以幾何規劃方式求解矽鍺異質接面雙極性電晶體摻雜輪廓最佳化之研究

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ႝ඲ᡏᄞᚇ፺ᄂന٫ϯϐࣴز!

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Doping Profile Optimization of Silicon-Germanium

Heterojunction Bipolar Transistors via Geometric

Programming

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ࡰᏤ௲௤Ǻ׵ကܴ!௲௤!

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以幾何規劃方式求解矽鍺異質接面

雙極性電晶體摻雜輪廓最佳化之研究

Doping Profile Optimization of Silicon-Germanium Heterojunction

Bipolar Transistors via Geometric Programming

研究生：陳英傑 Student：Ying-Chieh Chen

指導教授：李義明博士 Advisor：Dr. Yiming Li

國立交通大學

電信工程研究所

碩士論文

A Thesis

Submitted to Institute of Communications Engineering

College of Electrical Engineering and Computer Engineering

National Chiao Tung University

in partial Fulfillment of the Requirements

for the Degree of

Master

in

Electrical Engineering

Augest 2010

Hsinchu, Taiwan

中華民國九十九年八月

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̼ʳCopyright by Ying-Chieh Chen 2010

All Rights Reserved

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а൳ՖೕჄБԄ؃ှޖⷿ౦፦ௗय़ᚈཱུ܄ႝ඲ᡏᄞᚇ፺ᄂന٫ϯϐࣴز

ᏢғǺഋमണ

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୯ҥҬ೯εᏢႝߞπำࣴز܌ᅺγ੤

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ᒿ๱ଯᓎႝၡᔈҔሡ؃ϐගϲǴޖⷿ(SiGe)౦፦ϟय़ᚈཱུ܄ႝ඲ᡏ(HBTs)

ޑᄒЗᏹբᓎ౗(Cut-off frequency)Ψ΋ޔόᘐޑ΢ϲǶԶᙖҗፓ᏾୷ཱུޑᄞᚇ

፺ᄂǴёаቚуޖⷿ౦፦ϟय़ᚈཱུ܄ႝ඲ᡏޑᏹբೲࡋǶฅԶǴ೛ी୷ཱུޑᄞ

ᚇ፺ᄂ۳۳ሡा࿶җ࿶ᡍݤ߾ϸᙟӦ჋၂Զ઻຤೚ӭਔ໔ᆶߎᒲԋҁǶ

൳ՖೕჄ(Geometric programming)ࣁ΋ᅿኧᏢޑന٫ϯୢᚒǴ߈ԃٰதத

೏ᔈҔӧࣽᏢᆶπำୢᚒǶᙖҗсڄኧᙯඤᆶჹଽۓ౛ǴаϷϣᗺݤᄽᆉБݤ

(Interior point method)ᆶႝတ߈൳ΜԃٰႝတीᆉૈΚޑගϲǴךॺёаِೲ

؃ှڀεೕኳᡂኧ(Optimal variable)ᆶज़ڋԄ(Constraints)ޑ൳ՖೕჄୢᚒǴ٠

Ъ؃ளӄୱന٫ှ(Global solution)Ƕ

ҁፕଞჹ୷ཱུޖⷿᄞᚇ፺ᄂ೛ीቪԋ΋൳ՖೕჄୢᚒǶ२Ӄஒޖⷿ౦፦ϟ

य़ᚈཱུ܄ႝ඲ᡏޑᄒЗᓎ౗ኧᏢኳࠠ௢Ꮴԋᆶޖⷿᄞᚇ፺ᄂԖᜢޑೱុᑈϩ

ڄኧǶௗ๱ǴஒԜೱុᑈϩڄኧ଺ᚆණ(Discretization)Ǵ٠ஒޖⷿᄞᚇ፺ᄂ߄

Ңࣁᆶ୷ཱུՏ࿼(Base region)Ԗᜢϐᚆණޑޜ໔ᡂኧǶ೸ၸа΢ޑ՗ीǴךॺ

ёаעᄒЗᓎ౗߄Ңԋ΋੝ਸޑڄኧ—҅ӭ໨Ԅ(Posynomial) ǴԶёஒԜߚጕ

܄ޑന٫ϯୢᚒᙯԋ΋൳ՖೕჄୢᚒǴ٠аϣᗺݤ؃ှǶӧόѨπำྗዴ܄ޑ

ा؃ΠǴԜБݤԖਏӦගٮΑזೲޑޖⷿᄞᚇ፺ᄂ๧ڗǶ

i

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ࣁΑᡍ᛾ന٫ϯኳࠠޑྗዴࡋǴךॺаΒᆢࡋ(2-D)ϡҹኳᔕᏔჹ౦፦ϟ

य़ᚈཱུ܄ႝ඲ᡏኳࠠύୖኧᆶϡҹ੝܄଺߃؁ਠ՗Ƕ่݀ᡉҢ 23%ᐚࡋޑఊ

ࠠ׎ⷿᄞᚇ፺ᄂёаᡣႝࢬቚ੻(Current gain)നεǴځॶऊࣁ 1100ǴК 0%ᐚ

ࡋ(໺಍ᚈཱུ܄ႝ඲ᡏ)ޑႝࢬቚ੻(ऊࣁ 200)εǹԶ 12.5%ޑⷿᐚࡋᆶΟف׎ޑ

ᄞᚇ፺ᄂёаᡣႝ඲ᡏၲډ 254GHz ޑᄒЗᏹբᓎ౗ǴΨගϲΑ໺಍ᚈཱུ܄ႝ

඲ᡏޑᄒЗᏹբᓎ౗(ऊࣁ 71GHz)Ƕ

ᕴϐǴҁࣴزςၮҔ൳ՖೕჄኧᏢБݤٰ؃ှޖⷿ౦፦ௗय़ᚈཱུ܄ႝ඲ᡏ

ന٫ᄞᚇ፺ᄂǶԜࣴزԖշܭႝတኳᔕᏔޑന٫ϯфૈ೛ीǴჹܭъᏤᡏεቷ

ϐמೌૈΚගϲԖ҅य़ϐշ੻Ƕ

ii

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Abstract

A

s the the need of high frequency circuits, the speed of silicon-germanium (SiGe) heterojunction bipolar transistors (HBTs) has been dramatically increased. It is

known that the speed of HBTs is dominated by the base transit time, which is influenced

by the doping profile in the base region and the Ge concentration. However, to design the

doping profile of HBTs requires lots of empirical experiences and time-consuming

try-and-error rounds.

Geometric programming (GP) is a type of mathematical optimization problem, recently

used in applied science and engineering widely. Based on the transformation of the

geo-metric programming into convex form and the duality theory, also benefited from the

inte-rior point method and nowadays computing power, we can solve geometric programming

problem with large scale optimal variables and constraints efficiently and globally.

In this study, the design of the doping profile and Ge-dose concentration for SiGe HBTs

are mathematically formulated and solved by the technique of geometric programming. At

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first, we derive the cut-off frequency model as an integral of Si doping profile and Ge-dose.

Then, then discretization of the integral function according to the base region, is applied to

obtain the discretized optimal variables of doping profile. Base upon the aforementioned

approximation, we could derive the cut-off frequency model as a posynomial function;

after that, the interior point method is employed to solve the well-formulated geometric

programming. This methodology provides an efficient mechanism to extract the Si doping

profile and Ge-dose.

The solution calculated by the GP method is guaranteed to be a global optimal. The

accuracy of the adopted numerical optimization technique is first confirmed by comparing

with a two-dimensional device simulation. The result of this study shows that a 23 % Ge

fraction have the maximum current gain, about 1100, which higher than the 0 % Ge fraction

(BJT), about 200. Furthermore, a 12.5% Ge may maximize the cut-off frequency for the

explored device, where a 254 GHz cut-off frequency is achieved, high than the 0 % Ge

fraction case, about 71 GHz.

In summary, we have successfully optimized the doping profile of SiGe HBTs using

GP method. The results of this study may benefit the technology computer-aided design

tool in semiconductor industry.

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ҁፕЎளаֹԋǴाགᖴ೚ӭΓǶ२ाགᖴৱৣ ׵ကܴ௲௤Οԃٰޑ܎

ܘᆶࡰᏤǴдόӢךࢂځдᏢଣрيǴϝฅᜫཀԏך଺дޑᏢғǴЪᡣךӳӳ

วචၸѐ܌ᏢϐБݤፕǴᔈҔӧπำሦୱ΢ǶԴৣவ΋໒ۈࡰۓךᜢᗖፕЎޑ

ᒧ᠐Ǵ᏾঺Бݤፕޑࡌҥ᏾ӝǵࣴزޑБӛǵჴሞޑᔈҔǵࣗԿ׫ዺޑЎӷǴ

Դৣ೿вಒ࣮ၸǶВதғࢲύǴԴৣΨࡐᜢЈךޑيᡏ଼நǵᏢಞރݩǵࣗԿ

࿶ᔮଆۚୢᚒǶԶԴৣ଺ࣴزϐᝄᙣᆒઓǵᑈཱུޑΓғᢀᆶೀ٣ᄊࡋǵ୺๱ᆶ

࡫ࠂǴคፕᡣךӧፐ཰ǵ஑཰מೌǵΓғᢀ΢Ǵ܌ᏢؼӭǶΟԃٰᆶԴৣޑර

δ࣬ೀǴᚂ΢౳࿊ǴᐕᐕӧҞǹԴৣӷӷ௲ᇧǴు૶ӧЈǴ΋قᜤᅰǶ

ךᗋाགᖴၮᆅسޑڑ૽ᄪ௲௤ǴεѤ౥஑ਔࡰᏤךǴ٠עך௢ᙚډႝᐒ

Ꮲଣௗڙ૽ግǹࡰᏤᏢғය໔ΨࡐᜢЈךޑᏢಞރݩǴΨ௲ᏤΑך೚ӭޑࣴز

БݤǶΨགᖴα၂ہ঩݅Ў଻Դৣᆶ׵ػ҇ԴৣаϷڬѴࣔԴৣǴӧݹ዗ޑϺ

਻ΠǴୖᆶ٠ࡰᏤᏢғα၂٠๏ᏢғፕЎࡌ᝼Ƕ

ࡑӧჴᡍ࠻ޑΟԃ໔ǴགᖴҬ೯εᏢႝηၗૻύЈѳՉᆶࣽᏢीᆉჴᡍ࠻

܌ගٮޑࣴزၗྍǴளаᡣᏢғԖ๤፾ޑᕉნᏢಞǶӧჴᡍ࠻ύǴགᖴς౥཰

ޑԿᗶᏢߏࡰᏤᏢғፕЎቪբᆶϡҹނ౛ǵᖴᖴᏢۊඁЎதத዗ЈᔅԆϷᜢЈ

ךǶᖴᖴӕืӳ϶ሎᗶ௲ךາϡҹኳᔕᏔϷ௲ךϡҹᢀۺǵ୯ᇶ௲ך٬Ҕႝၡ

ኳᔕᆶႝၡ౛ፕǵ྆๔ᆶइᏣࡰᏤךቪำԄǹΨགᖴᏢ׌ℱঅதഉך૸ፕന٫

ϯ୷ᘵ౛ፕᆶ୷ӢᄽᆉݤǴᏢۂӵᖚ዗ЈޑᔅԆךೀ౛ࣴزҬᒤ٣୍Ƕ

ךᗋाᖴᖴᆶך่ࡨ 13 ԃޑӳ϶߷ԀǴததӣ୷ໜਔǴջ٬ךѝԖ΋ᗺ

ᗺਔ໔ǴΨᜫཀഉך᚛ًѐ֌॥Ǵ෧጗ᓸΚǹᖴᖴ࠻϶ڷᑉǴၟך΋ଆՐΑΟ

ԃǴதத௲ךቪำԄǴᗋा৒೚ךԐᅵӃᅝᐩѺڥǹᖴᖴ࠻϶ோૈǴεᏢࣴز

܌ᇡ᛽ΑϤԃǴததၟך૸ፕфፐǴᆶךຑፕਔ٣ǹᖴᖴϤԃӳ϶܃࣑Ǵதத

ᖱઢ၉๏ך᠋ǴၟךಠϺӞ໭ǹᖴᖴଯύޑܻ϶Ǵ⍌൛ǵੇ҇ǵ໡๩ǴଷВத

தפךѐᆫᓓǴࠅᡏፊךததؒॅऊǹᖴᖴၮᆅسԳᏢߏۆǴᏢ׌ۂǴૈᡣך

ΑှډѺԳౚࢂӭሶז኷ޑ΋ҹ٣Ǵӧࣴزᎁၶ౟ᓍϐਔǴନΑ൨؃Դৣڐ

շǴᗋૈӧၮ୏ύόޕό᝺གྷډ؃ှБݤǹΨᖴᖴ݅а҅ᙴৣǴ೭΋ԃъٰǴ

ჹךੰ௃΋ޔ๏ךߞЈǴࣁΑךޑੰ௃੝ձᙌ᎙ђਜ؃ှǴჹךيᡏвಒፓ

ᎦǴᡣךϡ਻ᅌᅌࡠൺǴࣴز΢ωૈ߇ӣॉၰǶ

നࡕǴૈ୼΋ޔوډԜೀǴନΑа΢ፏՏܻ϶ǵӕᏢޑЍ࡭Ǵ੝ձགᖴ҆

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рǴςϷᔉ٣ޑۂۂӧך᠐ࣴز܌аٰ΋ޔჹךޑЍ࡭Ǵᡣךૈ஑ݙܭፐ཰ࣴ

زύǶӧԜᙣஒךޑᅺγፕЎǴ᝘๏܌ԖᜢЈǴႴᓰϷᔅշၸךޑΓǶ

ҁ ፕ Ў ག ᖴ ୯ ࣽ ཮ ी ฝ NSC-97-2221-E-009-154-MY2 ک

NSC-96-2221-E-009-210ϷڻऍႝηӀႝިҽԖज़Ϧљ 2008-20011 ࣴزीჄ

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List of Tables

3.1 The adopted parameters for the cut-off frequency model. The WB is the

base width, Gmax is the maximin value of Ge-content, CJ,BC is the base–

collector junction capacitance, RC is the collector resistance, q is the

elec-trical charge, ABE is the area of the base-emitter junction, k is the

Boltz-mann constant, T is the temperature (Kelvin), n_i0 is the intrinsic carrier

concentration in a undoped Si, N_minis the background doping concentra-tion, N_maxis the maximum doping concentration VBEis the applied voltage across the emitter-base junction, Vbiis the built-in potential voltage, εSiis

the permittivity of Si, and b is the constant. . . . 33

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xii LIST OF TABLES

3.2 The adopted parameters for the forward and base transit time model. The

WBCis the base–collector depletion width, vsatis the saturation velocity of

electrons, WE is the width of the emitter region, PEq,E is the equilibrium

concentration of holes in the emitter, γ is the ratio of the effective density

of states in SiGe to the effective density of states in silicon. The kSiGe, γ2

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List of Figures

1.1 (a) The circuit diagram of HBTs (BJTs). (b) The illusion of HBTs (BJTs).

(c) The designed doping profile. . . 5

1.2 A general classification of optimization problems. . . 8

2.1 Solution proceeding of geometric programming. . . 24

3.1 Illustration of the two-dimensional device structure of the explored SiGe

HBT. . . 39

3.2 The discretization and variables transformation of integral (3.15). . . 46

4.1 Optimized doping profile with and without gradient constraint of doping

profile, where the Ge-dose concentration is set to be zero. . . 58

4.2 The doping profile obtained from GP model and the 2D device simulation.

The doping profile of TCAD simulation is obtained by three different ion

implantation processes. . . 59

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xiv LIST OF FIGURES

4.3 Doping profile and the corresponding cut-off frequency with 2%, 8%, and

12.5% Ge-dose concentration. . . 61

4.4 The Ge profiles for HBTs with 2%, 8%, and 12.5% Ge-dose concentration. 62

4.5 Cut-off frequency with various Ge-dose concentrations. . . 63

4.6 Doping profile of decreasing background doping to 3× 1016cm−3for 3% Ge content. . . 64

4.7 Doping profile of Ge for different background doping concentration of Si. . 65

4.8 The cut-off frequency as a function of Ge-dose and background doping

concentrations. . . 66

4.9 The maximized current gain constraint can add for 0% to 23% Ge content. . 70

4.10 The maximum current gain constraint, which is added for every Ge content

and background doping to maintain sufficient cut-off frequency. . . 71

4.11 Optimal Si and Ge doping profile for cut-off frequency maximize and

max-imize current gain constraint. . . 72

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Chapter 1 Introduction

This chapter first briefs the history of Silicon-germanium (SiGe) heterojunction bipolar

transistors (HBTs) and the background of SiGe HBTs doping profile design. In the section

2, the history of optimization and the classification of optimization problems are discussed.

Finally, we present the motivation and introduce the study of this thesis.

1.1 History and Background of SiGe HBTs

In this section, we brief the history of SiGe HBTs, and discuss the background of doping

profile design of SiGe HBTs.

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2 Chapter 1 : Introduction

1.1.1 History of SiGe HBTs

SiGe technology is SiGe heterojunction bipolar transistors, has undergone substantial

development for nearly two decades. SiGe HBT structure was first proposed in 1987 [1].

In 1990, a SiGe HBTs with 75 GHz cut-off frequency is investigated; in the same year, the

circuit application using this SiGe HBT’s device were demonstrated [2-3].

The first SiGe BiCMOS circuit was demonstrated in 1992 [4] and the first large-scale

integrated circuit based on this topology was sequently reported in 1993 [5]; The 100 GHz

frequency response SiGe HBTs were demonstrated in 1993-1994 [6-8], and the first SiGe

HBT technology fabricated on 200-mm wafers were in 1994 [9]. During this ten years

development, various SiGe HBT technologies had been demonstrated based on different

SiGe epitaxial growth techniques [10-18], and in the 1994-1998, the practical digital and

microwave high frequency applications had been proposed based on the SiGe epitaxial

growth technologies [19-28], the detailed review paper about the aforementioned history

of HBTs could be found in [29].

Recently, the SiGe HBTs have demonstrated cut-off frequency higher than 200 GHz

[30-31], and the high frequency, great power performance and low noise amplifier

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1.1 : History and Background of SiGe HBTs 3

1.1.2 Background of SiGe HBTs Doping Profile Design

The basis of SiGe technology is HBTs, which exhibits various merits over conventional

Si bipolar junction transistors (BJTs) and silicon metal-oxide-semiconductor field effect

transistors (MOSFETs) for implementation of high-frequency circuits [50-51]. Fig. 1.1(a)

shows the circuit diagram of HBTs (BJTs), which is a three-terminal electronic device with

doped semiconductor material and could be used in amplify or switching circuits. The

structure of HBT (BJT) devices are shown in Fig. 1.1(b), composed by the emitter, base,

and collector regions; The charge flow in a HBT (BJT) is due to bidirectional diffusion of

charge carriers across a junction between two regions of different charge concentrations

(emitter and base, or base and collector). The operation speed of HBTs are mainly

domi-nated by the transit time of base region, which is strongly influenced by the doping profile

and Ge concentration as shown in Fig. 1.1(c) in the base region (If the base is doped only

one semiconductor material, the device is the so-called BJT) [35-52].

The determination of the doping profile and Ge concentration of the base region and

thus is crucial for optimal design of SiGe HBTs in advanced high frequency communication

circuits. Diverse engineering and theoretical approaches have been proposed to optimize

the base transient time through optimization of the base doping profile [36-47]. An

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of diffusion coefficient on base doping concentration was derived [37]. The analytical

ap-proach has been extended to consider the dependence of intrinsic carrier concentration on

base doping concentration [38]. An iterative approach has also been proposed to obtain the

optimum base doping profile [39], where the dependence of mobility and bandgap

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1.1 : History and Background of SiGe HBTs 5

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Figure 1.1: (a) The circuit diagram of HBTs (BJTs), which is a three-terminal electronic devices. (b) The illusion of HBTs (BJTs). The structure of HBT (BJT) devices are composed by the emitter, base, and collector regions; The charge flow in a BJT is due to bidirectional diffusion of charge carriers across a junction between two regions of different charge concentrations. (c) The designed Si and Ge doping profiles, which significantly influence the base transit time and sequentially raise the cut-off frequency and operate speed of HBT (BJT). If the base is doped only one semiconductor material, the device is the so-called BJT.

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1.2 History and Background of Optimization

In this section, we list several important issues in the history of optimization theory and

summary the classification of optimization problems.

1.2.1 History of Optimization

Optimization is the mathematical discipline to find the maxima and minima of functions,

possibly subject to constraints. The first optimization algorithms are presented in 19th

century. In 1826, J. B. J. Fourier formulated LP-problem for solving engineering problems,

twenty years later; A. L. Cauchy presents the gradient method to search the solution in the

minimum of functions. In 1947 G. Dantzig presented simplex method for solving

problems and in the same years, Von Neumann established the theory of duality for

LP-problems. In 1951, Karush-Kuhn-Tucker theorem (KKT theorem) was proposed. The

algorithms for unbounded optimization problems, such as quasi-Newton and conjugate

gradient methods, were developed in 1954. In 1960s, the geometric programming had

been known (Detailed introduction about geometric programming could be found in the

chapter 2). In 1980s, the computers became more efficient, heuristic algorithms (such as

genetic algorithm) for global optimization and large scale problems had greatly developed.

In 1990s, the theory of interior point methods was established, and the algorithm to solve

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1.2 : History and Background of Optimization 7

nowadays [55, 57, 58].

1.2.2 Classification of Optimization Problems

A general classification of optimization problems for practical applications are shown

in Fig. 1.2. The formulated optimization problems are first divided into two parts: model

dependent and independent. The model independent problems could be solved using

evolu-tionary algorithm, such as genetic algorithm [84]. The model dependent problems are

gen-eral using search algorithm based on mathematical theory. According to the characteristic

of the established model, the mathematical programming is classified as linear

program-ming and nonlinear programprogram-ming. If the nonlinear programprogram-ming satisfied the convexity,

we have the global solution, such as general convex programming and quadratic

program-ming (least square problems) [58]. The geometric programprogram-ming is one of the nonlinear

program, however, we could transformed it as an convex programming (Detailed

mathe-matical theory could be found in the section 2.3). Based on formulating (transforming)

the practical problems as the aforementioned optimization problems, the well-developed

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Figure 1.2: A general classification of optimization problems.The optimization problem could be mainly divided into two parts: model dependent (mathematical

programming)/independent. The evolutionary algorithm could be employed to solve model independent problems, such as genetic algorithm. The model dependent problems are general using search algorithm based on mathematical theory. The mathematical programming is classified as linear programming and nonlinear programming. If the nonlinear programming satisfied the convexity, the global solution could be obtained, such as general convex programming and quadratic programming. The applied geometric programming in this thesis is one of the nonlinear programming which could be transformed into convex programming and solved efficiently.

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1.3 : Motivation of this Thesis 9

1.3 Motivation of this Thesis

Due to the urgent demand of high-speed and large gain electron circuits, the GP approach is

advanced to pursue the optimal Ge-dose as well as the doping profile as shown in Fig. 1.1(c)

for the high cut-off frequency, or the high current gain in SiGe HBTs. In the previous work,

the doping profile design for bipolar-junction transistor to optimize cut-off frequency and

gain via GP had been proposed [36], however, the doping profile optimization for obtaining

the electrical specifications for SiGe HBTs is laked. As a result, we provide a method to

explore the SiGe HBTs doping profile optimization problem, and try to obtain higher

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1.4 Objectives

In this thesis, the design of HBTs is first expressed as a special form of an optimization

problem, the so-called GP, which can be transformed to a convex optimization problem,

and then solved efficiently. The background doping profile is adjustable to improve the

cut-off frequency and current gain. The result shows that a 23% Ge fraction may maximize

the current gain, where a factor, current gain divided by the emitter Gummel number, of

1100 is attained. Furthermore, to maximize the cut-off frequency of HBTs, a Ge-dose

con-centration of 12.5% is used, where the cut-off frequency can achieve 254 GHz. Note that

the accuracy of the developed optimization technique has been confirmed by comparing

it with that of a two-dimensional (2D) device simulation [84-88]. This study successfully

considers the device characteristics and manufacturing limitation as a GP model and the

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1.5 : Outline 11

1.5 Outline

The thesis is organized as follows. In chapter 2, we brief the history and background

of GP. In chapter 3, GP formulation of the design of HBTs and manufacturing limitation

are described. In chapter 4, the optimized cut-off frequency and current gain are discussed

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Chapter 2 The Geometric Programming

T

his chapter introduces the background of geometric programming. The content starts form giving the definition of the specifically types of functions of monomial

and posynomial and geometric programming in standard form. Section 2 presents the

con-vex problems converted from the geometric programming with many desirable properties

and have a duality theory with them. The modern interior-point method is also briefed

and the proceeding of solving the geometric programming is also discussed in this section.

After that, some available tools to solve geometric programming as well as GP in convex

form, and the sensitive/trade-off analysis are investigated. Finally, we list some practical

applications of geometric programming.

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2.1 : Background and History 13

2.1 Background and History

This section firstly introduces the background of geometric programming, and then its

his-tory and development is briefed.

2.1.1 Background of Geometric Programming

Geometric programming is one of the optimization approaches which is characterized

by objective and constraint functions with special forms, i.e., they are posynomial functions

of the optimal variables. The name of geometric programming was from the original

math-ematical theory made extensive use of arithmetic-geometric mean inequality between sums

and products of positive numbers. During the mathematical transformation, the geometric

programming can be easily cast as convex programming (CP) problems. There are several

advantages for the fact that GP can be reformulated as CP. For example, any starting point

can find the global solution if the formulated optimization problem is feasible, on the other

hand, if the problem is infeasible, a certificate proves infeasibility is found. For the real

world problem, the most important characteristic of the GP may be the recently developed

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14 Chapter 2 : The Geometric Programming

2.1.2 History of Geometric Programming

The geometric programming has been known since 1961, when Clarence Zener found

that many cost minimization problems in engineering had a special form [53]. At this same

time, a duality theory for nonlinear programming, and a mathematical framework of

ge-ometric programming based upon its duality theory is proposed [54]. In 1967 to 1970,

three books: Geometric Program [55], Engineering Designed by Geometric Programming

[56] and Applied Geometric Programming [57] discussed the theoretical and practical

as-pects of geometric programming and established the fundamental groundwork of

geomet-ric programming. In recent years, GP has been applied to solve the electgeomet-rical engineering

problems (see Sec.2.4).

2.2 The Terminology of Geometric Programming

This section starts from the definition of different functions related to geometric

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2.2 : The Terminology of Geometric Programming 15

2.2.1 Monomial Functions

Let f denote n real positive variables, and x = (x₁, . . . , xn) a vector composed by xi. A real valued function f of x, with the form:

f (x) = cxa1

1 xa22...xann, (2.1)

where c > 0 and aiare real numbers, is called a monomial function, or a monomial. Note

that exponents can be any real numbers, including fractional or negative which is different

with the standard definition from algebra in which the coefficients must be nonnegative

integers. We refer to the constant c as the coefficient of the monomial, and we refer to the

constants a₁, . . . , anas the exponents of the monomial. For example:

5.33x1.3 1 x−1.22

is a monomial of the variables x₁and x₂with coefficient 5.33 and the exponents are 1.3 and -1.2 for x₁and x₂. We list some composition rules for monomial:

1. any positive constant is a monomial, as is any variable;

2. monomials are closed under multiplication and division: if f and g are both monomials,

then so are f g and f /g; (This includes scaling by any positive constant.) and

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2.2.2 Posynomial Functions

A sum of one or more monomials, i.e., a function of the form:

g (X) =

k

k=1

ckxa₁1kxa₂2k...xannk, (2.2)

where ck > 0, is called a posynomial function or, more a posynomial with k terms of the

variables x₁, ..., xn. The term ‘posynomial’ is meant to suggest a combination of ‘positive’ and ‘polynomial’. We list some composition rules for monomial:

1. any monomial is also a posynomial;

2. posynomials are closed under addition, multiplication, and positive scaling;

3. posynomials can be divided by monomials: If g is a posynomial and f is a monomial,

then f /g is a posynomial; and

4. if γ is a nonnegative integer and f is a posynomial, then fγalways makes sense and is a

posynomial.

For example:

2x1.3

1 x−1.22 + 1.5x3+ x1x5.54

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2.2 : The Terminology of Geometric Programming 17

2.2.3 The Standard Form of Geometric Programming

A geometric program in standard form is an optimization problem of the form:

Min f₀(x)

s.t. fi(x) ≤ 1, i = 1, ..., m

gi(x) = 1, i = 1, ..., p

, (2.3)

where fiare posynomial functions, giare monomials, and xiare the optimization variables.

(There is an implicit constraint that the variables must be positive, i.e., xi> 0.) We defined

the problem (2.3) as a geometric programming in standard form, to distinguish it from

extensions we will describe later. In a standard form of GP:

1. the objective must be posynomial (and it must be minimized); and

2. the equality constraints can only have the form of a monomial equal to one, and the

inequality constraints can only have the form of a posynomial less than or equal to one.

For example, consider the problem:

Min x23₁ x3₂+ x6₁x5.2₃ s.t. _x2 1+ x52x13≤ 1 x₂+ x₃≤ 1 x₂x3₂= 1 ,

with variables x₁, x₂and x₃. This is a GP in standard form, with n = 3 variables, m = 2

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2.2.4 The Extension of Geometric programming

Several extensions are readily handled:

1. if f is a posynomial and g is a monomial, then the constraint f (x)≤ g(x) can be handled by expressing it as f (x)/g(x) ≤ 1 (since f/g is posynomial). This includes as a special case a constraint of the form f (x)≤ b, where f is posynomial and b > 0;

2. if g₁ and g₂ are both monomial functions, then we can handle the equality constraint

g₁(x) = g₂(x) by expressing it as g₁(x)/g₂(x) = 1 (since g₁/g₂is monomial); and

3. we can maximize a nonzero monomial objective function, by minimizing its reciprocal

(which is also a monomial).

As an example, consider the problem:

Min xz/y

s.t. _{2 ≤ z ≤ 5} x + y≤ z xz = y2

.

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2.3 : Solving the Geometric Programming 19 form of GP: Min (xz)−1y s.t. _z/5_{≤ 1} 2z−1 _{≤ 1} xz−1+ yz−1 ≤ 1 xzy−2= 1 .

2.3 Solving the Geometric Programming

This section shows how to solve the geometric programming. First we introduce the

ge-ometric programming in convex form, and the dual problem is discussed. Section 2.3.3

describes the interior-point methods to solve the prime and dual problem of geometric

pro-gramming in convex form. Section 2.3.5 shows the softwares or tools to solve the geometric

programming; finally, the trade-off and sensitivity analysis are introduced.

2.3.1 Geometric Programming in Convex Form

A geometric program can be reformulated as a convex optimization problem, i.e., the

problem of minimizing a convex function subject to convex inequality constraints and

lin-ear equality constraints. This is the key to our ability to globally and efficiently solve

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a posynomial f , we can further obtain:

h(y) = log[f (ey1_{, ..., e}yn)] = log(

t

k=1

eaTky+bk), _(2.4)

where aT_k = [ak₁, ..., ak_n] and bk = log ck. It can be shown that h is a convex function of the new variable y: for all y, z∈ Rnand 1≤ λ ≤ 1, we have:

h(λy + (1− λ)z) ≤ λh(y) + (1 − λ)h(z). (2.5)

Note that if the posynomial f is a monomial, then the transformed function is affine,

i.e., a linear function plus a constant. We can convert the standard geometric programming

(2.3) in 2.2.3 into a convex programming by expressing it as:

Min log f₀(ey1_{, ..., e}yn) = log(

k0 k=1 eaT0ky+b0k) s.t. log fi(ey1, ..., eyn) = log( ki k=1 eaTiky+bik) ≤ 0, i = 1, ..., m log gi(ey1, ..., eyn) = aTiy + bi= 0, i = m + 1, ..., m + p . (2.6)

This is the geometric programming in convex form. Convexity of the convex form

geo-metric programming (2.6) has several important implications: we can use efficient

interior-point methods to solve them, and there is a complete and useful duality, or sensitivity theory

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2.3 : Solving the Geometric Programming 21

2.3.2 Dual Problem of Geometric Programming in Convex Form

The dual problem of GP in convex form (2.6) can be written as [54-55],

Max bT₀v₀− k0 j=1 v_0,jlog v_0,j+ m i=1 (bT ivi− ki j=1

vi,jlog(vi,j/λi)) + p i=1 b_m+i,1ui s.t. v₀≥ 0, 1Tv₀= 1 vi≥ 0, 1Tvi= λi, i = 1, ..., m λi≥ 0, i = 1, ..., m m i=0A T ivi+ p i=1A T m+iui= 0 , (2.7)

where vi = (vi,1, ..., vi,ki)T, ci,jis the coefficient of the jthmonomial term of the ith

con-straint and AT_i = [a_i,1, ..., ai,ki] is a matrix ∈ Rki×n whose column vectors ai,j are the

exponents corresponding to the jthmonomial term of the ithconstraint. In this optimization problem, there are

m

i=0

ki+ p optimal dual variables. The variables vi,j are associated with the jthinequality constraint. The variables μk are associated with the kthconstraint. The

dual problem of GP in convex form has some advantages from a computational point of

view:

1. the concave object function whereas the object is maximized and the constraints are

linear; and

2. the dual problem of this type has a significantly impact on the computational methods

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2.3.3 Interior-Point Methods

The foundation for modern interior-point methods, are based on the barrier methods.

For solving the constrained nonlinear optimization problems, the penalty and barrier

meth-ods, which have a common motivation: finding an unconstrained minimizer of a composite

function that reflects the original objective function as well as the presence of constraints.

The interior-point methods are based on transforming constrained optimization to

uncon-strained optimization problem via logarithmic barrier function, is defined as:

B(x, μ)≡ f0(x) − μ

m+p

i=1

lnfi(x), (2.8) where μ is a positive scalar, called the barrier parameter. An important feature of B(x, μ)

is that it retains the smoothness properties of f₀(x) and fi(x) as long as fi(x) > 0. If

μ > 0 and μ→ 0 , the characteristic of B(x, μ) is like f0(x). Intuition then suggests that

minimizing B(x, μ) for a sequence of positive μ values converging to zero will cause the

unconstrained minimizers of B(x, μ) to converge to a local constrained minimizer of the

original problem [59-60].

2.3.4 The Procedure for Solving the Geometric Programming

In the solution procedure, we first formulate our problem as a geometric programming.

Then we could transform the GP into convex form, and the corresponding dual problem

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we apply the logarithmic barrier function transformation to convert the constrained

opti-mization problems into unconstrained ones. Finally, we could employ the general search

algorithm such as gradient-method and Newton-method to solve this unconstrained

opti-mization and the original solution could be inversely obtained. We give an algorithm as an

example: given a strictly feasible point y, set μ(n=0) > 0, β > 1, and error tolerance ε > 0.

1. Centering step (gradient method for unconstrained optimization):

Set k = 0 and error tolerance θ > 0.

1(a) Based on logarithmic barrier function (2.8), (2.6) could be transformed as:

Min φ(y) = log(

k0 k=1 eaT0ky+b0k) − μ m+p i=1 ln log( ki k=1 eaTiky+bik) .

1(b) Compute∇φ(y(k)), αk = arg min φ(y(k)-α∇φ(y(k))), α > 0. 1(c) Update: y(k+1)= y(k)+ αk∇φ(y(k)).

1(d) Stopping criterion: Quit ify(k+1)− y(k) ≤θ.

else

1(e) Back to step 1(a).

2. Update: y = y(μ(n+1)).

3. Stopping criterion: Quit ify(μ(n+1)) − y(μ(n)) ≤ε.

else

4. Decrease μ(n+1)= _β1μ(n). 5. Back to step 1.

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Figure 2.1: In the solution proceeding, we first formulate our problem as geometric programming. Then we transform the GP in convex form and also find its dual problems. After that, the logarithmic barrier function transformation of the prime and dual problem of GP in convex form is employed and this unconstrained optimization problem could be solved efficiently by general search algorithms.

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2.3.5 Tools and Softwares for Geometric Programming

The nonlinear optimization solver using efficient interior-point algorithms [59] was

de-veloped since 1994, including geometric programming. Recently, the primal-dual

interior-point methods are applied to solve the geometric programming [60]. The software: MINOS

[61], LOQO [62] or LANCELOT [63], is also possible to solve the convex form problem

with smooth objectives and constraints. These software could always obtain the global

op-timal solution based in the convex theory. In this work, we use the package ggplab [64] to

solve our doping profile optimization problem in GP’s form, and then we could obtain the

solution efficiently and robustly.

2.3.6 Trade-Off Analysis

Suppose the right-hand sides of constraints are modified in the geometric programming

(2.3) as follows:

Min f₀(x)

s.t. _f_i_{(x) ≤ u}_i_{, i = 1, ..., m} gi(x) = vi, i = 1, ..., p

. (2.9)

If all of uiand viare one, this modified geometric programming return to the original

one. If ui≤ 1, then the constraint fi(x) ≤ uiwith a tighter restriction than the original ith constraint; conversely if ui ≥ 1, it represents a loosening restriction of the constraint. For

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example, the change in the specification ui= 0.9 means that the ithconstraint is tightened

10%, whereas ui= 1.1 means that the ithconstraint is loosened 10%. Supposed the f₀∗(u, v) represent the optimal objective value of the modified geometric programming (2.9), as a

function of the parameters u = [u₁, u₂, ..., um] and v = [v1, v2, ..., vm], so the original objective value is f₀∗(1, 1). In trade-off analysis, we observe the variation of objective function f₀∗as a function of small u and v. Then the change of objective function respect to the variation of constraint can be expressed as:

Si= ∂f ∗ 0/f0∗ ∂ui/ui, Ti= ∂f₀∗/f₀∗ ∂vi/vi .

These sensitivity numbers are dimensionless, since they express fractional changes per

fractional change.

2.3.7 Sensitivity Analysis

Sensitivity analysis considers how small changes in the optimal variables affect the

optimal objective value. Supposed the f₀(x∗) represent the optimal objective value, we observe the variation of objective function f₀∗ as a function of small perturbation of x∗, then the change of objective function respect to the variation of optimal variable can be

expressed as:

Zi= ∂f ∗ 0/f0∗

∂xi/xi.

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solved, and has a small sensitivity, then small changes in the optimal variables won’t affect

the optimal value of the problem much. On the other hand, a solution set has a large

sensitivity is one that (for small changes) will greatly change the optimal value, and the

solution may be instable. Roughly speaking, an optimal value with a small sensitivity can

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2.4 Practical Applications of Geometric Programming

There are wide varieties of application of geometric programming ranging from civil

engi-neering to economics since 1960s:

1. civil engineering: optimal structural design [65], optimization of cofferdam problem

[66];

2. environmental engineering: optimal wastewater treatment plants design [67], water

quality management [68];

3. chemical engineering: Williamsotto process optimization [69], condenser designed

[70];

4. mechanical engineering: space trusses design [71], optimal helical springs design [72];

5. nuclear engineering: nuclear systems design [73];

6. economics: marketing-mix problem [74], EOQ inventory model [75]; and

7. electrical engineering: CMOS op-amp design [76-78], gate sizing in digital circuits

[79], LNA circuit parameters optimization [80-81] and temperature-aware floorplanning

[82].

From the listed applications, we can know that GP has many contributions on many

areas, although the GP is a very restrictive type of optimization problem. The detailed

references about the aforementioned background of GP in the section 2 could refer [58,

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Chapter 3 Problem Formulation and Solution

Method

I

n this chapter, we first formulate the optimal doping profile problem for SiGe HBTs, then the cut-off frequency model and GP formulation for the doping profile

optimiza-tion is discussed. In Secoptimiza-tion 3, we show how to solve the formulated GP problem and list

the corresponding implemented codes.

3.1 Problem Formulation

Figure 3.1 shows the studied SiGe HBTs device for the doping profile and Ge-dose

con-centration co-design, and also for a 2D device simulation. Mathematically, a doping profile

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30 Chapter 3 : Problem Formulation and Solution Method

tuning problem for the high frequency property optimization of SiGe HBTs can be

formu-lated as an optimization problem:

Max ft s.t. _N_min_{≤ N}_A_{(x) ≤ N}_max_{, 0}_{≤ x ≤ W}_B 0 ≤ G(x) ≤ Gmax, 0≤ x ≤ WB GeAV G = _W1 B W_B 0 G(x)dx NA(x) = bxm, 0≤ x ≤ 0.05WB , (3.1)

where ftis the cut-off frequency; NA(x) and G(x) are the base doping profiles for silicon and germanium, which are spatial-dependent positive functions over the interval 0≤ x ≤

WB and x is depth from the interface of base and emitter into substrate. The base doping

profile of silicon is lower than the doping level of emitter-base junction, N_max, and higher than background doping, N_min. The base doping profile of germanium is less than the maximum value Gmax, and GeAV G is the average value of Ge fraction, which can be a

given parameter ranging from 0 to 0.23 [36]. Assuming the manufacturing limitation, the

maximum value of Ge fraction should be less or equal to the solubility of Ge atoms in

silicon, such as 0.23 [36, 50-51]. In the present work, a peak base doping N_max of 1× 1019 cm−3 at emitter edge of base and a minimum base doping N_min of 5× 1016 cm−3 at collector edge of base have been chosen to include the heavy doping induced band gap

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3.1 : Problem Formulation 31

narrowing effect in the entire base region [36, 50-51]. WBis the base width of the transistor,

in which a neutral base width of 100 nm is chosen. And without loss of generality, we may

assume the doping profile to be the form [36]:

NA(x) = bxm, 0≤ x ≤ 0.05WB.

Here we assume m = 0 for a liner doping within 5% of the base width near the

emitter-base junction.

3.1.1 Cut-off Frequency Model

For a SiGe HBT, the cut-off frequency ftof a HBT is given by [50-51]:

1 2πft

= CJ,BE + CJ,BC

gm

+ RCCJ,BC + τF, (3.2) where CJ,BE is the base–emitter junction or depletion layer capacitance, CJ,BCis the base–

collector junction or depletion layer capacitance, gm is the transconductance, RC is the

collector resistance and τF is the forward transit time. The gmand CJ,BEin Eq. (3.2) could

also be expressed as a function of doping profile:

gm= q2ABEn2_i0 KT GB exp( qVBE kT ), (3.3) and CJ,BE = ABE qεSiNA(0) 2 (Vbi− VBE) 0.5 , (3.4)

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where VBEis the applied voltage across the emitter-base junction, Vbiis the built-in

poten-tial voltage, n_i0is the intrinsic carrier concentration in a undoped Si, εSiis the permittivity

of Si, ABE is the area of the base-emitter junction, k is the Boltzmann constant, and T is

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Table 3.1: The adopted parameters for the cut-off frequency model. The WBis the base width, Gmaxis the maximin value of Ge-content, CJ,BCis the base–collector junction capacitance,

RCis the collector resistance, q is the electrical charge, ABE is the area of the base-emitter junction, k is the Boltzmann constant, T is the temperature (Kelvin), n_i0is the intrinsic carrier concentration in a undoped Si, N_minis the

background doping concentration, N_maxis the maximum doping concentration VBE is the applied voltage across the emitter-base junction, Vbiis the built-in potential voltage, εSi is the permittivity of Si, and b is the constant.

Symbol Value WB 100 nM Gmax 0.25 b 1 CJ,BC 0.8× 10−12F RC 0.4 kΩ q 1.6× 10−19C ABE 0.25 μm−2 k 8.617× 10−5eV / K T 300o K n_i0 1.4× 1010cm−3 Nmin 5× 1016cm−3 Nmax 1× 1019cm−3 VBE 1 V Vbi 1.1 V εsi 1.04× 10−12F / cm2

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3.1.2 Forward Transit Time Model

The forward transit time in Eq. (3.2) are approximately composed by three components:

τF = τB+ τE+ τBC, (3.5) where τE is the emitter delay time and τBC is the base–collector depletion region transit

time. The τBCcould be expressed as:

τBC =

WBC

2vsat, (3.6)

where the base–collector depletion width WBC is determined by the collector doping

con-centration near the base–collector junction which we assume to be lower than the base

doping concentration, and vsat is the saturation velocity of electrons. The τE could be

expressed as: τE = (WE_2nPEq,E₂ i0 )( γ−1 1 + kSiGeGeAV G )GB, (3.7) where WEis the width of the emitter region, PEq,Eis the equilibrium concentration of holes

in the emitter, γ is the ratio of the effective density of states in SiGe to the effective density

of states in silicon and kSiGeare constants [36]. The GBis the base Gummel number, which

is also a function of NA(x) [50-51]: GB = WB 0 NA(x)n2_i0 Dn(x)n2i(x) dx, (3.8)

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where ni(x) is the intrinsic carrier concentration in Si and Dn(y) is the carrier diffusion coefficient of Si, and both could be express as the function of doping profile:

ni(x)2= n2_i0(NA(x) N ref) γ2_, _(3.9) and Dn(x) = Dn0 NA(x) N ref −γ1 , (3.10)

where Nref, D_n0and γ₂are constants [36]. Substituting Eqs. (3.9) and (3.10) into Eq. (3.8), we have: GB = 1 N refγ1−γ2_D_n0 WB 0 NA(x)1+γ1−γ2dx. (3.11)

3.1.3 Base Transit Time Model

The base transit time model in the optimization problem is given by [50-51], as shown

in below: τB= WB 0 n2_i,SiGe(x) NA(x) ( WB x NA(y)

n2_i,SiGe(y)Dn,SiGe(y)dy)dx, (3.12)

where ni,SiGe(x) is the intrinsic carrier concentration in SiGe and Dn,SiGe(y) is the carrier diffusion coefficient of SiGe. The x- and y- directions in Eq. (3.12) are indicated in Fig.

3.1. The ni,SiGe(x) and Dn,SiGe(y) depend on the profile of Si and Ge-dose [36, 50-51]:

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and

Dn,SiGe(y) = (1 + kSiGeGeAV G)Dn0

NA(x)

N ref

γ2

, (3.14)

substituting Eqs. (3.13) and (3.14) to Eq. (3.12), we have:

τB = 1 N refγ1_D_n0(1 + k_SiGe_Ge_{AV G}) WB 0 exp(μG(x))NA(x)γ2−1( WB x NA(y)1+γ1−γ2 exp(μG(y)) dy)dx. (3.15)

3.1.4 Cut-off Frequency Model as A Function of Doping Profile

Substitute GB of Eq. (3.11) into Eqs. (3.7) and (3.3), τBC of Eq. (3.6) and τE of Eq.

(3.7) into τF of Eq. (3.5), and τBof Eq. (3.15), as well as Eqs. (3.3) to (3.5) into cut-off

frequency model of Eq. (3.2), we have:

1 2πft = 1 Nrefγ1Dn0(1+kSiGeGeAV G) WB 0 exp(μG(x))NA(x)γ2−1( WB x NA(y)1+γ1−γ2 exp(μG(y)) dy)dx +(WEP Eq,E 2n2 i0 )( γ−1 1+kSiGeGeAV G)( 1 Nrefγ1−γ2Dn0 WB 0 NA(x)1+γ1−γ2dx) + kT εSiGe n2 i02q3(Vbi−VBE) 1/2 exp(−qV_BE kT ) × NA(0)1/2(Nrefγ1−γ21 Dn0 WB 0 NA(x)1+γ1−γ2dx)( γ −1 1+kSiGeGeAV G) + CJ,BCkT q0.5A_BEn2 i0 exp(− qV_BE kT )( γ−1 1+kSiGeGeAV G)( 1 Nrefγ1−γ2Dn0 WB 0 NA(x)1+γ1−γ2dx) +W_2vBC_sat + RCCJ,BC , (3.16)

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3.1.5 SiGe HBTs Doping Profile Nonlinear Optimization

After substituting the cut-off frequency model of Eq. (3.16) into the original HBTs

doping profile designed problem of Eq. (3.1), we will have:

Max ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 2π Nrefγ1Dn0(1+kSiGeGeAV G) WB 0 exp(μG(x))NA(x)γ2−1( WB x NA(y)1+γ1−γ2 exp(μG(y)) dy)dx +2π(WEP Eq,E 2n2 i0 )( γ−1 1+kSiGeGeAV G)( 1 Nrefγ1−γ2Dn0 WB 0 NA(x)1+γ1−γ2dx) +2π kT εSiGe n2 i02q3(Vbi−VBE) 1/2 exp(−qV_BE kT )NA(0)1/2(Nrefγ1−γ21 Dn0 WB 0 NA(x)1+γ1−γ2dx) ( γ−1 1+kSiGeGeAV G) + 2πCJ,BCkT q2ABEn2i0 exp(− qV_BE kT )( γ−1 1+kSiGeGeAV G) ( 1 Nrefγ1−γ2Dn0 WB 0 NA(x)1+γ1−γ2dx) +πW_v_satBC + RCCJ,BC ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ −1 s.t. _N_min_{≤ N}_A_{(x) ≤ N}_max_{, 0}_{≤ x ≤ W}_B 0 ≤ G(x) ≤ Gmax, 0≤ x ≤ WB GeAV G = _W1 B W_B 0 G(x)dx NA(x) = bxm, 0≤ x ≤ 0.05WB , (3.17)

where the objective function is composed by a two-dimension integral, and the Ge-dose is

in the exponential term, which is a nonlinear continues function and is hard to solve

us-ing the general optimization solver. For example, if we apply an evolutionary algorithm,

the doping profile function is hard to encode to solve; if using the nonlinear optimization

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result, the technique of geometric programming transformation is employed in the

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Figure 3.1: Illustration of the two-dimensional device structure of the explored SiGe HBT. The doping profile and Ge-dose concentration co-design, and also for a 2D device simulation are implemented in this specific structure.

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3.2 GP Formulation for SiGe HBTs Doping Profile

Opti-mization

In this section, we show how to formulate the general nonlinear SiGe HBTs doping profile

optimization problem (3.17) into GP’s form.

3.2.1 Taking Reciprocal for the Objective Function

For GP transformation, the nonlinear optimization problem (3.17) is formulated as:

Min _Nref_γ1_D 1 n0(1+kSiGeGeAV G) WB 0 exp(μG(x))NA(x)γ2−1( WB x N_A(y)1+γ1−γ2 exp(μG(y)) dy)dx +2π(WEP Eq,E 2n2 i0 )( γ−1 1+kSiGeGeAV G)( 1 Nrefγ1−γ2Dn0 WB 0 NA(x)1+γ1−γ2dx) + kT ε_SiGe n2 i02q3(Vbi−VBE) 1/2 exp(−qVBE kT )NA(0)1/2(Nrefγ1−γ21 Dn0 WB 0 NA(x)1+γ1−γ2dx)( γ −1 1+kSiGeGeAV G) +CJ,BCkT q2ABEn2i0 exp(− qV_BE kT )( γ−1 1+kSiGeGeAV G)( 1 Nrefγ1−γ2Dn0 WB 0 NA(x)1+γ1−γ2dx) +W_2vBC_sat +RCCJ,BC s.t. _N_min_{≤ N}_A_{(x) ≤ N}_max_{, 0}_{≤ x ≤ W}_B 0 ≤ G(x) ≤ Gmax, 0≤ x ≤ WB GeAV G = _W1_B WB 0 G(x)dx NA(x) = bxm, 0≤ x ≤ 0.05WB . (3.18)

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3.2 : GP Formulation for SiGe HBTs Doping Profile Optimization 41

The effectiveness of maximizing the ftis equal to minimizing the reciprocal of ft.

3.2.2 Discretizing the Continuous Doping Profile Function

In the base transit time model (3.15) of SiGe HBT, the doping profile is continuous.

To represent the doping profile as the discretized variables to be solved, the base region in

Eq. (3.15) is first discretized to M regions, xi = iWB/M, i = 0, 1, ..., M − 1, and the continuous doping profile functions NA(x) and G(x) can be transformed to NA(xi) and

G(xi), i = 0, 1, ..., M − 1, as shown in Fig. 3.2; τB = W 2 B M2N refγ1_D_n0(1 + k_SiGe_Ge_{AV G}) M−1 i=0 exp(uG(xi))NA(xi)γ2−1 M−1 j=i NA(xj)1+γ1−γ2 exp(uG(xj)) . (3.19)

Problem (3.19) is not a valid posynomial since it contains the optimal variables G(xi) in the exponential term. Fortunately, we can use the variable transformation as shown in

Fig. 3.2:

L(xi) = exp(G(xi)), i = 0, 1, ..., M − 1, (3.20) then Eq. (3.19) could be reexpressed as:

τB = W_B2 M2N refγ1_D_n0(1 + k_SiGe_Ge_{AV G}) M−1 i=0 L(xi)uNA(xi)γ2−1 M−1 j=i L(xj)−uNA(xj)1+γ1−γ2, (3.21)

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(GeAV G= _M1

M−1

i=0 G(xi)) is discretized and reformulated as:

1 ≤ L(xi) ≤ exp(Gmax), i = 0, 1, ..., M − 1, (3.22) and exp(MGeAV G) = exp _M−1 i=0 G(xi) =M−1 i=0 exp G(xi) = M−1 i=0 L(xi) , (3.23) respectively.

3.2.3 Derive the Summation Function of Doping Profile as Posynomial

For the summation of optimal variables of doping profiles Eq. (3.21) and GB of Eq.

(3.11), we define: Si= M−1 j=i L(xj)−uNA(xj)1+γ1−γ2, i = 0, 1, ..., M− 1 Wi= M−1 j=i L(xj) u_N A(xj)γ2−1Sj, i = 0, 1, ..., M − 1 bi= M−1 j=i NA(xj)1+γ1−γ2, i = 0, 1, ..., M− 1 , (3.24)

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and the above equations can also be expressed as backward recursions:

S_i+1+ L(xi)−uNA(xi)1+γ1−γ2 ≤ Si, i = 0, 1, ..., M − 2 W_i+1+ L(xi)uNA(xi)γ2−1Si≤ Wi, i = 0, 1, ..., M − 2 b_i+1+ NA(xi)1+γ1−γ2 ≤ bi, i = 0, 1, ..., M− 2 W_M−1−u NA(xM−1)1+γ1−γ2 = SM−1 W_M−1u NA(xM−1)γ2−1SM−1= WM−1 NA(xM−1)1+γ1−γ2 = bM−1 . (3.25)

During the above representation for the summation functions, we have the recursive

posynomial inequality constraints (for every constraints, the left-hand sides of the

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3.2.4 SiGe HBTs Doping Profile Optimization in GP’s Form

In problem (3.18), we express the summation of Eqs. (3.21) by (3.24) and (3.25),

replace the Ge-dose constraints by Eqs. (3.22) and (3.23) and then we have:

Min AW₀+ B₁NA(x0)0.5b0(1 + KSiGeGeAV G)−1+ B2b0(1 + KSiGeGeAV G)−1+ C

s.t. Nmin≤ NA(xi) ≤ Nmax, i = 0, 1, ..., M− 1 S_i+1+ L(xi)−uNA(xi)1+γ1−γ2≤ Si, i = 0, 1, ..., M − 2 W_i+1+ L(xi)uNA(xi)γ2−1Si≤ Wi, i = 0, 1, ..., M− 2 b_i+1+ NA(xi)1+γ1−γ2≤ bi, i = 0, 1, ..., M − 2 W_M−1−u NA(xM−1)1+γ1−γ2 = SM−1 W_M−1u NA(xM−1)γ2−1SM−1= WM−1 NA(xM−1)1+γ1−γ2 = bM−1 1 ≤ L(xi) ≤ exp(Gmax), i = 0, 1, ..., M − 1 NA(xi) = bxmi , i = 0, 1, ..., 0.05M exp(MGeAV G) = M−1 i=0 L(xi) , (3.26)

where A, B and C are collected doping profile independent constants. Note that NA(xi) is the discretized variables of doping profile in base region; i, ranging between zero and M -1,

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NA(xi), L(xi), Si, Wi and bi we can co-optimize the doping profile of Si and Ge in the base region for different given GeAV Granged from 0 to 0.23. Problem (3.26) is a GP since

the coefficients of objective function A, B and C are positive, and thus it is a posynomial

function; the left-hand sides of the inequalities are posynomials and the respect right-hand

sides are monomial functions; and the equality is a monomial equality. For justifying the

solution of the formulated GP of Eq. (3.26), we apply the variable transformation as section

2.3.1: NA(xi) = exp(αi), i = 0, 1, ..., M − 1 L(xi) = exp(βi), i = 0, 1, ..., M − 1 Si= exp(ηi), i = 0, 1, ..., M − 1 Wi= exp(πi), i = 0, 1, ..., M − 1 bi= exp(λi), i = 0, 1, ..., M − 1 , (3.27)

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Figure 3.2: The discretization and variables transformation of integral (3.15). The base region in (3.15) is first discretized to M regions, xi= iWB/M, i = 0, 1, ..., M − 1, and the

continuous doping profile functions NA(x) and G(x) can be transformed to NA(xi) and G(xi), i = 0, 1, ..., M − 1. Second we assume

L(xi) = exp(G(xi)), i = 0, 1, ..., M − 1, and then the discretized doing profile function could be obtained.