge in the perpendicular direction to the surface (the transverse direction). The critical parameter in this direction is the gate-oxide thickness, which for a nano scale MOSFET device is, as noted earlier, on the order of 2 nm. Another aspect, which determines device characteristics, is the carrier transport along the channel (lateral direction) [6,9,10].

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Because of the two-dimensional (2D), and/or one dimensional (1D) in the case of narrow-width devices, confinement of carriers in the channel, the mobility (or microscopically speaking, the carrier scattering) will be different from the three-dimensional (3D) case. Theoretically speaking, the 2D/1D mobility should be larger than its 3D counterpart due to reduced density of states function, i.e. reduced number of final states the carriers can scatter into, which can lead to device performance enhancement.

A well known approach that takes this effect into consideration is based on the self-consistent solution of the 2D Poisson–1D Schrödinger–2D Monte Carlo, and requires enormous computational resources as it requires storage of position dependent scattering tables that describe carrier transition between various energy subbands. More importantly, these scattering tables have to be re-evaluated at each iteration step as the Hartree potential (the confinement) is a dynamical function and slowly adjusts to its steady-state value [6,9,10,12]. It is important to note, however, that in the smallest size devices (10 nm feature size), carriers experience very little or no scattering at all (ballistic limit), which makes this second issue less critical when modeling these nano scale devices.

On the other hand, the dynamical quantum effects in nanoscale MOSFETs, associated with energy dissipating scattering in electron transport can be physically much more involved. There are several other fundamental problems one must overcome in this regard. For example, since ultra-small devices, in which quantum effects are expected to be significant, are inherently three-dimensional (3D), one must solve the 3D open- Schrödinger equation [6].

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Another question that becomes important in nano-scaled devices is the treatment of scattering processes. Within the Born approximation, the scattering processes are treated as independent and instantaneous events. It is, however, a nontrivial question to ask whether such an approximation is actually satisfactory under high temperature, in which the electron strongly couples with the environment (such as phonons and other carriers) [6, 9].

In fact, many dynamical quantum effects, such as the collisional broadening of the states or the intra collisional field effect, are a direct consequence of the approximation employed for the scattering kernel in the quantum kinetic equation. Depending on the orders of the perturbation series in the scattering kernel, the magnitude of the quantum effects could be largely changed. Density matrices, and the associated Wigner function approach, Green’s functions, and Feynman path integrals all have their application strengths and weaknesses [9,11].

A general feature of electron devices is that they are of use only when connected to a circuit, and to be connected any device must possess at least two terminals, contacts, or leads. As a consequence, every device is an open system with respect to carrier flow. This is the overriding fact that determines which theoretical models and techniques may be appropriately applied to the study of quantum devices. For example, the quantum mechanics of pure, normalize states, such as those employed in atomic physics, does not contribute significantly to an understanding of devices, because such states describe closed systems.

To understand devices, one must consider the not normalizable scattering states, and/or describe the state of the device in terms of

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statistically mixed states, which casts the problem in terms of quantum kinetic theory. As a practical matter of fact, a device is of use only when its state is driven far from thermodynamic equilibrium by the action of the external circuit. The non-equilibrium state is characterized by the conduction of significant current through the device and/or the appearance of a non-negligible voltage drop across the device.

In classical transport theory, the openness of the device is addressed by the definition of appropriate boundary conditions for the differential (or integro-differential) transport equations. Such boundary conditions are formulated so as to approximate the behavior of the physical contacts to the device, typically Ohmic or Schottky contacts.

In the traditional treatments of quantum transport theories, the role of boundary conditions is often taken for granted, as the models are constructed upon an unbounded spatial domain. The proper formulation and interpretation of the boundary conditions remains an issue, however. It should be understood that, unless otherwise specified, all models to be considered here are based upon a single-band, effective mass open-system Schrodinger equation.

After the most widely used device model approaches were reviewed, in this thesis we point out to an alternative strategy for the simulation of nano-scaled semiconductor devices with atomistic properties.

The most used way to simulate nano-scaled devices is with the use of device simulators, such as; Atlas from Silvaco, Medici, GtsFramework, etc. All these device simulator programs are expensive and they are still based on a

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particle approach that does not allow having a look at the wave nature of carrier transport. The approach taken in this thesis, is taking the wave behavior of the carriers entirely, using a full wave simulator for this purposes, and then compare both results.

1.3 Organization of This Document

This document is organized in four chapters. Chapter II describes Maxwell and Schrödinger as Helmholtz equations, studying the necessary conditions for them to be equivalent. Chapter III presents simulations of both Maxwell and Schrödinger equations in different structures, using the closed and open boundary conditions, and also presenting the similarities between them. Chapter IV presents the reaches and limitations of this new technique to analyze nanostructures and future work.

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2 Analogy between Maxwell and

Schrödinger equations

In this chapter, an analogy between both Maxwell and Schrödinger equations will be reviewed, explaining the necessary conditions for both of them to provide similar eigen value solutions.

2.1 The Helmholtz equation

The Helmholtz equation, named after German scientist Hermann von Helmholtz (1821–1894) is the time-independent equation for wave descriptions in space as a solution to the wave function of the source of the waves. The scalar Helmholtz equation is

∇7IJ = K7IJ = 0, 7= L' LM'+ L

'

LN'+ L '

LO' (2.1)

where ∇7IJ is a complex scalar function (potential) defined at a spatial point J = , Q, R ∈ TU and k is some real or complex constant. It has a significant impact on acoustics, hydrodynamics, and electromagnetics [48].

This equation naturally appears from general conservation laws of physics and can be interpreted as a wave equation for monochromatic waves (wave equation in the frequency domain). The Helmholtz equation can also be derived from the heat conduction equation, Schrödinger equation, telegraph and other wave type, or evolutionary, equations. From a mathematical point of view; it appears also as an eigen value problem for the Laplace operator ∇7.

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17 2.2 Maxwell Equations

The Maxwell equations are the set of four fundamental equations governing electromagnetism (i.e., the behavior of electric and magnetic fields).

Now, we will derive how the Maxwell equations can be put in the form of Helmholtz equation; let’s consider the equations describing propagation of electromagnetic waves. For a medium free of charges and imposed currents, these equations can be written as:

∇ × W = − LXLY, ∇ × X = −LW

LY, ∇ ∙ W = 0, ∇ ∙ X = 0 (2.2)

where W and X are the Electric and Magnetic field and and are the magnetic permeability and electric permittivity in the medium, respectively. In the vacuum we have

= Z, = Z, D = Z ZG/7 (2.3)

where D is the speed of light and D ≈ 310]@/F.

Taking the curl of the first equation in (2.2) and using the second equation in (2.3) we have

∇ × ∇ × W = − ZLYL ∇ × X = −Z ZL

'W

LY' = −^G'L 'W

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due to the following well-known identity for the 7 operator and the third equation in equation (2.2)

∇7W = ∇∇ ∙ − ∇ × ∇ × W = −∇ × ∇ × W (2.5)

we obtain

∇7W = G ^'L

'W

LY' (2.6)

And similarly,

∇7X = G ^'L

'X

LY' (2.7)

Thus, both the electric and magnetic field vectors satisfy the vector wave equation. Transformation of this equation into the frequency domain,

using LYL → −`a we obtain:

∇7+ k7c = 0, 7+ k7d = 0, K =e

^, (2.8)

Note that the number of scalar equations (2.8) in three dimensions is six (each Cartesian component of c or d satisfies the scalar Helmholtz equation), while the original formulation (2.2) provide eight relations for the same quantities. The missing relations are equations stating that the divergence of the electric and magnetic fields is zero. This imposes limitations

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on the components of the electric and magnetic field vectors, since they should be constrained to satisfy these additional equations. So any of these fields is described by the following system of equations:

∇7+ k7c = 0, ∇ ∙ W = 0, (2.9)

where ccan be replaced by d.

2.3 The Schrödinger equation

In classical mechanics the state of motion of a particle is specified by giving the particle’s position and velocity. In quantum mechanics the state of motion of a particle is specified by giving the wave function. In either case the fundamental question is to predict how the state of motion will evolve as time goes by, and in each case the answer is given by an equation of motion. The classical equation of motion is Newton’s second law, f = @g if we know the particle’s position and velocity at time = 0. Newton’s second law determines the position and velocity at any other time. In quantum mechanics the equation of motion is the time-dependent Schrödinger equation. If we know a particle’s wave function at = 0, the time-dependent Schrödinger equation determines the wave function at any other time.

The Helmholtz equation can also be obtained from the Schrödinger equation for the wave function of a particle in quantum mechanics [52],

− ħ'

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20

where ħ= h

7i is the reduced Planck’s constant, @ is the mass of the particle,

∇7 is the Laplacian operator, I is the wave function, is the potential energy

of the particle and is the total energy of the particle. The reference for the electron energy E is the bottom edge of the conduction band.

This equation can be put in the form of:

∇7+ k7I = 0 (2.11)

where K =j7>kl

ħ .

The scalar quantum wave function I in (2.10) correctly describes the state of a ballistic electron in a single band, with no coupling to other bands; therefore, it describes the state of an electron in a semiconductor heterostructure. The coupling between different electron bands, in particular between the conduction and valence band can be accounted for by introducing a wave function with as many components as the number of coupled bands [49, 50]. We will consider only scalar electron wave functions and do not explicitly take into account spin properties.

2.4 Analogy between Maxwell and Schrödinger equations

Now that we have established that both Maxwell and Schrödinger equations can be put in the form of the Helmholtz equation, we can assume that there are some similarities between them.

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First, it is important to mention that sinceψ in (2.11) is scalar and E (or H) in (2.9) is a vector, the analogy can only be carried out between ψ and one component of E (or H), and even then only when the electromagnetic wave preserves its polarization [49, 50].

The analogy between propagation of electromagnetic waves in dielectrics and of electrons in various time-independent potentials is the main subject of this work. The basis of this analogy is the fact that both wave equations for electromagnetic (with well-defined frequency), obtained directly from the Maxwell equations, and the time-independent Schrodinger equation are Helmholtz equations; when specific restrictions (like behavior at infinity and boundary conditions) are imposed, they generate similar eigen values problems.

According to D. Dragoman [50], “electrons in bulk semiconductors or heterostructures do not behave like waves unless:

1. The interactions between electrons and the periodic crystal lattice, and

between different electrons, do not appear explicitly in the electron

equation of motion.

2. The collisions between different electrons as well as the interference

between the wave functions of different electrons can be neglected”.

The first requirement is solved by assigning the electron an effective mass instead of the free electron mass, and by incorporating all interactions in the potential energy V. Second requirement holds only in mesoscopic systems (<90 nm), for which the length of the heterostructure is smaller than the Fermi wavelength, the phase relaxation and the mean free path [9].

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22

In mesoscopic devices with dimensions less than the phase relaxation length, the transport is coherent, which means that the electron is transmitted across the structure in a single process. Colissionless transport is guaranteed when the length of the structure is smaller than the mean free path of the electron, which is the distance the electron can travel before its initial momentum is lost.

The propagation of ballistic electrons in mesoscopic conductors has many similarities with electromagnetic wave propagation in waveguides and resonant cavities [50], in fact, according to [53], there is a complete analogy between a cavity resonator and the 2D particle in a box Schrödinger problem. We will further review this.

2.5 Propagation of electromagnetic waves in waveguides and resonant cavities

The propagation of electromagnetic waves in hollow metallic structures is important for its applications in telecommunications. We shall consider that the metal is a perfect conductor; if the structure is infinite, we will call this metallic structure waveguide; if it has end faces, we shall call it resonant cavity.

The cross section of the structure is always the same, along the axis. With a time dependence exp (-iwt), the Maxwell equations (2.2) for the fields inside the structure are:

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23

For a structure filled with a uniform non-dissipative medium having permittivity and permeability .

∇7+ a7 nW

mo = 0 (2.13)

The specific geometry suggests us to single out the spatial variation of the fields in the z direction and assuming

pW, Q, R, m, Q, R, q = pW,Qexp ±`KR − `am, Qexp ±`KR − `aq (2.14)

The wave equation is reduced to two variables:

v∇w7+ a7− K7x yczq = 0 (2.15)

where ∇w7 is the transverse part of the Laplacian operator

∇w7= ∇7− L

'

LO' (2.16)

It is convenient to separate the fields into components parallel to and transverse the z axis:

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24 {|is as usual, a unit vector in the z−direction. Similar definitions hold for the magnetic field B. The Maxwell equations can be expressed in terms of transverse and parallel fields as:

LW}

LO + `a{| × mY= ∇YO, {| ∙ ∇Y× WY = `a~O (2.18)

Lm}

LO − ` a{| × WY = ∇Y~O, {| ∙ ∇Y× mY = −` aO (2.19)

∇Y∙ WY = −LkLO, ∇Y∙ mY = −L0LO (2.20)

According to the first equations in (2.18) and (2.19), if O and ~O are known, the transverse components of E and B are determined, assuming the

z dependence is given by (2.14). Considering that the propagation in the positive z direction and that at least one O and ~O have non-zero values, the transverse fields are

cY= ",e€'' vK∇YO− a{| × ∇Y~Ox (2.21)

zY =",e€'' vK∇Y~O+ a {| × ∇YOx (2.22)

Let us notice the existence of a special type of solution, called the transverse electromagnetic (TEM) wave, having only field components transverse to the direction of propagation. From the second equation in (2.18)

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25

and the first in (2.19), results that O = 0 and ~O = 0 implies that % = %k‚ satisfies:

Y× %k‚ = 0, ∇Y%k‚ = 0 (2.23)

So, %k‚ is a solution of an electrostatic problem in 2D. There are 4 consequences:

1. The axial wave number is given by the infinite-medium value

K = KZ = a√ (2.24)

as can be seen from (2.15).

2. The magnetic field deduced form the first equation in (2.19) is ~%k‚ = ±√ {| × %k‚ (2.25)

for waves propagating as exp (±ikz). The connection between ~%k‚ and %k‚ is just the same as for plane waves in an infinite medium

3. The TEM mode cannot exist inside a single, hollow, conductor of infinity conductivity. The surface is an equipotential; the electric field therefore vanishes inside. It is necessary to have two or more surfaces to support the TEM mode. Coaxial cables and parallel-wire

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26

transmission lines are structures for which the TEM mode is the dominant mode.

4. The absence of a cutoff frequency: the wave number (2.24) is real for all ω.

In fact, two types of field configuration occur in hollow cylinders. They are solutions of the eigenvalue problems given by the wave equation (2.15), solved with the following boundary conditions, to be fulfilled on the structure surface:

ƒ × W = 0, ƒ ∙ m = „ (2.26)

where n is a normal vector at the surface S. From the first equation of (2.26):

ƒ × W = ƒ × −ƒY + {|O = ƒ × {|O = 0 (2.27)

so:

qO|F = 0 (2.28)

Also, from the second one:

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27

with this value for ~Y in the component of the first equation (2.26) parallel to n, we get:

q

L0

L‡ F = 0 (2.30)

where L

L is the normal derivative at a point on the surface. Even if the wave

equation for O and ~O is the same (2.13), the boundary conditions on O and ~O are different, so the eigenvalues for O and ~O will in general be different.

The fields thus naturally divide themselves into two distinct categories:

Transverse magnetic (TM) waves:

~O= 0, everywhere; boundary condition qO|F = 0 (2.31)

Transverse electric (TE) waves:

O = 0, everywhere; boundary condition qL0L‡ F = 0 (2.32)

For a given frequency ω, only certain values of wave number k can

occur (typical waveguide situation), or, for a given k, only certain ω values are

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28

The various TM and TE waves, plus the TEM waves if it can exist, constitute a complete set of fields to describe an arbitrary electromagnetic disturbance in a waveguide or cavity.

2.5.1 Waveguides

Fig. 2.1 Rectangular Waveguide

For the propagation of waves inside a hollow waveguide of uniform cross section, it is found from (2.21) and (2.22) that the transverse magnetic fields for both TM and TE waves are related by:

ˆY = ±G‰ {| × WŠ (2.33)

where Z is called the wave impedance and is given by

‹ = Œ 

Ž"e = !,

,e

 = !, )

q

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29

and KZ is given by (2.24). The ± sign in (2.33) goes with z dependence, ’±`KR. The transverse fields are determined by the longitudinal fields, according to (2.21) and (2.22):

TM waves:

Y = ±“€' ∇wI (2.35)

TE waves

ˆY = ±“€' ∇wI (2.36)

where I ’ ±`KR is O ˆO for TM (TE) waves, and ”7 is defined below. The scalar function I satisfies the 2D wave equation (2.15):

∇w+ ”7I = 0 (2.37)

where

”7 = •7 − K7 (2.38)

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30 qO|F = 0 or qL0L‡ F = 0 (2.39)

for TM (TE) waves.

Equation (2.37) for I, together with boundary condition (2.39), specifies an eigenvalues problem. The similarity with non-relativistic quantum mechanics is evident [50].

2.5.2 Modes in a rectangular waveguide

According to the theory, we will consider propagation of TE waves in a rectangular waveguide (the corners of the rectangle are situated in (0, 0), (a, 0), (a, b), (0, b).

In this case, is easy to obtain explicit solutions for the fields. The wave equation for I = ˆO is:

–LML''+LNL''+ ”7— I = 0 (2.40)

with boundary conditions L˜

L = 0 at = 0, a and y = 0, b. The solution for I

is easily find to be:

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31

with ” given by:

”>7 = ž7–>' œ' +

'

8'— (2.42)

with @ and as integers. Consequently, from (2.38),

K>7 = a7− ”>7 = a7− a>7 , a>7 = “Ÿ9

'

", (2.43)

As only for • > a>, K> is real, so the waves propagate without attenuation; a> is called the cutoff frequency. For a given •, only certain values of K, namely K>, are allowed. For TM waves, the equation for the field I = O will be also (2.44), but the boundary condition will be different: I = 0 at = 0, g and Q = 0, <. The solution will be:

I>, Q = Zsin –>iMœ — sin –iN8 — (2.44)

with the same result for K>.

In a more general geometry, there will be a spectrum of eigenvalues ”£7 and corresponding solutions I£, with ¤ taking discrete values, which can be integers or sets of integers, like in (2.42). These different solutions are called the modes of the guide. For a given frequency •, the wave number K is determined for each value of ¤ :

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32 K £7 = a7− ”

£7 (2.45)

Defining a cutoff frequency •£,

•£ = √",“¥ !a7− a£7 (2.46)

then the wave number can be written:

K£ = √ !a7− a£7 (2.47)

2.5.3 Modes in a resonant cavity

Fig. 2.2 Rectangular resonant cavity

In a resonant cavity, a structure with metallic, perfect conductive ends perpendicular to the z axis, the wave equation is identical, but the eigenvalue problem is somewhat different, due to the restrictions on K. Indeed, the

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33

formation of standing waves requires a z−dependence of the fields having the form:

¦ F` KR + ~ D§F KR (2.48)

So, the wavenumber K is restricted to:

K = i*, = 0,1,2 …. (2.49)

and the condition (2.40) imposes a quantization of • :

a£7 = i

*7+ ” £7 (2.50)

So, the existence of quantized values of k implies the quantization of

ω.

2.6 The 2D Schrödinger equation

The time independent Schrödinger equation for a particle equation moving in more than one dimension:

− ħ'

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34

where ħ= h

7i is the reduced Planck’s constant, @ is the mass of the particle,

∇7 is the Laplacian operator, I is the wave function, is the potential energy

of the particle and is the total energy of the particle.

First, the Laplacian is expanded and the equation is written as:

− ħ'

7> L'˜

LM' +L '˜

LN' + , QI = I (2.52)

For a particle in a two-dimensional box of length L and height H, the potential energy function is

, Q = y0 0 < < g 0 < Q < ˆ ’¬F’aℎ’A’q (2.53)

This implies that the particle can only exist inside the box where , Q = 0. Using this fact and K7 = 7>k

ħ' letting allows us to rewrite the

equation

L'˜

LM' +L '˜

LN' = −K7I (2.54)

The result is a homogeneous 2nd order partial differential equation with constant coefficients. The separation of variables method is used to solve the above equation. Assume that the wave function I, Q is separable into two ® and ¯Q functions. For brevity ,I = ®¯.

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35

Therefore L'˜

LM' = X′′Y and L '˜

LN' = XY′′. This allows us to rewrite the equation (2.54) as

®′′

¯ + ®¯′′

= −K7®¯ (2.55)

Dividing both sides by ®¯ yields

²′′

² + ³′′

³ = −K7 (2.56)

The variables are separated by shifting the Y term to the right-hand side of the equation:

²′′

² = −³

′′

³ − K7 = −¤7 (2.57)

Since the variables have been fully separated, we can set both equations equal to the constant −¤7.

First, we solve for ®

²′′

² = −¤7 N€´µ*¶·¸¸¸¹ ®

′′

+ ¤7® = 0 (2.58)

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36 ® = DGsin¤ + D7cos ¤ (2.59)

Since the particle cannot be outside the box:

®0 = DGsin0 + D7cos0 = 0N€´µ*¶·¸¸¸¹ D7 = 0 N€´µ*¶·¸¸¸¹ ® = DGsin¤

(2.60)

and:

® = DGsin¤ = 0 N€´µ*¶·¸¸¸¹ ¤ = ž N€´µ*¶·¸¸¸¹ ¤ = i. (2.61)

where is a positive integer. Therefore:

® = Dsin –iM. — (2.62)

Now, it is time to solve for ¯:

³′′

³ − K7 = −¤7 N€´µ*¶·¸¸¸¹ ¯

′′+ K7+ ¤7¯ = 0

(2.63)

Again, the only non-trivial solution is:

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37

As before, the particle cannot be outside the box, so the solutions take the form of:

¯0 = DUsin0 + D=cos0 = 0N€´µ*¶·¸¸¸¹ D= = 0 N€´µ*¶·¸¸¸¹ ¯Q = DUsinn√K7− ¤7o

(2.65)

and

¯ˆ = DUsinn√K7− ¤7ˆo 0N€´µ*¶·¸¸¸¹ √K7− ¤7ˆ = ž N€´µ*¶·¸¸¸¹ √K7− ¤7 =iº

(2.66)

where is a positive integer. Therefore:

¯Q = Dsin –iNº — (2.67)

Since I = ®¯, then we have:

I = Dsin –iM. — sin –iNº — (2.68)

Note that D = DD . Here the wave function varies with integer values of and .

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38

Since »I, Q»7 is the probability function and the particle must be somewhere in the box, we get »I, Q»7 = 1 for 0 < < and 0 < Q < ˆ, therefore:

D7 ¼ ¼ sinZ. Zº 7–iM. — sin7–iNº — Q = 1 (2.69)

We can separate the integrals as follows (this is possible because the x and y variables are independent):

D7 ¼ sinZ. 7–iM. — ¼ sinZº 7–iM. — Q = 1 (2.70)

then,

D7 –.7— –º7— = 1N€´µ*¶·¸¸¸¹ D =√.º7 (2.71)

therefore,

I =√.º7 sin –iM. — sin –iNº — (2.72)

This is the solution for the wave equation for the particle in the two dimensional box.

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3 Closed and Open Boundary Conditions

3.1 Closed boundaries

As seen in last chapter, and according to [53 equivalence between the

in a box with infinitely high walls and a resonant cavity with cylindrical symmetry, including the boundary conditions. In this analogy

corresponds to the wave function, and

Fig 3. 1 Cylindrical symmetry between a resonant cavity and the 2D Scrödinger

Using the software COMSOL Multi physics, a resonant cavity and a structure in 2D to solve the Schrödinger equation

proof of this analogy is provided. The structures selected to examine the analogy were both a rectangular and a cylindrical resonant cavities and their quantum counterparts.

The first structure selected is a rectangular resonan purposes, we choose the x

propagation since there is no unique “longitudinal direction” (A TE mode with

3 Closed and Open Boundary Conditions

Closed boundaries

ast chapter, and according to [53]: “There is a complete between the two-dimensional Schrodinger equation for a particle box with infinitely high walls and a resonant cavity with cylindrical

including the boundary conditions. In this analogy corresponds to the wave function, and K7 to the eigen energy.”

Cylindrical symmetry between a resonant cavity and the 2D Scrödinger equation

Using the software COMSOL Multi physics, a resonant cavity and a solve the Schrödinger equation were designed so further proof of this analogy is provided. The structures selected to examine the analogy were both a rectangular and a cylindrical resonant cavities and their quantum counterparts.

The first structure selected is a rectangular resonant cavity; for our purposes, we choose the x-axis as the reference for the direction of propagation since there is no unique “longitudinal direction” (A TE mode with

39

3 Closed and Open Boundary Conditions

]: “There is a complete dimensional Schrodinger equation for a particle box with infinitely high walls and a resonant cavity with cylindrical including the boundary conditions. In this analogy , Q )

Cylindrical symmetry between a resonant cavity and the 2D Scrödinger

Using the software COMSOL Multi physics, a resonant cavity and a were designed so further proof of this analogy is provided. The structures selected to examine the analogy were both a rectangular and a cylindrical resonant cavities and their

t cavity; for our axis as the reference for the direction of propagation since there is no unique “longitudinal direction” (A TE mode with

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40

respect to the x-axis can be a TM mode to the z-axis). Actually, the existence of conducting end walls at z=0 and at z=d gives rise to multiple reflections and set up standing waves; no waves propagates in an enclosed cavity.

The resonant frequencies are obtained with the following expression:

½> =^7!–>œ— 7

+ –8—7+ –^—7 ˆR (3.1)

where D is the speed of light, @, , and are the modes and g, <, and D are the dimensions of the rectangular resonant cavity.

The structure designed is a rectangular resonant cavity; using equation 3.1, we calculate the resonant frequency for the TE101 mode, which is the first mode to be activated. The structure, as well as the electric field in the z-direction, is shown in fig. 3.2 and fig 3.3.

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Fig 3. 3 First mode of the Electric field in the z

After solving for the resonant cavity, the 2D Schrödinger equation with hard walls is solved. The structure has to be the same, so the results could be attainable and comparable with each other. The structure in this case is a square. The structure and the w

Fig 3. 2 Rectangular resonant cavity

First mode of the Electric field in the z-direction viewed from the xy

After solving for the resonant cavity, the 2D Schrödinger equation with . The structure has to be the same, so the results could be attainable and comparable with each other. The structure in this case is a

The structure and the wave function are seen in fig 3.4 and fig 3.5

41

direction viewed from the xy - plane

After solving for the resonant cavity, the 2D Schrödinger equation with . The structure has to be the same, so the results could be attainable and comparable with each other. The structure in this case is a

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Fig 3. 5 Wave function of the 2D Schrödinger equation

As seen at first sight, both solutions seem very similar, except for the absolute value of each; nevertheless, since this is an eigen value problem, it is possible to multiply both solutions for a scalar numbe

Figures 3.6 and 3.7 show the results, and figure 3.8 both results.

Fig 3. 4 Box with infinite high walls

Wave function of the 2D Schrödinger equation

As seen at first sight, both solutions seem very similar, except for the absolute value of each; nevertheless, since this is an eigen value problem, it is possible to multiply both solutions for a scalar number and compare again. show the results, and figure 3.8 shows a subtraction of

42

As seen at first sight, both solutions seem very similar, except for the absolute value of each; nevertheless, since this is an eigen value problem, it r and compare again. shows a subtraction of

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43

Fig 3. 6 Wave function of the 2D Schrödinger equation

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44

Fig 3. 8 Subtraction of the wave equation and the electric field

As seen by figure 3.8, the analogy holds, since both results are the same for both structures, since the error is less than 1%. Now it is time to analyze the higher transmission modes, since those correspond to a new sub band of energy in the Schrödinger equation. The second and third modes of each structure are shown in figures 3.9 through 3.12.

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Fig 3. 10 Wave function of the second sub band of Energy in the 2D Schrödinger

Fig 3. 11 Subtraction of the wave equation and the electric field (Second mode) Wave function of the second sub band of Energy in the 2D Schrödinger

equation

Subtraction of the wave equation and the electric field (Second mode)

45

Wave function of the second sub band of Energy in the 2D Schrödinger

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46

Fig 3. 12 Third mode of the electric field in the z-direction viewed from the xy - plane

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47

Fig 3. 14 Subtraction of the wave equation and the electric field (Third mode)

Each mode of the resonant cavity corresponds to an energy sub band of the 2D Schrödinger equation.

Another structure will be analyzed to ensure that the analogy still is viable with other structures. In this case a cylindrical resonant cavity and a circle for the 2D Schrödinger with hard walls will be analyzed. The results are shown in fig. 3.15 and 3.16.

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48

Fig 3. 15 First mode of the electric field in the z-direction in a cylindrical resonant cavity viewed from the xy-plane

Fig 3. 16 Wave function of a 2D Schrödinger equation in a circle

Now, the second higher mode and the second sub band of energy will be simulated. The results are shown in figures 3.17 and 3.18.

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49

Fig 3. 17 Propagation of the second mode of the Electric field in the z-direction viewed from the xy - plane

Fig 3. 18 Wave function of the second sub band of Energy in the 2D Schrödinger equation

As seen from figures 3.3 to 3.18, we can prove beyond doubt that both equations produce similar eigen value results under these conditions (hard walls or closed boundaries). Nevertheless, even when this is an important development; in real life, we never usually find closed systems, so, apart from

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the theoretical value; we need to go further down so we ca to real structures (e.g. a transistor

3.2 Open Boundaries

After proving that the analogy holds for closed systems, it is time to “open” those systems; but why is it important to do so? First, since we are trying to analyze a transistor, the most ideal way of make a scheme in this analogy it will be trough figures 3.19 and 3.20

Fig 3. 20 Scheme of a MOS transistor using Maxwell

The best way to fulfill this analogy is by adding waveguide ports to a resonant cavity, so it can be excited; and open the 2D Schrödinger system too.

Drain

Waveguide port

the theoretical value; we need to go further down so we can apply this theory to real structures (e.g. a transistor).

Open Boundaries

After proving that the analogy holds for closed systems, it is time to “open” those systems; but why is it important to do so? First, since we are trying to analyze a transistor, the most ideal way of make a scheme in this

y it will be trough figures 3.19 and 3.20.

Fig 3. 19 Scheme of a MOS Transistor

of a MOS transistor using Maxwell – Schrödinger analogy

The best way to fulfill this analogy is by adding waveguide ports to a resonant cavity, so it can be excited; and open the 2D Schrödinger system

Drain Channel Source

Waveguide Waveguide

port Waveguide

50

n apply this theory

After proving that the analogy holds for closed systems, it is time to “open” those systems; but why is it important to do so? First, since we are trying to analyze a transistor, the most ideal way of make a scheme in this

Schrödinger analogy

The best way to fulfill this analogy is by adding waveguide ports to a resonant cavity, so it can be excited; and open the 2D Schrödinger system

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The structure designed is a resonant cavity with waveguide ports located at the half of it, that extend

have a cutoff frequency of 3 GHz for the TE10 mode, which is the first mode activated in a rectangular waveguide. The frequenc

wavelength can be seen. The structure, Electric Field in the

z-for solving the 2D Schrödinger equation with open boundaries, is a cavity with waveguide ports as shown in fig 3.22

3.24.

Fig 3.

The structure designed is a resonant cavity with waveguide ports located at the half of it, that extend all through the z-axis. It is designed to have a cutoff frequency of 3 GHz for the TE10 mode, which is the first mode activated in a rectangular waveguide. The frequency is chosen

wavelength can be seen. The structure, as well as the propagation

-axis, is shown in fig 3.21 and 3.23. The structure, used for solving the 2D Schrödinger equation with open boundaries, is a

guide ports as shown in fig 3.22. The solution is shown in

Fig 3. 21 Resonant cavity with waveguides ports

51

The structure designed is a resonant cavity with waveguide ports axis. It is designed to have a cutoff frequency of 3 GHz for the TE10 mode, which is the first mode y is chosen so a full as well as the propagation of the . The structure, used for solving the 2D Schrödinger equation with open boundaries, is a quantum . The solution is shown in

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Fig 3.

Fig 3. 23 Propagation of the electric f

Fig 3. 22 Quantum structure with waveguide ports

Propagation of the electric field in the resonant cavity with waveguide in the z- axis

52

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53

Fig 3. 24 Wave function of the quantum cavity with waveguide ports

Fig 3.25 shows the superposition of the wave function and the Ez component of the Electric field in the resonant cavity with waveguide ports.

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As shown in fig 3.2

amazing; but, how such a similarity can be explained?

First of all, it has been explained that both resonant cavity and 2D Schrödinger equation with closed boundaries produce the same

using this knowledge, new s

It is important to note that a system with closed boundaries far from the point of analysis, behaves like an open

Fig 3. 26

Then we proceed to

interest, first by a distance L, and then by a distance 2L system resembles an open system. The

shown in fig. 3.27 trough 3.

As shown in fig 3.25, the resemblance between both results is amazing; but, how such a similarity can be explained?

First of all, it has been explained that both resonant cavity and 2D Schrödinger equation with closed boundaries produce the same

, new structures can be designed.

It is important to note that a system with closed boundaries far from the analysis, behaves like an open system, as shown in fig. 3.26.

26 Closed system behaving like an open system

Then we proceed to back away the boundaries from the point of first by a distance L, and then by a distance 2L so the new closed system resembles an open system. The structures and the simulations are shown in fig. 3.27 trough 3.32.

54

, the resemblance between both results is

First of all, it has been explained that both resonant cavity and 2D Schrödinger equation with closed boundaries produce the same result; so

It is important to note that a system with closed boundaries far from the 3.26.

back away the boundaries from the point of so the new closed structures and the simulations are

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Fig 3. 28 Resonant cavity

Fig 3. 27 Resonant cavity

Resonant cavity with boundaries backed away by a distance L

55

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Fig 3. 29 Resonant cavity with boundaries

Fig 3. 30 Electric field in the z direction in the resonant cavity Resonant cavity with boundaries backed away by a distance 2L

Electric field in the z direction in the resonant cavity

56

backed away by a distance 2L

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