雙閘極暨絕緣層上矽場效應電晶體有效位勢量子修正模式的線性迴歸

國

立

交

通

大

學

統計學研究所

碩

士

論

文

雙 閘 極 暨 絕 緣 層 上 矽 場 效 應 電 晶 體

有 效 位 勢 量 子 修 正 模 式 的 線 性 迴 歸

Application of Linear Regression to Effective Potential of Double-Gate and

Silicon-on-Insulator Metal-Oxide-Semiconductor Field-Effect Transistors

研 究 生：張景嵐

指導教授：周幼珍 博士

李義明 博士

Application of Linear Regression to Effective Potential of Double-Gate and

Silicon-on-Insulator Metal-Oxide-Semiconductor Field-Effect Transistors

研 究 生：張景嵐 Student：Ching-Lan Chang

指導教授：周幼珍 博士 Advisor：Dr. Yow-Jen Jou

李義明 博士 Advisor：Dr. Yiming Li

國 立 交 通 大 學

統 計 學 研 究 所

碩 士 論 文

A Thesis

Submitted to Institute of Statistics National Chiao Tung University in partial Fulfillment of the Requirements

for the Degree of Master in Statistics July 2006 Hsinchu, Taiwan

中華民國九十五年七月

i

All Rights Reserved

v

學生：張景嵐

指導教授：周幼珍 博士

李義明 博士

國立交通大學 統計 學研究所 碩士班

摘

要

現今在系統晶片積體電路中的半導體元件尺寸已經縮小到奈米刻度的尺

寸，隨著尺寸的縮小，因應各種特殊設計，半導體元件中氧化層的厚度也

隨之變薄。因為氧化層厚度變薄的因素，在氧化層及通道的界面便產生了

能量井，量子效應也就產生。在模擬上我們該如何考慮所謂的量子效應，

是一個重要的議題。

傳統上，為了模擬量子效應會加入水丁格(Schrödinger equation)方程式在半

導體方程式中。然而，水丁格方程式在數值計算上相當耗時及會有數值收

斂上的麻煩，在二維度或三維度空間中邊界條件的設定也不容易。為了避

免此方程式在模擬上的困難，許多替代的量子修正模型也陸續被提出，在

這許多的模型中，大都還是存在著偏微分方程式。近年來被提出的有效位

勢(effective potential)理論，是一個簡單的積分方程式。除此之外，在演算

法中也大大的改善了耗時的缺點。不過在有效位勢模型中，存在著一個具

有不確定性的變數(波包的標準差，standard deviation of wave packet)。隨著

標準差的變化，所模擬得到的結果也會有所差異。為了得到正確的值，吾

人利用波松-水丁格方程式的結果為基準，調整波包的標準差以達到兩者的

結果最為接近。而另外一個問題隨之出現，隨著元件外加不同的條件(偏

壓、氧化層厚度…等等)，標準差的值也會隨之變化。

在此論文中，所探討的元件結構為雙閘極以及絕緣層上矽金屬氧化物半導

體場效電晶體為主，探討不同的條件對波包的標準差的影響為何。吾人在

各種不同的外加條件下，以波松-水丁格方程式的結果為基準，求出各個不

同的波包標準差值。接著利用統計的方法，建立出波包標準差以及各外加

條件的模型。首先，我們以散佈圖觀察各外加條件對波包的標準差的關係

圖，發現之間並沒有複雜的關係，所以我們建立一個二階的線性模型。經

過變數轉換得到不錯的結果。

在此提出的模型在結構，外加條件上有所限制，可以將此模型的適用性擴

展到更多結構、或是特性相似的半導體元件上。文章中所提出的統計方法

可以廣為應用在其他的半導體元件特性分析上。

vi

vii

student：Ching-Lan Chang

Advisors：Dr. Yow-Jen Jou

Dr. Yiming Li

Department of Institute of Statistics National Chiao Tung University

ABSTRACT

Within the next decade or so, it is expected that gate lengths will shrink to 45 nm or less in devices found in integrated circuits. Quantum effects are known to occur in the channel region of MOSFET devices, in which the carriers are confined in a triangular potential well at the semiconductor-oxide interface. How might we expect quantum mechanics to arise in the transport through these small devices?

Typically, these effects are quantified by a simultaneous solution of the Schrödinger and Poisson equations, which can be a very time consuming procedure if it needs to be incorporated in realistic device simulations. Besides, different methods are proposed to include quantization effects in simulation of carrier transport in nanoscale devices. For instance, Hansch, MlDA, Van Dort, Density Gradient model … etc. Among these approaches, Density Gradient method are used generally. However, the quantum potential is defined in terms of the second derivative of the square root of local density. Such and approach is highly sensitive to noise in the determination of the local carrier density. Recently, Ferry propose an efficient method, effective potential, to include quantum effects. This approach avoids complex computation. Later, an more complicated effective potential is develop, but it is not included in our discussion.

viii

Effective potential method is quite convenient to calculate. However, one variable, standard deviation of wave packet, in the model influence the results quite significantly. Unfortunately, value of this parameter is not known exactly. How to determine the value is an interesting problem.

In this thesis, we do some simulations with various conditions to calibrate value of the variable by Schrödinger equation. And try to establish a model of standard deviation of wave packet by using statistical methods. First, we draw the scattering plots and find that correlations between outer conditions and value of standard deviation of wave packet are simple. So we just establish a second order multiple linear model. We get results which are satisfied through power transformation. The model is established corresponding to double-gate and silicon-on-insulator (SOI) MOSFET structures. Though the model is not suitable for any structure, conditions of devices. This method can be expanded to establish other models more generally.

可以繼續完成感興趣的研究，並在課業以及生活上不吝惜的指導及支持。其次，我要感謝 我另ㄧ共同指導老師 李義明老師，感謝老師指導我論文方向脈絡，以及研究的能力激發有 著深厚的影響。 論文口試期間，承蒙清華大學統計學研究所周若珍教授、以及國家高速網路與計算中 心羅仕京博士撥冗細審，並會予寶貴意見與殷切指正使本論文更臻完備。 研究室方面，我要感謝周宏穆學長、建松、煒昕、柏賢、宏榮教導我許多元件物理的 觀念，真的讓我獲益匪淺；紹銘學長、陳璞學長、傳盛學長、正凱學長、彥羽學長在程式 上以及論文給予相當大的幫助。 統研所方面，感謝豐洋、育仕、玉均、雅靜、穎劭、婉文；不僅在統計的知識上會給 予幫助，也在我學習的路上給予許多的建議。 我還要感謝我的好朋友們，哲名、信揚、義富、全豐、智國、穎慈、岱融。雖然現在 大家都各分東西，但是總在最恰當的時候，給予最恰當的鼓勵、安慰、歡笑。並且在論文 中各方面的問題提供協助。 最後最應該感謝的是我的爸爸，媽媽，和我的姐姐以及金龍，支持我在統計之外，往 半導體學習。這期間即便我承受在大的壓力、不開心、不耐煩、及挫折，還是一直在鼓勵 我、幫我調適心情。現在終於完成論文畢業了，真的很感謝你們。 感謝這段期間大家對我的包容、關懷與愛護。這篇論文，這個工作，以及我在交大的 一切成長，沒有你們，是完全沒有辦法達成的，一切的功勞都歸因於全部的人。謝謝大家 一直挺我鼓勵我，使我順順利利的度過這段非凡且精采的日子。在此將這篇論文獻給所有 關心我以及我所愛的人，謝謝你們。 本論文感謝行政院國家科學委員會(計畫編號 NSC-93-2115-E-492-008、NSC-94-2115- E-009-084)、卓越沿續計畫(計畫編號 NSC-94-2752-E-009-003-PAE、NSC-95-2752-E-009- 003-PAE)、五年五百億計畫、經濟部科專計劃(計畫編號 93-EC-17-A-07-S1-0011)以及台 灣積體電路製造股份有限公司之資助。 張景嵐 謹誌 中華民國九十五年七月 于風城交大 ix

Abstract (in Chinese) . . . v

Abstract (in English) . . . vii

Acknowledgments (in Chinese) . . . ix

List of Tables . . . xv

List of Figures . . . xvi

1 Introduction 1 1.1 Background . . . 1

1.2 Motivation . . . 3

1.3 Outline . . . 4

2 Classical and Quantum Mechanical Transport Models 5 2.1 Double-Gate and Silicon-On-Insulator Metal-Oxide-Semiconductor Field-Effect Transistors . . . 6

xii CONTENTS

2.2 Classical Drift-Diffusion Model . . . 11

2.3 Quantum Mechanical Model . . . 14

3 Effective Potential 20 3.1 Fundamental of the Effective Potential . . . 21

3.2 Ferry’s Effective Potential Approach . . . 24

4 The Linear Regression 32 4.1 Scattering Plot . . . 33

4.2 Multiple Linear Regression . . . 34

4.2.1 Model Expression . . . 34

4.2.2 Estimation of The Model Parameters . . . 35

4.2.3 Hypothesis Testing in Multiple Linear Regression . . . 39

4.2.4 Variable Selection in Regression Analysis . . . 42

4.3 Residual Analysis . . . 45

4.3.1 Normal Probability Plot . . . 45

4.3.2 Plot of Residuals against the Fitted Values . . . 46

4.4 Power Transformation . . . 47

5 Results and Discussion 49 5.1 Calibration of Ferry’s Effective Potential . . . 50

5.2 Data Collection by Using Device Simulation Tool . . . 53

5.3 Modelling and Simulation Results . . . 54

5.3.1 Modelling Double-Gate MOSFET . . . 54

5.3.2 Accuracy of Model of Double-Gate MOSFET . . . 74

5.3.3 Modelling Double-Gate and SOI MOSFETs . . . 77

5.3.4 Accuracy of Model of Double-Gate and SOI MOSFET . . . 90

5.4 Discussion . . . 95 6 Conclusions 96 6.1 Summary . . . 97 6.2 Future Work . . . 98 References . . . 99 Appendix A Effective Potential Source Code of Matlab . . . 107

Appendix B ISE Commands for Classical Transport and Schr¨odinger Equation . . . 109

B.1 MDraw Commands . . . 110

B.2 Dessis Commands . . . 111

xiv CONTENTS

Appendix C

Energy Band of Double-Gate MOSFET . . . 116

Appendix D

4.1 ANOVA table for significance of regression in multiple regression . . . 41

5.1 Correlation Table . . . 56

5.2 ANOVA table for significance of regression in multiple regression in double-gate MOSFET . . . 56

5.3 Residual statistics in double-gate MOSFET . . . 57

5.4 Coefficients table in double-gate MOSFET . . . 57

5.5 ANOVA table for significance of regression in multiple regression . . . 77

5.6 Residual statistics . . . 78

5.7 Coefficients table . . . 78

List of Figures

2.1 3D double-gate schematic diagram . . . 7

2.2 2D double-gate schematic diagram . . . 8

2.3 A energy band profile for the double-gate MOSFET used in the simulation . 9 2.4 3D SOI schematic diagram . . . 10

2.5 Flow chart of Poisson and Continuity equations in ISE . . . 13

2.6 Flow chart of SP Equation in ISE . . . 17

2.7 Potential of SP equations . . . 18

2.8 Electron density of SP equations . . . 19

3.1 Flow chart of Ferry’s effective potential . . . 27

3.2 Potential from Ferry’s effective potential . . . 28

3.3 Electron density of Ferry’s effective potential . . . 29

3.4 Calibration of electron density of Ferry’s effective potential by SP equa-tions - I . . . 30

3.5 Calibration of electron density of Ferry’s effective potential by SP

equa-tions - II . . . 31

4.1 Example of scattering plot . . . 33

4.2 The Box-Cox likelihood plot . . . 48

5.1 Classical electron density and electron density corrected by SP equations of double-gate MOSFET . . . 51

5.2 Electron density corrected by Ferry’s effective potential with various stan-dard deviation of wave packet of double- gate MOSFET . . . 52

5.3 Scattering Plot : Channel length vs. standard deviation of wave packet . . . 59

5.4 Scattering plot : Gate voltage vs. Standard deviation of wave packet . . . . 60

5.5 Scattering plot : Drain voltage vs. Standard deviation of wave packet . . . . 61

5.6 Scattering plot : Thickness of bulk vs. Standard deviation of wave packet . 62 5.7 Scattering plot : Thickness of oxide vs. Standard deviation of wave packet . 63 5.8 Scattering plot : Doping concentration vs. Standard deviation of wave packet 64 5.9 Normal plot I . . . 65

5.10 Scattering plot : Fitted value against residual I . . . 66

5.11 Likelihood plot of power transformation . . . 67

5.12 Normal plot II . . . 68

xviii LIST OF FIGURES

5.14 Normal plot III . . . 70

5.15 Scattering plot : Fitted value against residual III . . . 71

5.16 Normal plot IV . . . 72

5.17 Scattering plot : Fitted value against residual IV . . . 73

5.18 Comparison of electron density . . . 75

5.19 Scattering plot : Id-Vg curve . . . 76

5.20 Likelihood plot of power transformation . . . 80

5.21 Scattering plot : Thickness of oxide vs. Standard deviation of wave packet . 81 5.22 Scattering Plot : Channel length vs. standard deviation of wave packet . . . 82

5.23 Scattering plot : Thickness of bulk vs. Standard deviation of wave packet . 83 5.24 Scattering plot : Gate voltage vs. Standard deviation of wave packet . . . . 84

5.25 Scattering plot : Drain voltage vs. Standard deviation of wave packet . . . . 85

5.26 Scattering plot : Doping concentration vs. Standard deviation of wave packet 86 5.27 Scattering plot : Structure vs. Standard deviation of wave packet . . . 87

5.28 Normal plot . . . 88

5.29 Scattering plot : Fitted value against residual I . . . 89

5.30 Comparison of electron density . . . 91

5.31 Scattering plot : Id-Vg curve of double-gate MOSFET . . . 92

5.33 Scattering plot : Id-Vg curve of SOI . . . 94

C.1 Energy band plot of various thickness of bulk . . . 117

C.2 Energy band plot of various doping concentration . . . 118

C.3 Energy band plot of various gate voltage . . . 119

C.4 Energy band plot of various thickness of oxide . . . 120

C.5 Energy band plot of various drain voltage . . . 121

C.6 Energy band plot of various channel length . . . 122

D.1 SPSS-I . . . 124

D.2 SPSS-II . . . 125

D.3 SPSS-III . . . 126

D.4 SPSS-IV . . . 127

Introduction

1.1

Background

Study of advanced nanoscience and nanotechnology has recently been of great interest, in particular nanoscale semiconductor structures and devices [1][2]. In order to obtain high chip density, low power dissipation, and high speed for devices [3], the reduction of the gate oxide thickness (to around 1 nm) is necessary [4]. The ultra-thin oxide leads to a very large electric field at the SiO₂/Si interface. This results in a narrow and deep potential well at the semiconductor-insulator interface. According to quantum-mechanics (QM), elec-trons are now confined in such a potential well and then quantized to many discrete energy levels consequently force the motion of the electron in the direction perpendicular to the

2 Chapter 1 : Introduction

silicon-insulator interface [5][6]. Since the quantum effect becomes noticeable in the deep-submicron devices and a mere classical description of the physics is not sufficient for an accurate calculation of the inversion-layer charge, in order to understand the characteristics of a nanoscale device, it is important to take quantum mechanical effects into account. In principle, the Schr¨odinger-Poisson (SP) equations are the most accurate way to handle the problem of the inversion-layer charge density, but it is not suitable for engineering ap-plications especially for the two- and three-dimensional cases. This is not only because it is computationally expensive but also because it is difficult to generalize to the multi-dimensional case (e.g. how to deal with the boundary condition in the 3-D case). Thus it is important to find a method which can produce a result similar to the quantum mechanically calculated one but requires only about the same computation cost as that of the classical calculation. Over the last two decades, various quantum mechanical correction methods are proposed. Among these approaches, the effective potential has the easiest numerical computation, but is too sensitive to the fitted parameter. The value usually used is 5 ˚A. Is the value exact? It is suspect. So determination of this parameter is an important issue.

1.2

Motivation

There are several approaches have been proposed to replace solving Schr¨odinger equation to include the quantum effects. And accuracy is highly believable. However, computa-tion of algorism is also time-consuming. The effective potential Ferry proposed has been advanced which has the advantages of easy numerical implementation and almost guar-anteed convergence. And this approach are widely used and compared with other models [7][8][9][10]. We calibrate effective potential method by the results from SP equations to determine the suitable standard deviation of the wave packet, a parameter in the ef-fective potential formula. Value of the variable people usually use is 0.5 nm. However, result from effective potential approach is quite sensitive to the parameter. Different ap-plied voltage, thickness of oxide, doping and other conditions will cause different values of the parameter. In order to choose better value of this parameter in simulation with various conditions. The objective of this thesis is to model the correlation between the parameter and other conditions of devices. In this thesis, we try to use statistical method to analysis the model of double-gate and SOI (Silicon-on-Insulator) MOSFETs (Metal-Oxide-Semiconductor Field-Effect Transistor). And these statistical approaches may be extended to more general structures or devices with other conditions.

4 Chapter 1 : Introduction

1.3

Outline

There are six chapters in this thesis. In Chapter 2, classical drift-diffusion model and classi-cal quantum mechaniclassi-cal transport model-SP will be introduced. Chapter 3 presents Ferry’s effective potential, including its merit, shortcomings, and comparison between SP equa-tions and Ferry’s effective potential. Chapter 4 will present some statistical methods used in our analyzing. In the chapter 5, we will show the results and some discussion according to statistical analyzing results. Finally, we draw some conclusions and suggest the future works in Chapter 6.

Classical and Quantum Mechanical

Transport Models

I

n the last years, different techniques have been proposed to include quantization effects in simulation in nanoscale devices. In this chapter, we will present the classical trans-port (drift-diffusion model) and quantum mechanical model - Schr¨odinger-Poisson equa-tions. We simulate these models by software [ISE]. We will show the simulation procedure for these models. Besides, the command used in ISE is shown in appendix B.

6 Chapter 2 : Classical and Quantum Mechanical Transport Models

2.1

Double-Gate and Silicon-On-Insulator

Metal-Oxide-Semiconductor Field-Effect Transistors

Figure 2.1 and 2.2 show 3D and 2D schematic diagrams of double-gate MOSFET. Our sim-ulation focuses on this structure. Figure 2.3 shows the energy band profile for a double-gate MOSFET. From Fig. 2.3, we can find there are potential wells in the direction perpendicu-lar to the SiO₂/Si interface (Region1). Therefore, quantum effects are often considered in the direction which is confined [16]. In our simulation, we solve 2D Poisson equation and 2D electron current continuity equation [17]. All quantum mechanisms are considered in one-dimension (1D) along x direction. So, 1D Schr¨odinger equation is considered (in the direction x) [42][18]. Also, effective potential method presented in Chapter 3 is corrected in one-dimension.

For double-gate MOSFETs, if one of gate is increased thick enough and gates voltage equal to 0, then we can treat it as SOI (Fig. 2.4). Because of the thick oxide, there will be only one potential well.

drain

channel

source

gate

gate

oxide

oxide

x

y

Figure 2.1: 3D double-gate schematic diagram. Scales in our

simulation are following: thickness of oxide: 1 nm∼ 2 nm; channel length: 20 nm∼ 50 nm; thickness of film: about 0.5*channel length; doping of source and drain: 1e20 / cm3; doping of film: 1e16∼ 5e17 / cm3.

8 Chapter 2 : Classical and Quantum Mechanical Transport Models

channel

x

y

source

drain

_gate

oxide

Figure 2.2: 2D double-gate schematic diagram. In our research, we consider 2D model except quantum correction. And quantum correction will be consider in x direction for each

drain

channel

source

gate

gate

oxide

oxide

x

y

Figure 2.3: A energy band profile for the double-gate MOSFET in x direction. When voltage is applied on gate, there will be potential well (Region 1) if the oxide is thin enough. And then there will be quantum mechanism. E_limis the energy level corresponding to the classical regime.

10 Chapter 2 : Classical and Quantum Mechanical Transport Models

drain channel source

gate

gate

oxide

oxide x

y

Figure 2.4: 3D SOI schematic diagram. If we let one oxide in

double-gate MOSFET is thick enough, then it will become SOI structure. In this thesis, we set one of thickness of oxide is equal to 200 nm. And whose gate voltage is equal to 0 V. Other conditions are the same as Fig. 2.1.

2.2

Classical Drift-Diffusion Model

Essentially, the continuity equations and the Poisson equation have to be satisfied when we consider a device with nonequilibrium applied voltage

V = −ρ ε_s = q ε_s(n − p + D), (2.1) ∂n ∂t = − 1 q∇ · Jn+ (Gn− Rn), (2.2) ∂p ∂t = − 1 q∇ · Jp+ (Gp − Rp).

Eq. 2.1 is the Poisson equation, where V is potential, ρ is space charge density, ε_s is permittivity of silicon, n is electron density, p is hole density, and D is doping. Eq. 2.2 are continuity equations for electron and hole. Where J is current density, G and R are the generation term and recombination term, respectively [13].

In Poisson equation, Consider the Boltzmann relation. At thermal equilibrium the relation is given by n= ni· exp(EF − Ei kT ) ≡ ni· exp[ q(V − φ) kT ], (2.3) p= ni· exp(Ei− EF kT ) ≡ ni· exp[ q(φ − V ) kT ],

where φ is the potential corresponding to the Fermi level. When the voltage is applied, the relation becomes

n ≡ n_i· exp[q(V − φn)

kT ], (2.4)

p≡ n_i· exp[q(φp− V )

12 Chapter 2 : Classical and Quantum Mechanical Transport Models

where φ_nand φ_pare the quasi-Fermi levels for electrons and holes, respectively[12]. There-fore, Poisson equation becomes a function of potential and quasi Fermi-levels.

In continuity equations, assume(Gn− Rn) = (Gp− R_p) = 0 to simplify the equations. At the stable state, Eq. 2.2 becomes,

∇ · Jn = 0, (2.5)

∇ · Jp = 0,

where

J_n = −qµnn∇V + qD_n∇n, (2.6)

J_p = −qµ_pp∇V − qD_p∇p.

µ is mobility, D_nand D_p are diffusion coefficient for electron and hole.

Thus, we can get the potential, φ_n, and φ_p self-consistent by solving Eq. 2.1 and Eq. 2.5 repeatedly [12] [13] until the results are convergent [14][15]. The flow is shown as Fig. 2.5. Potential solved from these equations is classical, without considering any other mechanism.

Initial guesses of

potential and imref

Poisson equation

(imref fixed, get new potential)

Continuity equation

(potential fixed, get new imref)

Convergent

Result

yes

no

Figure 2.5: Flow chart of Poisson and Continuity equations in ISE (imref : quasi Fermi-levels). Here, potential and imref are unknown. First, we have initial guesses of both. Then we solve Poisson equation by given imref, and solve continuity equation by given potential. The algorithm will be

14 Chapter 2 : Classical and Quantum Mechanical Transport Models

2.3

Quantum Mechanical Model

As the size of devices decreasing, quantum effects are included in simulation. In principle, the Schr¨odinger Equation have to be considered to describe the quantum effects. However, it is not efficient to solve Schr¨odinger Equation.

Conventionally, we consider the Schr¨odinger Equation to include the quantum effects. Following is 1-D Schr¨odinger Equation,

∆ψ(x) + 2m∗

2 (E − V (x))ψ(x) = 0, (2.7)

where ψ is the wave function, m∗is the effective mass,
is the Planck’s constant, E is total energy, and V is potential. We can rewrite Eq. 2.7 as

[−
2 2m∗ ·

∂2

∂x2 + V (x)]ψ(x) = Eψ(x). (2.8)

From Eq. 2.8, we can know that the Schr¨odinger equation is an eigenvalue problem [29]. For given potential, we can get the subbands and wave function. Therefore, we can get charge density from subbands and wave function. Charge density calculated from Schr¨odinger equation includes quantum effects.

Based on the model described above, the charge density in the silicon layer is given by

For the p-type substrate, the hole density p is calculated by the Boltzmann approximation as before,

p= n_i· exp[q(φp− V ) kT ],

whereas the electron density n_q contained in the subbands which are lower then E_limand is given by [19][20][21][22][23][24]: n_q = qVT π
2 2
k=1 g_km_jk
j ln[1 + exp( EF−Ejk kBT ) 1 + exp(EF−Elim

kBT )

]|ψjk|2_{, (2.9)}

where E_F is the electron quasi-Fermi level, g_kis the degeneracy factor of the kth valley, ψ_jk is the wave function of the jth level in the kth valley and m_jkis the parallel effective mass in the kth valley. For (100) silicon, there is a two-fold degenerate pair of valleys with a larger effective mass (along the transverse direction), m∗ = 0.916m₀,which comprises the lowest subband. The four-fold degenerate valleys have a lighter effective mass m∗ = 0.190m₀, and lie higher in the subband ladder. E_limis the energy level corresponding to the classical regime. Treating the density of states classically above the energy level E_lim limits the

j and k values in the summation such E_jk
Elim in the polycryatalline layer, classical treatment is usually assumed.

Under quantum effects, We get new charge density from Schr¨odinger equation. However, potential will change simultaneously. In order to get potential under quantum effects, we have to solve Poisson equation by using new charge density. Because potential and charge

16 Chapter 2 : Classical and Quantum Mechanical Transport Models

density influence each other. So, we have to solve Poisson and Schr¨odinger equations repeatedly until convergent. Flow chart is shown as Fig. 2.6. Fig. 2.7, and Fig. 2.8 show the comparison of potential and carrier density between classical results and Schr¨odinger equation.

Initial guesses of

potential and imref

Poisson equation

Continuity equation

no

Schrödinger equation

(get new charge density)

Convergent

yes

Result

Figure 2.6: Flow chart of SP Equation in ISE. Steps of Poisson equation and continuity are the same as statement in Fig. 2.5. Here, we solve Schr¨odinger equation to correct carrier density to include quantum effect.

18 Chapter 2 : Classical and Quantum Mechanical Transport Models

Figure 2.7: Solid line is potential solved from classical transport, dash line is potential solved by SP equations. Potential after quantum corrected is mush higher than classical. We only show half curves because of symmetry of double-gate MOSFET. tox = 1 nm, Vd = Vg = 0.6 V, tsi = 10 nm, Lg = 20 nm, N = 1e22 / m3.

Figure 2.8: Solid line is electron density derived from classical transport; dash line is electron density derived from SP equations. The serious change nearby the SiO₂/Si interface arises from quantum effect. We only show half curves because of symmetry of double-gate MOSFET. tox = 1 nm, Vd = Vg = 0.6 V, tsi = 10 nm, Lg = 20 nm, N = 1e22 / m3.

Chapter 3

Effective Potential

In an effective potential approach, one replaces the quantum distribution function by a classical distribution function with a modified potential. Thus, all the quantum effects in the system are modelled solely through the forces acting on the electron. Effective potentials are derived from a quantum mechanical description, either directly from the Schr¨odinger Equation or from a quantum kinetic transport equation for the Wigner function.

3.1

Fundamental of the Effective Potential

The idea of quantum potential is quite old and originates from the hydrodynamic formula-tion of quantum mechanics, first introduced by de Broglie and Madelung, and later devel-oped by Bohm. one begins with the one particle Schr¨odinger Equation, of the form

i
∂ψ ∂t = −(

2

2m)∇2ψ+ V (x)ψ, (3.1)

The wave function is written in complex form in terms of its amplitude R(r, t) and phase

S(r, t) as

ψ(r, t) = R(r, t)exp[iS(r, t)/
]. (3.2)

When substituted back into the Schr¨odinger Equation, one arrives at the following coupled equations of motion for the density and phase

∂R(r, t) ∂t = − 1 2m[R(r, t)∇2S(r, t) + 2∇R(r, t) · ∇S(r, t)], (3.3) ∂S(r, t) ∂t = −[ [∇(r, t)]2 2m + V (r, t) −
2 2m ∇2_{R(r, t)} R(r, t) ]. (3.4)

It is convenient to write ρ(r, t) = R(r, t)2, where ρ(r, t) is the probability density. One then obtains ∂ρ(r, t) ∂t + ∇ · (ρ(r, t) 1 m∇S(r, t)) = 0, (3.5) −∂S(r, t) ∂t = 1 2m[∇S(r, t)]2+ V (r, t) + Q(ρ, r, t). (3.6)

22 Chapter 3 : Effective Potential

In the classical limit the above equations are subject to a very simple interpretation. The function S(r, t) is a solution of the Hamiltonian-Jacobi equation. If we consider an en-semble of particle trajectories which are solutions of the equations of motion, then from a well-known theorem of mechanics which states that if all of these trajectories are normal to any given surface of constant S, then they are normal to all surfaces of constant S, and

∇S(r, t)/m equals the velocity vector, v. Therefore, Eq. 3.5 can be rewritten as ∂ρ(r, t)

∂t + ∇ · [ρ(r, t)v] = 0. (3.7)

Since ρ(r, t) is the probability density, ρv is the mean current of particles in the ensemble, and Eq. 3.7 simply expresses conservation of probability or of particles in the ensem-ble (continuity equation). Also note that Eq. 3.5, 3.6 arising from this so -called Madelung transformation to the Schr¨odinger Equation, have the form of classical hydrodynamic equa-tions with the addition of an extra potential, often referred to as the quantum or Bohm potential, written as Q= −
2 2mR∇2R ≈ −
2 2m√n ∂2√n ∂x2 , (3.8)

where the density n is related to the probability density as n(r, t) = Nρ(r, t) = NR(r, t)2,

N being the total number number of particles in the ensemble. The Bohm potential

essen-tially represents a field through which the particle interacts with itself. Once we know the field functions, one can calculate the force, os that, if one knows the initial position and momentum of the particle, one can calculate its entire trajectory. This effective potential

approach has been used, for example, in the study of wave packet tunnelling through [35], where the effect of the quantum potential is shown to lower or smoothen barriers and hence allows for the particles to leak through.

An alternative form of the quantum potential was proposed by Iafrate, Grubin and Ferry [30], who derived a form of the quantum potential based on moments of the Wigner-Boltamann equation.the kinetic equation describing the time evolution of the Wigner dis-tribution function [31]. Their form for the quantum potential, based on moments of the Wigner distribution function in a pure state, and involving an expansion of order O(
2), is given by

V_Q= −
2

8m∆(ln n), (3.9)

and is referred to as the Wigner potential, or as the density gradient correction. This form of the Wigner potential is better thought of as a quantum pressure term, which works to modify the actual potential to allow charge penetration into the classically forbidden regions. Ferry and Zhou derived a form for a smooth quantum potential [36], based on the effective classical partition function of Feynman and Kleinert [28], by linearizing an equation for the equilibrium density matrix. The Feynman-Kleinert effective partition function involves a smoothed potential of the form

V_a2(x) =  dy √ 2πa2exp{− (x − y)2 2a2 }V (y), (3.10)

24 Chapter 3 : Effective Potential

where V is the classical potential energy, a2 ∝ β
2/m, β = 1/T is the inverse

tempera-ture, and m is the particle mass. The Ferry-Zhou effective stress represents the difference between the smoothed and the local quantum potential
2∇2n/8mn + V , where n is the

particle density. Their smoothing function is of the form exp(−(x−y)2/2a2)/|x−y|. Note

that the off-diagonal entries in the stress tensor are neglected in [36].

3.2

Ferry’s Effective Potential Approach

In analogy to the smoothed potential representations discussed above for the quantum hy-drodynamic models, it is desirable to define a smooth quantum potential for use in quantum particle-based simulations. Ferry [33] has suggested an effective potential that emerges from the wave packet description of particle motion, where the extent of the wave packet spread is obtained from the range of wavevectors in the thermal-distribution function. This form for the effective potential allows one to build in certain quantum effects that primarily arise from the non-zero size of the electron wave packet. One arrives at the final result by noting that the potential, in an inhomogeneous system enters the Hamiltonian as [33]

H_ν =



Using the non-local form(wave packet description) for the charge leads to V =  drV(r)
i n_i(r) (3.12) =  drV(r)
i  drexp(−|r − r _|2 α2 )δ(r _{− r} i) =
i  drδ(r − ri)  drV(r) exp(−|r − r _|2 α2 )

where the summation over i is a summation over the carriers themselves. The term in the primed integration is now the effective potential, V_{ef f}, and the finite size of the electron has been replaced by smoothing of the real potential. In essence, the effective potential, V_{ef f}, is related to the potential obtained from the Poisson equation, through an integral smoothing relation

V_{ef f}(x) =



V(x + y)G(y, a₀)dy (3.13)

where G is a Gaussian with the standard deviation a₀. In two dimensions, the formula becomes, V_{ef f}(x, y) = 1 2πaxay   V(x, y)exp[−(x − x ₎2 2a2 x − (y − y_2a2)2 y ]dx_dy _(3.14)

where V is the actual potential, and a_x,y are the standard deviations of the Gaussian wave packet [34][35][36]. The flow of computing the effective potential is shown as below. However, V_{ef f} is quite sensitive to the standard deviation of the wave packet. Fig. 3.2 shows potential derived from Ferry’s effective potential with various standard deviation

26 Chapter 3 : Effective Potential

of wave packet. Fig. 3.3 shows carrier density derived from Ferry’s effective potential. From these two figures, we can find that the influence of a is quite significant. So, the determination of its value is an important issue. We calibrate Ferry’s effective potential by Schr¨odinger equation to determine the value of a. Fig. 3.4 shows the comparison between SP equations and Ferry’s effective potential with five various values of a. Results from these two methods are closest when a = 5 under following condition: thickness of oxide = 20 nm, channel length = 40 nm, thickness of bulk = 24 nm, gate voltage = 0.9 V, and doping concentration = 5e+23 / m3. Unfortunately, the results are not close anymore when we change the gate voltage from 0.9 V to 1.0 V and keep a= 5. See Fig. 3.5.

In terms of computation, Ferry’s effective potential is a good approach, but not in terms of sensitivity.

Initial guesses of

potential and imref

Poisson equation

Continuity equation

no

Ferry’s effective

potential

Convergent

yes

Result

Figure 3.1: Flow chart of Ferry’s effective potential. Ferry’s effective potential formula don’t need to solved in the loop. It just correct the potential which is convergent at last. Algorithm in the loop we simulate by ISE, and calculate Ferry’s effective by using our own code.

28 Chapter 3 : Effective Potential

Figure 3.2: Ferry’s effective potential of double-gate MOSFET in the direction normal to the semiconductor/oxide interface with various standard deviation of wave packet, a( ˚A). We only show half curves because of symmetry. Potential shift down as a is increasing. tox = 1 nm, Vd = Vg = 0.6 V, tsi = 10 nm, Lg = 20 nm, N = 1e22 / m3.

Figure 3.3: Electron density from Ferry’s effective potential of double-gate MOSFET in the direction normal to the semiconductor/oxide interface with various standard deviation of wave packet, a( ˚A). We only show half curves because of symmetry. Carrier density shift down as a is increasing. tox = 1 nm, Vd = Vg = 0.6 V, tsi = 10 nm, Lg = 20 nm, N = 1e22 / m3.

30 Chapter 3 : Effective Potential

Figure 3.4: Calibration of electron density of Ferry’s effective potential by SP equations - I. We only show half curves because of symmetry. (with tox= 1 nm, Lg = 40 nm, tsi = 24 nm,

V g = 0.9 V, N = 5e+23 / m3). We set a = 3, 4, 5, 6, 7 ˚Ato solve carrier density by effective potential. Result is the most close to the result form SP equations when a = 5 ˚A.

Figure 3.5: Calibration of electron density of Ferry’s effective potential by SP equations - II. We only show half curves because of symmetry. (with tox= 1 nm, Lg = 40 nm, tsi = 24 nm,

V g= 0.9 V, N = 5e + 23 / m3). Result from effective is not close to result from SP equations if we set V g = 1 V and keep a = 5 ˚A.

Chapter 4

The Linear Regression

M

ost studies and experiments, scientific or industrial, large scale or small, produce data whose analysis is the ultimate object of the endeavor. Statistics is a pow-erful tool. Regression analysis is a statistical technique for investigating and modelling the relationship between variables. Applications of regression are numerous and occur in almost every field. It is the most important step in our analyzing. Besides regression analy-sis, there are several statistical methods to assist us in model establishing and checking. In this chapter, we will present statistical approaches will be used in our discussion. Follow-ing, we will present scattering plot, multiple linear regression, residual analysis and power transformation.

4.1

Scattering Plot

Scattering Plot is an important tool in statistical analysis. We can approximately know how the correlation between variables before doing further statistical analysis. For example, Fig. 4.1 shows population of U.S.A. from 1790 to 1990. The graph suggest the possibility of fitting a quadratic or exponential trend [37]. Of course, scattering plot is a important basis in statistical analysis.

Figure 4.1: Population of the U.S.A at ten-year intervals, 1790-1990. From figure, we find that population increasing as years increasing. And it appears quadratic or exponential trend.

34 Chapter 4 : The Linear Regression

4.2

Multiple Linear Regression

4.2.1

Model Expression

A regression model that involves more than on regressor variables is called a multiple regression model. Simple linear regression model is a special case of multiple linear re-gression with only one regressor variables.

In general, the response y may be related to k regressor or predictor variables. The model

y= β₀+ β₁x₁+ β₂x₂+ · · · + βkx_k+ ε, (4.1)

is so-called a multiple linear regression model with k regressors. The parameters β_j, j =

0, 1, . . . , k are called the regression coefficients. This model describes a hyperplane in the k

dimensional space of the regressor variables x_j. The parameter β_j represents the expected change in the response y per unit change in x_j when all of the remaining regressor variables

xi(i = j) are held constant. For the reason the parameters βj, j = 0, 1, . . . , k, are often

called partial regression coefficients.

Multiple linear regression models are often used as empirical models or approximating functions. That is, the true functional relationship between y and x₁, x₂, . . . , x_kis unknown, but over certain ranges of the regressor variables the linear regression model is an adequate approximation to the true unknown function [38].

4.2.2

Estimation of The Model Parameters

The method of least squares can be used to estimate the regression coefficients in Eq. 4.1. Suppose that n > k observations are available, and let y_i denote the ith observed response and x_ij denote the ith observation or level of regressor x_j. We assume that the error term ε in the model has E(ε) = 0, V ar(ε) = σ2, and that the errors are uncorrelated. Furthermore, we assume that the regressor variables x₁, x₂, . . . , x_kare fixed variables, measured without error.

We may write the sample regression model corresponding to Eq. 4.1 as

y_i = β₀+ β₁x_i1+ β₂x_i2+ · · · + βkx_ik+ εi (4.2) = β0+ k
j=1 β_jx_ij + ε_i , i= 1, 2, . . . , k.

The least-squares function is

S(β₀, β₁, . . . , β_k) = n
i ε2_i (4.3) =
n i (yi− β0− k
j=1 β_jx_ij)2.

The function S must be minimized with respect to β₀, β₁, . . . β_k. The least-squares estima-tors of β₀, β₁, . . . β_kmust satisfy

∂S ∂β₀|βˆ0, ˆβ1,..., ˆβk = −2 n
i=1 (yi− ˆβ0− ˆ k
j=1 β_jx_ij) = 0 , (4.4)

36 Chapter 4 : The Linear Regression and ∂S ∂β_j|βˆ0, ˆβ1,..., ˆβk = −2 n
i=1 (yi− ˆβ₀− ˆ k
j=1 β_jx_ij)xij = 0 , (4.5) j = 1, 2, . . . , k.

Simplifying Eq. 4.4 and Eq. 4.5, we obtain the least-squares normal equations

n ˆβ₀+ ˆβ₁ n
i=1 x_i1+ ˆβ₂ n
i=1 x_i2+ · · · + ˆβ_k n
i=1 x_ik = n
i=1 y_i, (4.6) ˆ β₀ n
i=1 x_i1+ ˆβ₁ n
i=1 x2_i1+ ˆβ₂ n
i=1 x_i1x_i2+ · · · + ˆβ_k n
i=1 x_i1x_ik = n
i=1 x_i1y_i, .. . ˆ β₀ n
i=1 x_ik+ ˆβ₁ n
i=1 x_ikx_i1+ ˆβ₂ n
i=1 x_ikx_i2+ · · · + ˆβ_k n
i=1 x2_ik = n
i=1 x_iky_i.

Note that there are p = k + 1 normal equations, one for each of the unknown regres-sion coefficients. The solution to the normal equations will be the least-squares estimators

ˆ

β₀, ˆβ₁, . . . ˆβ_k.

It is more convenient to deal with multiple regression models if they are expressed in ma-trix notation. This allows a very compact display of the model, data. and results. In mama-trix notation, the model given by Eq. 4.2 is

where y= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ y₁ y₂ .. . y_n ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , X = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 x11 x12 · · · x1k 1 x21 x22 · · · x2k .. . ... ... ... ... 1 xn1 x_n2 · · · x_nk ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ and β = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ β₀ β₁ .. . β_k, ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , ε= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ _{1 2} .. . _n ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . (4.8)

In general, y is an n× 1 vector of the observations, X is an n × p matrix of the levels of the regressor variables, β is a p×1 vector of the regression coefficients, and ε is an n×1 vector of random errors. We wish to find the vector of least-squares estimators, ˆβ, that minimizes

S(β) =

n

i

ε2_i = εε= (y − Xβ)(y − Xβ). (4.9)

Note that S(β) may be expressed as

S(β) = yy− βXy− yXβ+ βXXβ (4.10)

38 Chapter 4 : The Linear Regression

since βXy is a 1 × 1 matrix, or a scalar, and its transpose (βXy) = yXβ is the same

scalar. The least-squares estimators must satisfy

∂S

∂β|βˆ = −2Xy+ 2XX ˆβ = 0, (4.11)

which simplifies to

XX ˆβ= Xy. (4.12)

Eq. 4.12 are the least-squares normal equations. They are the matrix analogue of the scalar presentation in Eq. 4.6.

To solve the normal equations, multiply both sides of Eq. 4.12 by the inverse of X’X. Thus, the least-squares estimator of β is

ˆβ = (X_X)−1_X_{y, (4.13)}

provided that the inverse matrix (XX)−1 exists. The(XX)−1 matrix will always exist if the regressors are linearly independent, that is, if no column of the X matrix is a linear combination of the other columns.

The fitted regression model corresponding to the levels of the regressor variables x =

[1, x1, x1, . . . , xk] is ˆy = xˆβ = ˆβ 0+ k
j=1 ˆ β_jx_j. (4.14)

The difference between the observed value y_i and the corresponding fitted value ˆy_i is the residual

e_i = yi− ˆy_i. (4.15)

We may develop and estimator of σ2from the residual sum of squares

SS_Ees n
i=1 (yi− ˆyi)2 ₌
n i=1 e2_i = yy− ˆβXy. (4.16)

The residual sum of squares has n− p degrees of freedom associated with it since p para-meters are estimated in the regression model. The residual mean square is

M S_Res = SSRes

n− p, (4.17)

the expected value of M S_Resis σ2, so an unbiased estimator of σ2is given by

σ2 = MS_Res. (4.18)

4.2.3

Hypothesis Testing in Multiple Linear Regression

The test for significance of regression is a test to determine if there is a linear relationship between the response y and any of the regressor variables x₁, x₂, . . . , x_k. This procedure is often thought of as an overall or global test of model adequacy. The appropriate hypotheses are

40 Chapter 4 : The Linear Regression

H₁ : βj = 0, for at least one j.

Rejection of this mull hypothesis implies that at least one of the regressors x₁, x₂, . . . , x_k

contributes significantly to the model.

The procedure is a generalization of the analysis of variance (ANOVA). The total sum of squares SS_T is partitioned into a sum of squares due to regression, SS_R, and a residual sum of squares, SS_Res. Thus

SS_T = SSR+ SSRes.

If the null hypothesis is true, then SS_R/σ2 follows a χ2_k distribution, which has the same number of degrees of freedom as number of regressor variables in the model. Besides,

SS_Res/σ2 ∼ χ2_n−k−1and that SS_Res and SS_R are independent. By the definition of an F statistic,

F₀ = SSR/k

SS_Res/(n − k − 1) =

M S_R

M S_Res, (4.19)

follows the F_k,n−k−1 distribution. Therefore, to test the hypothesis H₀ : β₀ = β₁ = · · · =

β_k= 0, compute the test statistic F₀and reject H₀ if

F₀ > F_{α,k,n−k−1}, (4.20)

where α is significant level. The test procedure is usually summarized in an ANOVA table as follow:

Source of Variation Sum of Squares Degrees of Freedom Mean Square F₀

Regression SS_R k M S_R M SR

M SRes

Residual SS_Res n− k − 1 M S_Res

Total SS_T n− 1

Table 4.1: ANOVA table for significance of regression in multiple regression. SS_R, SS_RES, and SS_T are square error. From Degrees of Freedom, we can know there are n cases and k parameters have to be estimated. Value of F₀is the criterion to judge whether the model significant is.

becomes which one(s). Adding a variable to a regression model always causes the sum of squares for regression to increase and the residual sum of squares to decrease. We must decide whether the increase in the regression sum of squares is sufficient to warrant using the additional regressor in the model. The addition of a regressor also increases the variance of the fitted value ˆy, so we must be careful to include only regressors that are of real value in explaining the response. Furthermore, adding an unimportant regressor may increase the residual mean square, which may decrease the usefulness of the model.

The hypotheses for testing the significance of any individual regression coefficient, such as

β_j, are

H₀ : βj = 0,

42 Chapter 4 : The Linear Regression

If H₀ : β_j = 0 is not rejected, then this indicates that the regressor x_j can be deleted from the model. The test statistic for this hypothesis is

t₀ = βˆj ˆ σ₂C_jj = ˆ β_j se( ˆβj), (4.21)

where C_jj is the diagonal element of(XX)−1 corresponding to β_j. The null hypothesis

H₀ : βj = 0 is rejected if |t₀| > t_{α/2,n−k−1}. Note that this is really a partial or marginal test because the regression coefficient β_j depends on all of the other regressor variables

x_i(i = j) that are in the model. Thus, this is a test of the contribution of x_j given the other regressors in the model.

4.2.4

Variable Selection in Regression Analysis

Because evaluating all possible regressions can be burdensome computationally, various methods have been developed for evaluating only a small number of subset regression models by either add or deleting regressors one at a time. These methods are generally referred to as stepwise-type procedures. They can be classified into three broad categories: (1) forward selection, (2) backward elimination, and (3) stepwise regression, which is a popular combination of procedures 1 and 2.

(a) Forward Selection

This procedure begins with the assumption that there are no regressors in the model other than the intercept. An effort is made to find an optimal subset by inserting regressors into the model one at a time. The first regressor selected for entry into the equation is the one that has the largest simple correlation with the response variable y. Suppose that this regressor is x₁. This is also the regressor that will produce the largest value of the F statistic for testing significance of regression. This regressor is entered if the F statistic exceeds a preselected F value, say F_IN. The second regressor chosen for entry is the one that now has the largest correlation with y after adjusting for the effect of the first regressor entered (x₁) on y. We refer to these correlations as partial correlations. They are the simple correlations between the residuals from the regression ˆy = ˆβ₀ + ˆβ₁x₁ and the residuals from the regressions of the other candidate regressors on x₁, say xˆ_j = ˆα_0j + ˆα_ajx₁, j =

2, 3, . . . , k. Suppose that at step 2 the regressor with the highest partial correlation with y

is x₂. This implies that the largest partial F -statistic is

F = SSR(x2|x1)

M S_Res(x₁, x₂). (4.22)

If this F value exceeds F_IN, then x₂ is added to the model. In general, at each step the regressor having the highest partial correlation with y is added to the model if its partial

F -statistic exceeds the preselected entry level F_IN. The procedure terminates either when the partial F -statistic at a particular step does not exceed F_IN or when the last candidate

44 Chapter 4 : The Linear Regression

regressor is added to the model.

(b) Backward Elimination

Backward elimination begins with a model that includes all k a candidate regressors. Then the partial F -statistic is computed for each regressor as if it were the last variable to enter the model. The smallest of these partial F -statistics is compared with a preselected value,

F_{OU T}. If the smallest partial F value is less than F_{OU T}, that regressor is removed from the model. Now a regression model with k− 1 regressors is fit, the partial F -statistics for this new model calculated, and the procedure repeated. The backward elimination algorithm terminates when the smallest partial F value is not less than the preselected cutoff value

F_{OU T}.

(c) Stepwise Regression

The two procedure described above suggest a number of possible combinations. One of the most popular is the stepwise regression algorithm. Stepwise regression is a modification of forward selection in which at each step all regressors entered into the model previously are reassessed via their partial F -statistics. A regressor added at and earlier step nay now be redundant because of the relationships between it and regressors now in the equation. If the partial F -statistic for a variable is less than F_{OU T}, that variable is dropped from the model. Stepwise regression require two cut off values, F_IN and F_{OU T}. Some analysis prefer to

choose F_IN = FOU T, although this is not necessary. Frequently we choose F_IN > F_{OU T}, making it relatively more difficult to add a regressor than to delete one.

Among the three selection procedures, stepwise selection is known to be the most effective and is therefore recommended for general use [38][39].

4.3

Residual Analysis

Graphical analysis of residuals is a very effective way to investigate the adequacy of the fit of a regression model and to check the underlying assumptions. In this section, we introduce and illustrate the basic residual plots [41].

4.3.1

Normal Probability Plot

A very simple method of checking the normality assumption is to construct a normal prob-ability plot of the residuals. This is a graph designed so that the cumulative normal distribu-tion will plot as a straight line. Let e_[1] < e_[2] <· · · < e_[n]be the residuals ranked in increas-ing order. If we plot e_[i] against the cumulative probability P_i = (i − 1₂)/n, i = 1, 2, . . . , n, on the normal probability plot, the resulting points should lie approximately on a straight line. Substantial departures from a straight line indicate that the distribution is not normal. Usually normal probability plots are constructed by plotting the ranked residual e_[i]against the expected normal valueΦ−1[(i−1₂)/n], where Φ denotes the standard normal cumulative

46 Chapter 4 : The Linear Regression

distribution. This follows from the fact that E(e_[i]) Φ−1[(i − 1₂)/n].

4.3.2

Plot of Residuals against the Fitted Values

A plot of the residuals e_i versus the corresponding fitted values ˆy_i is useful for detecting several common types of model inadequacies. If the scattering plot indicates that the resid-uals can be contained in a horizontal band, then there are no obvious model defects. Else if the plot has strange pattern , then we may try to transform the response variable to improve the model. The outward-opening funnel pattern implies that the variance is and increasing function of response value. A curved plot which is nonlinearity may mean that other re-gressor variables are need in the model. Transformations on the rere-gressor or the response variable may also be helpful.

4.4

Power Transformation

Generally, transformation are used for three purposes: (1) stabilizing the response variance, (2) making the distribution of the response variable closer to the normal distribution, (3) improving the fit of the model to the data.

We often find that the power family (Eq. 4.23) of transformation is very useful. Box and Cox(1964) have shown how the transformation parameter λ may be estimated simul-taneously with the other model parameters. The theory underlying their method uses the method of maximum likelihood.

f(y) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ yλ−1 λ , λ = 0; ln y, λ = 0. (4.23)

Notice that we cannot select a value of λ by directly comparing the error sums of squares from analysis of y after transformation, because for each value of λ the error sum of squares is measured on a different scale.

In applying the Box-Cox method, we recommend using simple choices for λ, because the practical difference between λ = 0.5 and λ = 0.58 is likely to be small, but the square root transformation is much easier to interpret. Fig. 4.2 shows that Likelihood achieves maximum when λ= −0.148 and 95% confidence interval. By the reason above, we prefer choosing λ = 0 [40].

48 Chapter 4 : The Linear Regression

Figure 4.2: The Box-Cox likelihood plot. x axis is value of λ, and y axis is likelihood function. Likelihood achieves maximum when λ= −0.148. The three vertical dot line indicate 95% confidence interval. Plot by S-PLUS.

Results and Discussion

In the chapter, the simulation results are shown and the statistical results are also discussed. First of all, the calibration of Ferry’s effective potential is introduced. We use least square error to determine value of standard deviation of wave packet. Second, range of variables we simulation and some constraint of variables will be stated. There is collinearity phe-nomenon because of these constraint. At last, results will be shown and we will discuss according to the results, including double-gate and SOI MOSFET. And models will be presented. We will also compare the statistical results with simulation results.

50 Chapter 5 : Results and Discussion

5.1

Calibration of Ferry’s Effective Potential

Figure 5.1 shows the classical carrier density and carrier density corrected by SP equa-tions. Fig. 5.2 shows carrier density from Ferry’s effective potential with various standard deviation of wave packet. However, different values of a cause quite different results. We calibrate the value by comparing with result from Schr¨odinger equation. We determine the value of standard deviation of wave packet by achieving the criterion

min a n
i=1 (nSP,i− nEP,a,i)2 _(5.1)

where n_SP,iis carrier density of mesh i from SP equations, n_EP,a,iis carrier density of mesh

i from Ferry’s effective potential.

For every condition of device, we determine a value of a by above criterion. In simulation procedure, we scan a from 0 to 20 with step 0.001, unit is ˚A. Range of a we get is about in the range from four to twelve. The value generally used is 5 ˚A, and it is during the interval.

Figure 5.1: Classical electron density and electron density corrected by SP equations of double-gate MOSFET. And we treat result from SP equations as reference includes quantum effect.

52 Chapter 5 : Results and Discussion

Figure 5.2: Electron density corrected by Ferry’s effective potential with various standard deviation of wave packet of

double-gate MOSFET. We will choose value of a such that result is close to the result from SP equations.

5.2

Data Collection by Using Device Simulation Tool

Here, we set several different conditions to find the suit values of standard deviation of wave packet. Values of standard deviation of wave packet is the dependent variable. And we consider following independent variables: 1. channel length (Lg), 2. gate voltage (Vg), 3. drain voltage (Vd) , 4. thickness of bulk (tsi), 5. thickness of oxide (tox) , 6. doping concentration (N). Beside, every variable has limiting of range.

1. Channel length : 20, 30, 40, 50 (nm) [43] 2. Gate voltage : 0.6, 0.7, 0.8, 0.9, 1.0, 1.1 (V) 3. Drain voltage : 0.6, 0.7, 0.8, 0.9 (V), if Vg=0.6 0.7, 0.8, 0.9, 1.0 (V), if Vg= 0.7 V .. . 1.1, 1.2, 1.3, 1.4 (V), if Vg= 1.1 V (ensure device is in the saturation region) 4. Thickness of bulk : 8, 10, 12 (nm), if Lg= 20 nm

12, 15, 18 (nm), if Lg= 30 nm ..

.

20, 25, 30 (nm), if Lg= 50 nm

(only discuss devices whose ratio of Thickness of bulk over Channel length≈ 1/2

54 Chapter 5 : Results and Discussion

≈ 1/2 has better characters.)

5. Thickness of oxide : 1, 1.5, 2 (nm)

6. Doping concentration : 1e16, 5e16, 1e17, 5e17(/cm3)

Here, we only discuss symmetric double-gate MOSFET. Besides, we will set thickness of one oxide is equal to 200 nm with gate voltage 0 V. And we will treat the latter be SOI structure.

5.3

Modelling and Simulation Results

5.3.1

Modelling Double-Gate MOSFET

In this part, we only consider double-gate MOSFET. First, Table 5.1 shows the correla-tion between every independent variables. Because of our design of experiments, we can find that there is collinearity phenomenon among some variables. Figs. 5.3, 5.4, 5.5, 5.6, 5.7, 5.8 show the scattering plots : each independent variable against standard deviation of wave packet. Obviously, we can find that there are not only linear correlations but also quadratic correlations between dependent variable and some independent variables. So, we will add square terms into the model. Because there is collinearity phenomenon, effect of some variables may be weakened so that variables are not significant. Besides, we also add interaction terms to check whether there are interaction effects or not.