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Strain Effect on the Valence Band of Silicon: Piezoresistance in P-Type Silicon and Mobility Enhancement in Strained Sil...

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STRAIN EFFECTS ON THE VALENCE BA ND OF SILICON: PIEZORESISTANCE IN P-TYPE SILICON AND MOBILITY ENHANCEMENT IN STRAINED SILICON PMOSFET By KEHUEY WU A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005

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Copyright 2005 by Kehuey Wu

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iii TABLE OF CONTENTS Page LIST OF TABLES...............................................................................................................v LIST OF FIGURES...........................................................................................................vi ABSTRACT....................................................................................................................... ..x CHAPTER 1 INTRODUCTION...................................................................................................1 1.1 Relation of Strained Silicon CMOS to Piezoresistive Sensor and Piezoresistivity.............................................................................................1 1.2 Motivation....................................................................................................4 1.3 Valence Band Structure—A Preview..........................................................8 1.4 Focus and Organization of Dissertation.....................................................10 2 STRAIN EFFECTS ON THE VALENCE BAND AND PIEZORESISTANCE MODEL.................................................................................................................12 2.1 Introduction................................................................................................12 2.2 Review of Valence Band Theory a nd Explanations of Strain Effects on Valence Band.............................................................................................13 2.3 Modeling of Piezoresistance in p-Type Silicon.........................................23 2.3.1 Calculations of Hole Tr ansfer and Effective Mass........................23 2.3.2 Calculation of Relaxation Time and Quantization-Induced Band Splitting..........................................................................................30 2.4 Results and Discussion..............................................................................37 2.5 Summary....................................................................................................45 3 HOLE MOBILITY ENHANCEMENT IN BIAXIAL AND UNIAXIAL STRAINED-SILICON PMOSFET........................................................................49 3.1 Introduction................................................................................................49 3.2 Mobility Enhancement in Strained-Silicon PMOSFET.............................52 3.3 Discussion..................................................................................................58 3.4 Summary....................................................................................................60

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iv 4 WAFER BENDING EXPERIMENT AND MOBILITY ENHANCEMENT EXTRACTION ON STRAINED-SILICON PMOSFETS....................................61 4.1 Introduction................................................................................................61 4.2 Wafer Bending Experiments on pMOSFETs............................................62 4.2.1 Four-Point Bending for A pplying Uniaxial Stress.........................62 4.2.2 Concentric-Ring Bending for Applying Biaxial Stress.................67 4.2.3 Uncertainty Analysis......................................................................74 4.3 Extracting Threshold Voltage, Mobil ity, and Vertical Effective Field.....85 4.4 Summary....................................................................................................91 5 RESULTS AND DISCUSSIONS..........................................................................92 5.1 Mobility Enhancement and Coefficient versus Stress............................92 5.2 Discussion..................................................................................................99 5.2.1 Identifying the Main Factor C ontributing to the Stress-Induced Drain Current Change....................................................................99 5.2.2 Internal Stress in the Channel......................................................108 5.2.3 Stress-Induced Mobility Enhancement at High Temperature......108 5.2.4 Stress-Induced Gate Leakage Current Change............................116 5.3 Summary..................................................................................................118 6 SUMMARY, CONTRIBUTIONS, AND RECOMMENDATIONS FOR FUTURE WORK.................................................................................................119 6.1 Summary..................................................................................................119 6.2 Contributions............................................................................................120 6.3 Recommendations for Future Work.........................................................121 APPENDIX A STRESS-STRAIN RELATION...........................................................................124 B PIEZORESISTANCE COEFFICIEN T AND COORDINATE TRANSFORM.129 C UNCERTAINTY ANALYSIS............................................................................132 LIST OF REFERENCES.................................................................................................138 BIOGRAPHICAL SKETCH...........................................................................................146

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v LIST OF TABLES Table Page 1 Values of the inverse mass band paramete rs and deformation potentials used in the calculations.........................................................................................................29 2 Calculated zeroth-order longit udinal ( || ) and transverse ( ) stressed effective masses of the heavy and li ght holes (in units of m0) for [001], [111], and [110] directions..................................................................................................................30 3 Inand out-of-plane effective masses of the heavy and light holes for uniaxial compression and biaxial tension..............................................................................54 4 Stiffness cij, in units of 1011Pa, and compliance sij, in units of 10-11Pa-1, coefficients of silicon.............................................................................................128 5 Longitudinal and transverse coefficients for [001], [111], and [110] directions................................................................................................................131 6 Experimental data used in Fig. 36..........................................................................135 7 Mobility enhancement experimental data and uncertainty for uniaxial longitudinal stresses...............................................................................................136 8 Mobility enhancement experimental data and uncertainty for uniaxial transverse stresses....................................................................................................................137 9 Mobility enhancement experimental data and uncertainty for biaxial stresses......137

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vi LIST OF FIGURES Figure Page 1 Definitions of longitudinal and tr ansverse directions for defining coefficients......3 2 Schematic diagram of the biaxial strained-Si MOSFET on relaxed Si1-xGex layer.......................................................................................................................... ..5 3 Schematic diagram of the uniaxial stra ined-Si pMOSFET with the source and drain refilled with SiGe a nd physical gate length 45nm............................................6 4 Strain effect on the valence band of silicon...............................................................9 5 Ek diagram and constant energy surfaces of the heavyand light-hole and split-off bands near the band edge, k =0, for unstressed silicon...............................15 6 E-k diagram and constant energy surf aces of the heavyand light-hole and split-off bands near the band edge, k =0, for stressed silicon with a uniaxial compressive stress applied along [001] direction....................................................18 7 E-k diagram and constant energy surf aces of the heavyand light-hole and split-off bands near the band edge, k =0, for stressed silicon with a uniaxial compressive stress applied along [111] direction....................................................19 8 E-k diagram and constant energy surf aces of the heavyand light-hole and split-off bands near the band edge, k =0, for stressed silicon with a uniaxial compressive stress applied along [110] direction....................................................21 9 Top view (observed from [110] directio n) of the constant energy surfaces with a uniaxial compressive stress ap plied along [110] direction....................................22 10 Stress-induced band splitting vs. stress for [001], [111], and [110] directions........31 11 The scattering times due to acoustic and optical phonons and surface roughness scatterings.................................................................................................................36 12 Calculated effective masses of heavy a nd light holes vs. stress using 66 strain Hamiltonian..............................................................................................................39 13 Model-predicted longitudinal coefficient vs. stress for [001] direction................40

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vii 14 Model-predicted longitudinal coefficient vs. stress for [111] direction................41 15 Model-predicted longitudinal coefficient vs. stress for [110] direction................42 16 The energies of the heavyand light-hol e and split-off bands vs. stress for [001] direction calculated using 44 and 66 strain Hamiltonians...................................46 17 Effective masses vs. stress for the hea vy and light holes for [001] direction calculated using 44 and 6 6 strain Hamiltonians..................................................47 18 Illustration of uniaxial strained-Si pMOSFET.........................................................50 19 Illustration of biaxial strained-Si pMOSFET...........................................................51 20 In-plane effective masses of the heavy and light holes vs. stress for uniaxial compression and biaxial tension..............................................................................55 21 Band splitting vs. stress for uniaxia l compression and biaxial tension....................56 22 Model-predicted mobility enhancement vs. stress for uniaxial compression and biaxial tension..........................................................................................................57 23 The underlying mechanism of mobility enhancement in uniaxialand biaxialstrained pMOSFETs.................................................................................................59 24 Apparatus used to apply uniaxial stress and schematic of four-point bending........63 25 Stress at the center of the upper surface of the substrate vs. the deflection of the top pins.....................................................................................................................66 26 Stress vs. position and schematic of bending substrate............................................68 27 Apparatus used to apply biaxial stre ss and schematic of concentric-ring bending.....................................................................................................................69 28 Stress vs. displacement and schematic of bending plate..........................................71 29 Finite element analysis simulation of the bending plate (substrate)........................72 30 Uncertainty analysis of the starting po int for the four-point and concentric-ring bending experiments................................................................................................75 31 Uncertainty analysis of the misalignment of the substrate with respect to the pins for the four-point bending experiment..............................................................77 32 The experimental setup for calibrating the uniaxial stress in the four-point bending experiment..................................................................................................79

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viii 33 Extracted displacement vs. position curv es on the upper surface of the substrate in the four-point bending experiment.......................................................................80 34 Extracted radius of curv ature vs. position on the upper surface of the substrate in the four-point bending experiment.......................................................................82 35 Extracted uniaxial stress vs. position on the upper surface of the substrate in the four-point bending experiment.................................................................................83 36 Calibration of the four-poi nt bending experiment....................................................84 37 Uncertainty analysis for the con centric-ring bending experiment...........................86 38 Illustration of extracting threshold voltage..............................................................88 39 Effective hole mobility vs. effective field before and after bending with uniaxial longitudinal tensile and compre ssive stresses at 226MPa........................................93 40 Effective hole mobility vs. effective field before and after bending with uniaxial transverse tensile and comp ressive stresses at 113MPa...........................................94 41 Effective hole mobility vs. effective field before and after bending with biaxial tensile stress at 303MPa and co mpressive stress at 134MPa...................................95 42 Mobility enhancement vs. stress..............................................................................96 43 coefficient vs. stress..............................................................................................98 44 Average effective channel length ratio and variance < 2> vs. gate voltage shift for longitudinal stress..................................................................................102 45 Average effective channel length ratio and variance < 2> vs. gate voltage shift for transverse stress.....................................................................................103 46 Schematic diagram of doping concentra tion gradient and current flow pattern near the metallurgical junction between the source/drain and body......................105 47 Simulation result of the internal st ress distribution in a pMOSFET......................109 48 Mobility enhancement vs. stress at room temperature and 100C.........................111 49 Before and after bending drain current ID and gate current IG vs. gate voltage VG. The after bending gate current coin cides with the before bending one...........113 50 Before and after bending drain current ID and gate current IG vs. gate voltage VG. The after bending gate current is much higher than the before bending one..........................................................................................................................11 4

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ix 51 Before and after bending drain current ID and gate current IG vs. gate voltage VG. The after bending gate current is extr emely higher than the before bending one..........................................................................................................................11 5 52 Stress-induced gate leakage current change vs. stress...........................................117 53 Definitions of uniaxial stresses, x, y, and z, and shear stresses, xy, yx, xz, zx, yz, andzy........................................................................................................125 54 Definitions of strain................................................................................................126 55 Measurement errors in X........................................................................................133 56 Random and bias errors in gun shots.....................................................................134

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x Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy STRAIN EFFECT ON THE VALENCE BAND OF SILICON: PIEZORESISTANCE IN P-TYPE SILICON AND MOBILITY EN HANCEMENT IN STRAINED-SILICON PMOSFET By Kehuey Wu December 2005 Chair: Toshikazu Nishida Cochair: Scott E. Thompson Major Department: Electrical and Computer Engineering This dissertation explores strain effects on the valence band of silicon to explain and model piezoresistance effects in p-type silicon and mobility enhancements in strained-Si pMOSFETs. The strain effects are manifested as changes in the valence band when applying a stress, including band structur e alteration, heavy and light hole effective mass changes, band splitting, and hole repopulation. Using the 44 kp strain Hamiltonian, the stressed effective masses of the heavy and light holes, band splitting, and hole repopulation are used to analyti cally model the conductivity and effective mobility changes and the piezoresistance coefficients. The model predictions agree well with the experiments and other published works. Mobility enhancements and coefficients are extracted from four-point and concentric-ring wafer bending experiments used to apply external stresses to pMOSFET devices. The theoretical results show that the piezoresistance coefficient is stress-dependent in agreement with the measured

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xi coefficients. The analytical model predictions for mobility enhancements in uniaxial and biaxial strained-Si pMOSFETs are consistent with experiments as well as published experimental data and numerical simulations. In addition, for biaxial tensile stress, the model correctly predicts mobility degrad ation at low biaxial tensile stress. The main factor contributing to the stress-induced linear drain current increase is identified as mobility enhancement. The c ontribution from the change in effective channel length is shown to be negligible. The temperature dependence of stress-induced mobility enhancement is also considered in the model. At high temperature, the hole repopulation is smaller than at room temp erature, causing smaller mobility change whereas stress-induced band splitting suppre sses the interband op tical phonon scattering which reduces the mobility degradation.

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1 CHAPTER 1 INTRODUCTION Aggressive scaling of complementar y-metal-oxide-semiconductor (CMOS) technology has driven the performance improve ment of very larg e scale integrated (VLSI) circuits for years. However, as CM OS technology advances into the nanometer regime, scaling down the channel length of CM OS devices is becoming less effective for performance improvement mainly due to mobi lity degradation resu lting from the high channel doping density, and hence high vertical effective field, in the channel. StrainedSi CMOS provides a very promising approach for mobility enhancement and has been extensively investigated recently [1-14]. In addition, as CMOS technology advan ces into the deep submicron regime, process-induced stresses, for example, shallo w trench isolation [ 15], contact etch stop nitride layer [16], and source and drain silici de [17], etc., may affect device performance and reliability. 1.1 Relation of Strained Silicon CMOS to Pi ezoresistive Sensor and Piezoresistivity Silicon has been widely used in mechanical stress and pressure sensors for a long time due to its high sensitivity, good linearity and excellent mechanical properties [18, 19]. The strain effects responsible for the transduction physics of micromachined piezoresistive sensors is closely related to mobility enhancement in strained silicon CMOS. The strain effect on the valence ba nd of silicon can be used to explain and quantify the piezoresistance effect in p-type st rained silicon as well as the hole mobility

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2 enhancement in strained-Si pMOSFETs. Details of the strain-stress relation in a material with cubic symmetry such as silic on are discussed in Appendix A. The piezoresistance effect in strained silicon, first discovered by Smith [20] fifty years ago, is the stress-induced re sistance change. A coefficient, used to characterize the piezoresistance is defined as [18] 0 0 01 1 R R R R R ( 1 ) where 0R and R are the unstressed and stressed resistances and is the uniaxial stress. Since L A R / where is the resistivity, A is the cross-sectional area, and L is the length, the resistance change may be due to a combination of resistivity change and geometry change. However, in semiconducto rs, the contribution from the geometrical change may be neglected because it is 50 times smaller than the resistivity change [19]. Hence, the coefficient may be expressed in terms of resistivity change or conversely in terms of conductivity change as follows: 0 0 01 1 1 R R, ( 2 ) where and are the resi stivity and conductiv ity respectively. This conductivity change is directly related to mobility change since qp where q is the electronic charge, is the hole mobility, and p is the valence hole concen tration. Two types of uniaxial stresses are defined in Fig. 1 in order to disti nguish two kinds of coefficients, longitudinal and transverse [18]. Further de tails on the piezoresistance coefficient and coordinate transformations for an arbitr ary direction are give n in Appendix B.

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3 Figure 1. Definitions of longitudinal a nd transverse directions for defining coefficients (adapted from Ref. [18]). The strain-str ess relation is discussed in more detail in Appendix A.

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4 Longitudinal means that the uniaxial stress, elec tric field, and electri c current are all in the same direction. Transverse means that the electric field is parallel to the electric current but normal to the uniaxial stress. Recently, biaxial strained-Si pMOSFETs using a thick relaxed Si1-xGex layer to stretch the Si channel has been studied extensively because hi gh biaxial tensile stress can increase the hole mobility [1-8]. Figure 2 [9] is a schematic diagram of the biaxial strained-Si MOSFET using relaxed Si1-xGex. Alternatively, applying uniaxial compressive stress along the channel can al so enhance the hole mobility [10-12]. Two approaches have been used to apply uniax ial compressive stress; one method employs source and drain refilled with SiGe [10-12], another uses highly compressive stress SiN layer [13]. Figure 3 [14] show s a pMOSFET with the source a nd drain refilled with SiGe and a physical gate length of 45nm. Although both biaxial tensile and uniaxial compressive stresses can improve the hole mobil ity, the efficacies of the two stresses are different. For uniaxial compressive stress, about 50% hole mobility enhancement can be achieved with about 500MPa [11, 12]; however, the biaxial tensile st ress needs more than 1GPa to be able to increase the hole mobility In fact, at low biaxial tensile stress, the hole mobility is actually degraded [5, 8], which is contradictory to the theoretical prediction made by Oberhuber et al. [3]. 1.2 Motivation The piezoresistance effect in n-type strain ed silicon has been well explained by the many-valley model [21], while, in p-type strained silicon, the piezoresistance effect has not yet been fully understood and character ized; most of the previous theoretical

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5 Figure 2. Schematic diagram of the bi axial strained-Si MOSFET on relaxed Si1-xGex layer (adapted from Ref. [9]).

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6 Figure 3. Schematic diagram of the uniaxial strained-Si pMOSFET with the source and drain refilled with SiGe and physical gate length 45nm (adapted from Ref. [14]).

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7 works [19, 22, 23] only show th e longitudinal and transverse coefficients along [111] direction. There is also a need for extended measurem ent of the piezoresistance coefficients in silicon at higher stresses. The original data by Smith on piezoresistance of bulk silicon were obtained with applied stresses of 106 N/m2 (1 MPa) to 107 N/m2 (10 MPa) using loads attached to the free end of a clamped crystalline silicon or germanium sample with electrodes attached on parallel or perpendicular faces of the sample [20]. However, much higher fixed stresses on the order of 100 to 1000 MPa in the surf ace is required to significantly improve the performance of tr ansistor devices (n-channel metal-oxidesemiconductor field-effect transistors (nMOSFET) and p-channel metal-oxidesemiconductor field-effect transi stors (pMOSFET)) [5, 24, 25]. There is a need for a better fundamental understanding of the eff ect of these high stresses and corresponding st rains on the carrier transpor t properties in advanced MOSFETs and the influence of quantum conf inement in nanostructures as well as temperature. Since the pi ezoresistance effect is precise ly the conductivity enhancement obtained in strained-Si CMOS, understanding strain effects in semiconductors is vital for continuing performance enhancement in advanced CMOS technologies. An accurate model is needed to estimate the impact of process-induced stresses on the device performance. Such a model can also be us ed in process and device simulations to estimate the overall impact from vari ous process-induced stresses. Recently, due to the advance of silic on IC technology, the mass production of high precision sensors and the in tegration of mechanical sens ors and electronic circuits (system on chip SOC) are now possible [18]. To design high precision sensors or SOC, a

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8 more accurate model and better understanding of the piezoresistance effect in silicon are needed. 1.3 Valence Band Structure—A Preview The underlying physics of piezoresistance in p-type silicon and mobility enhancement in strained-Si pMOSFETs can be explained by the strain effect on the valence band of silicon. The valence band of silicon consists of three bands, heavyand light-hole and split-off bands. The heavyand light-hole bands are dege nerate at the band edge and the split-off band is 44meV below the band edge [26, 27] Applying stress to silicon will lift the degeneracy and alter the valence band structure. As a result, the effective masses of the heavy and light holes will change and holes will repopulate between the heavyand light-hole bands. To the first order approximation, the contribution from the split-off band can be ne glected [27-30] since it is 44meV below the band edge. These stress-induced changes in the valence band are collectively called the strain effect. Figure 4 is the illustration of the strain effects in silicon. Both the unstressed and stressed silicon (with a uniaxial stress ap plied along the [111] di rection) are shown in the figure. For unstressed silicon on th e left hand side of the figure, the E-k diagram shows the degeneracy of the heavyand light -hole bands at the ba nd edge and the splitoff band is 44meV below the band edge. The energy surfaces of three hole bands are shown next to the E-k diagram. On the right hand side of the figure are the E-k diagram and the constant energy surfaces of the three hole bands for stressed silicon, assuming a uniaxial compressive stress is applied along the [111] direction. As can be seen from the E-k diagram, the degeneracy at the band edge is lifted and the light-hole band rises above the heavy-hole band with a band splitting E, resulting in hole repopulation from the

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9 Figure 4. Strain effect on the valence band of silicon. On the left hand side are the E-k diagram and the constant energy surfaces of the heavyand light-hole and split-off bands fo r unstressed silicon. The degeneracy of the heavyand light-hole bands at the band edge is also shown. The E-k diagram and the constant energy surfaces of the three hole bands are shown on the right hand side. The degeneracy is lifted and the light-hole ba nd rises above the heavy-hole band with a band splitting E, causing the hole repopulation from the heavyto light-hole bands. The shapes of the constant energy surf aces of the heavyand light-hole bands are altered and effective masses of the heavy and light holes are also changed.

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10 heavyto light-hole bands. In the meantime, the band structures or the shapes of the constant energy surfaces of the heavya nd light-hole bands are altered, causing the changes in the effective masses of the heavy and light holes. Detail ed discussion of the strain effects will be presented in Chapter 2. 1.4 Focus and Organization of Dissertation This dissertation will mainly focus on creat ing a simple, analytical model that can be easily understood and provide quick, accurate predictions for the piezoresistance in ptype silicon and mobility enhancement in st rained-Si pMOSFETs. Most of the theoretical works on strain effects on metal-oxide-semi conductor field-effect transistors (MOSFETs) have employed pseudo-potential [2, 31] or kp full band numerical simulations [3, 7]. A simple, analytical model can provide a quick check for numerical simulations as well as provide physical insight. We develop a simple, analytic model using 44 kp strain Hamiltonian [27, 28, 32]. The split-off band will be neglected [27-30] bu t the influence from it is added into the model as a correction. In Chapter 2, the strain effects on the valence band of silicon will be explained and quantified using 44 strain Hamiltonian. The stressed effective masses of the heavy and light holes, band splitting be tween the heavyand light-hole bands, and the amount of hole repopulation will be calculated. Then, th e longitudinal and transverse coefficients along three major crystal axes, [001], [111], and [110] directions will be calculated. The result of the strain effects cal culated using 44 strain Hamiltonian will be compared with the result using 66 Hamiltonian to estimate the valid stress range that the model is applicable. Chapter 3 is the calculation and comparison of mobility enhancements in uniaxialand biaxial-stra ined Si pMOSFETs usi ng the results of the

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11 strain effects developed in Chapter 2. Th e reason for the mobility degradation at low biaxial tensile stress will be explained in de tail. In Chapter 4, experiments designed to test the validity of the models are presente d. The four-point and concentric-ring wafer bending experiments are used to apply the unia xial and biaxial stresses respectively. The approaches used to extract the threshold volta ge, mobility, and vertical effective field are described. In Chapter 5, the hole mobility and the mobility enhancement will be extracted from the drain current in the linear region. The mobility enhancement vs. stress will be plotted and compared with the model. The coefficients will then be calculated from the mobility enhancement vs. stress. The main factor contributing to the linear drain current increase will be identified from analyzing the variables in the linear drain current equation. The internal channe l stress will be estimated using the process simulator FLOOPS-ISE. The mobility enhancement at high temperature will be discussed and compared with the mobility enhancement at room temperature. And, finally, Chapter 6 is the summary and the recommendation for future work.

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12 CHAPTER 2 STRAIN EFFECTS ON THE VALENCE BAND AND PIEZORESISTANCE MODEL 2.1 Introduction In this chapter, the strain effects on the valence band will be explained in detail. The valence band theory will be reviewed fi rst. The equations of the constant energy surfaces, effective masses, band splitting, and hole repopulation, which are derived from Kleiner-Roth 44 strain Hamiltonian [27, 32], will be presented and explained. The equations will then be used to model the piezoresistance in ptype silicon. The piezoresistance in nand p-type si licon was discovered by Smith 50 years ago [20]. The n-type silicon piezoresistance can be well explained by th e many-valley model [21, 23]. Recently, piezoresistance in p-type silicon has been modeled in terms of stressinduced conductivity change due to two key mechanisms [19, 22, 23]: (i) the difference in the stressed effective masses of the hea vy and light holes and (ii) hole repopulation between the heavyand light -hole bands due to the st ress-induced band splitting. However, previous works [19, 22, 23] only focus on the piezoresistance along the [111] direction. In this chapter, we will extend the previous works to model the piezoresistance along three major crystal axes, [001] [110], and [111] directions. In section 2.2, we review the valence band theory and explain th e strain effect on the valence band using KleinerRoth 44 strain Hamiltonian [27, 32]. The valence band structure, band splitting, a nd hole repopulation between the heavyand light-hole bands will be explained. In section 2.3, a model of piezoresistance in p-type silicon is presented. The longitudinal and transver se conductivity eff ective masses of the heavy and light

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13 holes, the magnitude of the ba nd splitting and hole repopulati on, and the corrections due to the influence of the split-off band are ca lculated. The relaxation times due to the acoustic and optical phonon scatterings are cal culated. Since the model is employed to estimate mobility enhancement on pMOSFETs, surface roughness scatte ring is also taken into account. In addition, the quantization effect in the i nversion layer of a pMOSFET is also considered. Section 2.4 is the result a nd discussion. Comparisons of the model with the previous works will be made. The valid st ress range in which the model is applicable is estimated by comparing the strain effects calculated using the 44 with the 66 strain Hamiltonians, which includes the split-off band. Finally, section 2.5 is the summary. 2.2 Review of Valence Band Theory and Expl anations of Strain Effects on Valence Band Single-crystal silicon is a cubi c crystal. Without strain or spin-orbit inte raction, the valence band at the band edge, k=0, is a sixfold degenerate p multiplet due to cubic symmetry [27]. The sixfold p multiplet is composed of three bands, and each band is twofold degenerate due to spin. The spin-orbit interaction lift s the degeneracy at the band edge, and the sixfold p multiplet is decomposed into a fourfold 2 3p multiplet, J=3/2 state, and a twofold 2 1p multiplet, J=1/2 state, with splitting energy =44meV between the two p multiplets [26, 27]. The 2 3p state consists of two twofold degenerate bands designated as heavyand light-hole band. The 2 1p state is a twofold degenerate band called spin-orbit split-off band. Near the band edge, k=0, the constant energy surface for the 2 3p state can be determined by kp perturbation [27, 33], approximated as

PAGE 25

14 ) ( ) (2 2 2 2 2 2 2 4 2 2x z z y y xk k k k k k C k B Ak k E ( 3 ) and the 2 1p state is given by 2) (Ak k E, ( 4 ) where A, B, and C are the inverse mass band parameters determined by cyclotron resonance experiments [27, 33, 34]. The upper and lower signs in Eq. (3) represent the heavyand light-hole band resp ectively. Figure 5 shows the E-k diagram and the constant energy surfaces of the heavyand light-hole and split-off bands near the band edge for unstressed silicon. As seen from the E-k diagram in Fig. 5, th e heavyand light-hole bands are degenerate at the band edge and th e split-off band is below the band edge with a splitting energy =44meV. Also seen from Fig. 5, th e constant energy surfaces of the heavyand light-hole bands are distorted, usua lly called “warped” or fluted,” due to the coupling between them. As for the split-off ba nd, it is decoupled from the heavyand light-hole bands and has a spheri cal constant energy surface. Applying stress to silicon will break the cubic symmetry and lift the degeneracy of the fourfold 2 3p multiplet at the band edge [27, 28]. If a uniaxial stress is applied along an axis with higher rotational symmetry, fo r example, [001] dir ection with four-fold rotational symmetry or [111] direction w ith three-fold rota tional symmetry, the 2 3p state will be decoupled into two ellipsoids. For the [001] direction, the constant energy surfaces of the heavyand light-hole bands become [27, 28] 0 2 || 2 2 3) ( ) 2 1 ( ) ( k B A k B A k E, ( 5 )

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15 Figure 5. E-k diagram and constant energy surfaces of the heavyand light-hole and split-off bands near the band edge, k=0, for unstressed silicon. The heavyand light-hole bands are degenerate at the band edge, as shown in the E-k diagram. The constant energy surfaces of the heavyand light-hole bands are distorted or warped. The split-off band is 44meV below the band edge and has a spherical constant energy surface.

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16 0 2 || 2 2 1) ( ) 2 1 ( ) ( k B A k B A k E ( 6 ) where ) (2 3k E and ) (2 1k E are the heavyand light-hole bands respectively, 2 2 2 y xk k k 2 2 || zk k The band splitting 001E between the heavyand light-hole bands is expressed as ) ( 3 2 2 212 11 0 001s s D Eu ( 7 ) where 0 is the energy shift for the heavyand light-hole bands for the [001] direction, is the uniaxial stress, uD is the valence band defo rmation potentials for [001] direction, 11s and 12s are compliance coefficients of si licon. The definition of compliance coefficients is given in Appe ndix A. Along the [111] directi on, the heavyand light-hole bands become [27, 28] 0 2 || 2 2 3) 3 1 ( ) 6 1 ( ) ( k N A k N A k E, ( 8 ) 0 2 || 2 2 1) 3 1 ( ) 6 1 ( ) ( k N A k N A k E, ( 9 ) where 2 2 2 1 2k k k 2 3 2 ||k k k1 and k2 are along the ] 0 1 1 [ and ] 2 11 [ directions respectively, k3 is along the [111] direction, and 2 2 23 9 C B N is an inverse mass band parameter. The band splitting 111E between the heavyand lig ht-hole bands is expressed as 2 3 2 2 244 ' 0 111s D Eu, ( 10 )

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17 where 0 is the energy shift for the heavyand light-hole bands for the [111] direction, 44s is a compliance coefficient of silicon, 'uD is the valence band deformation potential for the [111] direction. Fi gures 6 and 7 show the E-k diagrams and constant energy surfaces of the heavyand light-hole and splitoff bands for stressed silicon with uniaxial compressive stresses applied along the [001] an d [111] directions respectively. As seen from Figs. 6 and 7, the degeneracy of the heavyand light-hole bands is lifted with a band splitting energy E and the heavyand light-hole ba nds become prolate and oblate ellipsoids respectively with ax ial symmetry about the stress direction [27, 28]. When a uniaxial compressive stress is applied, the li ght-hole band will rise above the heavy-hole band and holes will transfer from the heavyto light-hole bands because the energy of the light-hole band is lower than the heavy-hole band and vice versa. Note that the energy axis E of the E-k diagrams represents the electron en ergy. The hole energy is the negative of the electron energy and in the opposite direction of the electron energy axis. Therefore, by valence band “rising” or “falling” it means that the hole energy in the valence band is decreasing or increasing respectively. When applying a uniaxial stress along the tw o-fold rotational symmetry axis, [110] direction, the situation is more complex. Th e energy surfaces of the heavyand light-hole bands still are ellipsoids yet have three unequal principal ax es and the constant energy surface of the heavyand light hole band become [27, 28] 2 1 2 0 2 0 2 3 3 2 2 2 2 2 2 1 1 23 2 1 2 2 2 ) ( k m k m k m k E ( 11 ) where

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18 Figure 6. E-k diagram and constant energy surf aces of the heavya nd light-hole and splitoff bands near the band edge, k=0, for stressed silicon with a uniaxial compressive stress applied along [001] direction. The degeneracy of the heavyand light-hole bands is li fted with a band splitting energy E and the light-hole band rise above the heavyhole band. The heavyand light-hole bands become prolate and oblate ellipsoids respectiv ely with axial symmetry about the stress direction.

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19 Figure 7. E-k diagram and constant energy surf aces of the heavya nd light-hole and splitoff bands near the band edge, k=0, for stressed silicon with a uniaxial compressive stress applied along [111] direction. The degeneracy of the heavyand light-hole bands is li fted with a band splitting energy E and the light-hole band rise above the heavyhole band. The heavyand light-hole bands become prolate and oblate ellipsoids respectiv ely with axial symmetry about the stress direction.

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20 2 1 1 22 2 2 N B A m ( 12 ) 1 2 22B A m ( 13 ) 2 1 3 22 2 2 N B A m ( 14 ) and 2 1 2 13 1 1 and 2 1 2 2 23 1 ( 15 ) with 0 0 The upper signs in Eqs. (11) – (1 4) belong to the heavy-hole band E3/2(k) and the lower signs belong to the light-hole band E1/2(k). In Eq. (11), k3 is the longitudinal direction along the [110] direction; k1 and k2 are two transverse directions along the ] 0 1 1 [ and [001] directions resp ectively. The band splitting 110E between the heavyand light-hole bands is expressed as 2 1 2 0 2 0 1103 E. ( 16 ) In Eq. (11), 2 32 1 2 0 2 0 is the energy shift for the h eavyand light-hol e bands for the [110] direction [27]. Figure 8 shows the E-k diagram and constant energy surfaces of the heavyand light-hole and split-off bands for stressed silicon with a uniaxial compressive stress applied along the [110] direction. Fi gure 9(a) and (b) are the top views of the constant energy surfaces of the lightand heav y-hole bands respectively as observed from the [110] direction. As can be seen from Figs. 8 and 9, the constant energy surfaces of the

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21 Figure 8. E-k diagram and constant energy surf aces of the heavya nd light-hole and splitoff bands near the band edge, k=0, for stressed silicon with a uniaxial compressive stress applied along [110] direction. The degeneracy of the heavyand light-hole bands is li fted with a band splitting energy E and the light-hole band rise above the heavyhole band. The heavyand light-hole bands still are ellipsoids yet ha ve three unequal principal axes.

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22 Figure 9. Top view (observed fr om [110] direction) of the co nstant energy surfaces with a uniaxial compressive stress applied along [110] direction. (a) Light-hole band. (b) Heavy-hole band.

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23 heavyand light-hole bands are ellipsoids with three unequal principal axes and the lighthole band rises above the heavy-hole band. 2.3 Modeling of Piezoresistance in p-Type Silicon 2.3.1 Calculations of Hole Transfer and Effective Mass From Eq. (2) and the illustrations in Fi g. 1, the longitudinal and transverse coefficients can be defined as [23] 0 0 0 01 1 1 l l l l, ( 17 ) 0 0 0 01 1 1 t t t t, ( 18 ) where l and t are the longitudinal and transverse uniaxial stresses respectively, and 0 are the stressed and unstressed conductivity re spectively, and eff lh lh hh hhqp m p m p q ) (2, ( 19 ) where eff is the effective carrier mobility, q is the electron charge, is the hole relaxation time, hhm and lhm are the heavy and light hol e conductivity effective mass respectively. For silicon, the resistance change due to the geometrical change is 50 times smaller than the resistivity change [19], theref ore, in Eqs. (17) a nd (18), the contribution from the geometrical change is neglected. Usin g Eq. (19), Eqs. (17) and (18) can then be expressed as eff eff eff l eff l l l l 1 1 1 10 0 0, ( 20 )

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24 eff eff eff t eff t t t t 1 1 1 10 0 0, ( 21 ) where eff and eff are the stressed and unstressed effective mobility, respectively. In Eq. (19), lh hhp p p is the total hole concentration, hhp and lhp are the heavy and light hole concentrations respectively. In order to simplify the model, the contribution from the split-off band is neglected [27-30] because the split-off band is 44meV below the valence band edge [35]. For a non-degenerate p-type silicon, hhp and lhp are given by [36], ) exp( ) exp( ) 2 ( 22 3 2 *T k E E N T k E E h T k m pB v F vh B v F B hh hh ( 22 ) ) exp( ) exp( ) 2 ( 22 3 2 *T k E E N T k E E h T k m pB v F vl B v F B lh lh ( 23 ) where 0 *49 0m mhhand 0 *16 0m mlh [37, 38] are the densityof-state effective mass of the heavy and light hole respectively, 0m is the free electron mass, Bk is the Boltzmann constant, T is the absolute temperature, FE is the Fermi level, and vE is the energy at the valence band edge. For the unstressed case, th e valence band is dege nerate at the band edge, k=0, the heavy-hole band energy Evh and the light-hole band energy Evl are equal and Evh = Evl = Ev. Using Eqs. (22) and (23), the heavy and light hole concentrations can then be calculated from the doping density p [23]: p m m m plh hh hh hh2 3 2 3 2 3 and p m m m plh hh lh lh2 3 2 3 2 3 ( 24 )

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25 When stress is applied, energy splitting of the heavyand light-hole bands occur and holes repopulate between the heavyand light -hole bands. The concentration changes in the heavy-hole band, hhp, and the light-hole band, lhp can be obtained by differentiating Eqs. (22) and (23) [39] ) ( ) )( 1 )( exp(vh F B hh vh F B B vh F vh hhE E T k p E E T k T k E E N p ( 25 ) ) ( ) )( 1 )( exp(vl F B lh vl F B B vl F vl lhE E T k p E E T k T k E E N p ( 26 ) with 0 lh hhp p, ( 27 ) E E Evl vh ( 28 ) where vhE and vlE are stress-induced energy shifts of the heavyan d light-hole bands. Using Eqs. (25) – (28), we can get [23] 2 3 *) ( 1lh hh B hh hhm m E T k p p and 2 3 *) ( 1hh lh B lh lhm m E T k p p ( 29 ) In Eq. (29), the upper and lower signs are fo r uniaxial tensile and compressive stress respectively. The conductivity effective masses of the heavy and light holes, mhh and mlh, can be derived from the E-k dispersion relations described in sec tion 2.2, Eqs. (5), (6), (8), (9), and (11),

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26 2 2 2dk E d m ( 30 ) For a uniaxial stress applied along [001] directio n, if the inverse mass band parameters, A and B, are given in units of 0 22m, where m0 is the free electron mass, using Eqs. (5), (6), and (30), the longitudina l ( || ) and transverse ( ) effective masses of the heavy and light holes can be obtained as [27, 28] B A mhh 1] 001 [ || and B A mlh 1] 001 [ ||, ( 31 ) 2 1] 001 [B A mhh and 2 1] 001 [B A mlh ( 32 ) where the units of effective masses are normalized by m0. Along the [111] direction, if N is also given in units of 0 22m, then using Eqs. (8), (9), and (30), the effective masses are obtained as [27, 28] 3 1] 111 [ ||N A mhh and 3 1] 111 [ ||N A mlh ( 33 ) 6 1] 111 [N A mhh and 6 1] 111 [N A mlh ( 34 ) For [110] direction, the situa tion is more complex due to th e ellipsoids with three unequal axes. We define m1, m2, and m3 as the effective masses corresponding to k1, k2, and k3 as defined in Eq. (11). The longitudinal effectiv e masses of the heavy and light holes can be obtained using Eqs. (11), (14), and (30) [27, 28]

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27 2 1 ] 110 [ 32 2 1 N B A mhh and 2 1 ] 110 [ 32 2 1 N B A mlh ( 35 ) and the two transverse effectiv e masses of the heavy and light holes can be obtained from Eqs. (11) – (13) and (30) 2 1 ] 110 [ 12 2 1 N B A mhh and 2 1 ] 110 [ 12 2 1 N B A mlh ( 36 ) 1 ] 110 [ 21B A mhh and 1 ] 110 [ 21B A mlh ( 37 ) where 1 and 2 are defined in Eq. (15). At high stress, the influence from the spli t-off band is no longer negligible and the 44 strain Hamiltonian is subject to a correction [27, 28]. Th e correction can be expressed in terms of effective mass shift a dded to the stressed e ffective masses obtained in Eqs. (31)-(37), which are zeroth-order, stress-independent stressed effective masses 0m The experimentally measured effective mass *m can be expressed by an empirical formula [28] 0 0 *1 1 1 1 m m m m, ( 38 ) where is a parameter [27, 28]. For a special case that a uniaxial stre ss is applied along a higher rotational symmetry axis the [001] direction (four fo ld) or the [111] direction (three fold), the correction and the effectiv e mass shift only affects the light hole and the heavy hole will not be affected. The longitudina l and transverse effective mass shifts for the light hole for [001] and [111] directions are given by [27, 28],

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28 0 || ] 001 [ ||4 1 B m and 0 ] 001 [2 1 B m ( 39 ) 0 || ] 111 [ ||3 4 1 N m and 0 ] 111 [3 2 1 N m ( 40 ) In Eqs. (39) and (40), the upper and lower si gns are for uniaxial tensile and compressive stress respectively. For [110] direction, due to the lower rotational symmetry (two fold), both the heavy and light holes will experien ce effective mass shifts [27, 28]. For the heavy hole, the effective mass shifts, (1/ m1hh), (1/ m2hh), and (1/ m3hh), are expressed as [28] 0 2 1 2 0 2 0 0 0 0 2 1 2 0 2 0 2 0 2 0 1 ] 110 [ 13 2 2 3 3 2 1 N B mhh, ( 41 ) 0 2 1 2 0 2 0 2 0 2 0 2 ] 110 [ 23 3 1 B mhh, ( 42 ) 0 2 1 2 0 2 0 0 0 0 2 1 2 0 2 0 2 0 2 0 3 ] 110 [ 33 2 2 3 3 2 1 N B mhh. ( 43 ) For light holes, the three eff ective mass shifts are given by 0 2 1 2 0 2 0 0 0 0 2 1 2 0 2 0 2 0 2 0 1 ] 110 [ 13 2 2 3 3 2 1 N B mlh, ( 44 ) 0 2 1 2 0 2 0 2 0 2 0 2 ] 110 [ 23 3 1 B mlh, ( 45 )

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29 0 2 1 2 0 2 0 0 0 0 2 1 2 0 2 0 2 0 2 0 3 ] 110 [ 33 2 2 3 3 2 1 N B mlh. ( 46) In calculation of the stresse d effective mass and band splitting for the three major crystal axes, the values of the inverse mass band parameters and deformation potentials we use are listed in Table 1. The values of A B and N are given in units of 2/2m0 [27]. Table 1. Values of the inverse mass band para meters and deformation potentials used in the calculations. Parameters Symbol Units Values Inverse Mass Band Parameters A 0 22m -4.28 Ref.[27] Inverse Mass Band Parameters B 0 22m -0.75 Ref.[27] Inverse Mass Band Parameters N 0 22m -9.36 Ref.[27] Deformation Potentials uD eV 3.4 Ref.[40] Deformation Potentials 'uD eV 4.4 Ref.[40] The calculated effective mass and band splitt ing will then be substituted into the piezoresistance model to predict coefficients. Table 2 list s the zeroth-order stressed effective mass 0m calculated using Eqs. (31) (37) and Fig. 10 shows the stress-induced band splitting vs. stress for [ 001], [111], and [110] directions calculated using Eqs. (7), (10), and (16). In Table 2, for the [001] and [111] directions, m|| is the longitudinal effective mass, i.e., the effective mass calcu lated using Eq. (30) and the direction of k is parallel to the direction of the stress; m is the transverse effective mass, i.e., the effective mass calculated using Eq. (30) and the direction of k is perpendicular to the direction of the stress. For the [110] direction, m1, m2, and m3 are the effective masses with the

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30 Table 2. Calculated zeroth-order lo ngitudinal ( || ) and transverse ( ) stressed effective masses of the heavy and li ght holes (in units of m0) for [001], [111], and [110] directions. [001] heavy 0.28 m|| light 0.20 heavy 0.21 m light 0.26 [111] heavy 0.86 m|| light 0.14 heavy 0.17 m light 0.37 [110] heavy 0.16 m1 ] 0 1 1 [ light 0.44 heavy 0.21 m2 ] 001 [ light 0.26 heavy 0.54 m3 ] 110 [ light 0.15 direction of k along the directions of k1, k2, and k3 respectively and the direction of k3 is along the stress direction [110]. 2.3.2 Calculation of Relaxation Time and Quantization-Induced Band Splitting In a lowly doped surface invers ion channel, the relaxation time in Eq. (19) is due mainly to three scatte ring mechanisms: acoustic and optical phonon and surface

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31 0 0.005 0.01 0.015 0.02 0.025 0100200300400500Stress / MPaBand Splitting / e V [001] [110] [111] Figure 10. Stress-induced band splitting vs. st ress for [001], [111], and [110] directions.

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32 roughness scattering. Including surface roughness scattering is necessary because the model will be verified by experiments on hole inversion channel in pMOSFETs. The relaxation time can be calculated using approxi mate analytical equa tions based on certain assumptions. To calculate the acoustic and optical phonon scattering times, the following assumptions have been made: (i) the heavyand light-hole bands are assumed to be parabolic as shown in Eqs. (5), (6), (8), (9), and (11) [41], ( ii) the silicon is nondegenerate [41], (iii) the acoustic phonon scattering is el astic and the optical phonon scattering is inelastic and the corresponding scattering times depend only on hole energy [41, 42], and (iv) all holes are scattered isotropicly [41, 42]. For the acoustic phonon scattering, the scattering time ac is expressed as [41-43] 2 1 2 4 2 1 2 22 1 l B l t ac acu T k m m E ( 47 ) where is the energy, eV Eac3 5 [42] is the acoustic deform ation potential constant of the valence band, lm and tm are longitudinal and transverse effective mass, is the density of silicon, and lu is the longitudinal sound ve locity. The tota l acoustic phonon scattering time is given by [42] lh ac hh ac total ac, ,1 1 1 ( 48 ) where hh ac, and lh ac, are the acoustic phonon scatteri ng time in the heavyand light-hole bands respectively. For non-degenera te silicon, the average scattering time for the acoustic phonon can be obtained as [41]

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33 T k d T k T kB B B total ac total ac exp 3 42 3 0 ,. ( 49 ) Since acoustic phonon energy is very small comp ared to the carrier thermal energy [41], the acoustic phonon scattering mainly occurs in the intraband scattering and the stressinduced band splitting will not affect its scattering rate [44, 45]. For optical phonon scattering, without stress, the scattering time opu is given by [41-43] 2 1 2 1 2 2 1 2 2Re exp 2 1 1B B B q l t opuk T k k N m m D ( 50 ) where cm eV D / 10 6 68 [42] is the optical deforma tion potential constant of the valence band, K 735 is the Debye temperature and meV kB630 is the optical phonon energy [41], Nq is the Bose-Einstein ph onon distribution [42] and 1 1 01 exp 1 exp T T k NB q [41]. The “Re” in Eq. (50) means A A Re if A is real; 0 Re A if A is a complex number [41]. Ap plying stress, the stress-induced band splitting will change the scattering rate [44, 45], 2 1 2 1 2 2 1 2 2Re exp 2 1 1E k T E k k N m m DB B B q l t ops ( 51) where E is the band splitting energy. Us ing Eqs. (48) and (49) with ac replaced by opu and ops in Eqs. (50) and (51) respectively the average optical phonon scattering time can be obtained.

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34 For the surface roughness scatte ring, the scattering time sr is derived based on the assumption that the surface roughness causes pote ntial fluctuation to the carrier transport and resulting in carrier scattering. The roughness of the surface can be characterized by the power spectrum density S(q) [46] 4 exp4 2 2Lq L q S ( 52 ) where = 2.7 [46] is the r.m.s valu e of the roughness asperities and L = 10.3 [46] is the roughness correlation length, 2 sin 2 k q and k is the crystal momentum. Using the simple parabolic band approximation, k can be expressed as m k2. Then, the surface roughness scattering time sr is obtained as [46-48] d q S m E eeff sr 2 0 3 2 2cos 1 2 1, ( 53 ) where effE is the surface effective fiel d (normal to the channel), m is the conductivity effective mass along the channel. Considering th e effect of stress on the scattering time, for a 500MPa stress, the corresponding st rain is only abou t 0.3%, thus the L and in Eq. (52) is essentially unchanged. In addition, surfac e roughness scattering is independent of the stress-induced band splitting. Therefore, the surface roughness sc attering will not be affected by stress. Using Eqs. ( 48), (49), (51), and (52) with ac replaced by sr the average scattering time due to surface roughness scattering can be obtained. Then the total scattering time in Eq. (19) can be calculated: total sr total op total ac, ,1 1 1 1 ( 54 )

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35 Figure 11 shows the calculation results of the scattering times. It is assumed that a longitudinal tensile stress is applied along the [110] direction on a (001) wafer. In calculation of the surface roughness scattering, the vertical effective field is chosen at 0.7MV/cm because the mobility enhancemen t in the pMOSFET will be extracted at 0.7MV/cm. As seen from Fig. 11, the surface roughness scattering is the most significant compared to the acoustic and optical phonon scatterings. The optical phonon scattering time increases as the stre ss increases, while the surface roughness and the acoustic phonon scatterings are indepe ndent of stress. Since the mobility enhancement and piezoresistance will be extracted using pMOSFETs, in addition to the surface roughness scattering, we also must consider the quantization effect in the channel due to surf ace effective field. Like stress, the surface electric field can lift the degeneracy at the valence band edge and cause band splitting due to the difference in the effective mass of the heavy and light hole. This band splitting must be considered in addition to the stressinduced band splitting. We use the triangular potential approximation [49, 50] to estim ate the field-induced band splitting. The quantized energy subbands for the heavyan d light-hole bands can be approximated by [49, 50] 3 / 24 3 2 4 3 j m hq Ehh s jhh and 3 / 24 3 2 4 3 j m hq Elh s jlh, j=0,1,2,…., ( 55 ) respectively, where s is the surface electric field, mhh and mlh are the heavy and light hole effective mass normal to the surface, and h is the Planck’s constant. For our application, only non-degenerat e silicon is considered, ther efore, we only take into

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36 0 0.5 1 1.5 2 2.5 3 3.5 4 0100200300400500Stress / MPaScattering Time / ps Acoustic Phonon Scattering Optical Phonon Scattering Surface Roughness Scattering Total Figure 11 The scattering times due to acoustic and optical phonons and surface roughness scatterings. It is assumed that a long itudinal tensile stress is applied along [110] direction on a (001) wafer.

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37 account the first subband, i.e., j=0. Then the field-induced band splitting can be approximated as lh hhE E E1 1 ( 56 ) The field-induced band splitting will then be added to the stress-induced band splitting to calculate the hole repopulation a nd the total ba nd splitting is E E Estrain total ( 57 ) 2.4 Results and Discussion In this section, coefficients will be calculated and compared with the published data [18, 20]. Later in Chapter 5, the calculated coefficients along [110] and [001] directions will be compared with the expe rimental results. Using the definitions of coefficient and conductivity in Eqs. (17) – (19), hole concentrations in Eq. (24), hole repopulation in Eq. (29), the zeroth-order stre ssed effective masses in Eqs. (31) – (37), the effective mass shifts in Eqs. (39) – (46), the scattering time in Eq. (54), and the quantization effect in Eq. (56), the stressed and unstressed conductivities, and 0 can be calculated from lh lh lh hh hh hhm p p m p p q2 and lh lh hh hhm p m p q2 0, ( 58 ) and the longitudinal and transverse coefficients can be obtai ned from Eqs. (20) and (21). To calculate the unstressed conductivity 0 in Eq. (58), the stressed conductivity effective masses are used instead of unstressed ones. This is because, for extrinsic silicon, due to the lattice mismatch be tween the silicon a nd the dopant atom, there exists a small

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38 but not insignificant lattice st ress, estimated about 60kPa [5153]. This small lattice stress can lift the degeneracy at the valence band e dge and change the shap es of constant energy surfaces of the heavyand light-hole bands and the effective masses of the heavy and light holes. Figure 12 is a plot of calculated effective masse s of holes in top and bottom bands vs. stress using 66 strain Hamiltoni an [54], assuming a uniaxial compressive stress is applied along the [110] direction. On the left hand side of the figure, for silicon, the top band represents the h eavy-hole band and the bottom band represents the light-hole band and they are degenerate at the band ed ge. As uniaxial stress increases to about 60kPa, on the right hand side of the figure, th e degeneracy at the band edge is lifted and the top band now represents the light-hole band and the bottom band represents the heavy-hole band. The stressed effective masse s of heavy and light holes saturate. As a result, the stressed effective masses shoul d be used in calcula tion of unstressed conductivity due to the presence of small dopant-induced resi dual stress. However, this small lattice stress, 60kPa, only causes very sma ll band splitting, as can be seen from Fig. 10, thus the hole population is essentially unchanged. Figures 13, 14, and 15 show th e model-predicted longitudinal coefficient vs. stress for [001], [111], and [ 110] direction. The published da ta from Smith [18, 20] are also included for comparison. Note that Smith’s data were extracted with 1 to 10MPa uniaxial tensile stress. One important observa tion from Figs. 13, 14, and 15 is that the coefficients for uniaxial tensile and comp ressive stresses are different and stressdependent. The main reasons are two folds: (i) the stress-indu ced hole repopulation between the heavyand light-hole band and (ii) the correction to th e hole effective mass is stress-dependent as shown in Eq. (38). Th e discontinuities at ze ro stress are due to

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39 Figure 12. Calculated effective masses of hea vy and light holes vs. stress using 66 strain Hamiltonian [54], assuming a uniaxial co mpressive stress is applied along the [110] direction. The solid line represents the effectiv e masses of holes in the top band and the dashed line represents th e bottom band. On the left hand side of the figure, for the unstressed silicon, the top band represents the heavy-hole band and the bottom band represents the light-hole band. On the right hand side of the figure, for the stressed si licon, the top band represents the lighthole band and the bottom band represents the heavy-hole band. After about 60kPa stress, the degeneracy is lifted and the light-hole band rises above the heavy-hole band and the stressed effec tive masses of heavy and light holes saturate.

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40 5 10 15 20 25 -500-300-100100300500Stress / MPa||[001] / 10-11Pa-1Smith's data Compression Tension Figure 13. Model-predicted longitudinal coefficient vs. stress for [001] direction. Smith’s data [20] are included for comparison.

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41 80 100 120 140 160 -500-300-100100300500Stress / MPa||[111] / 10-11Pa-1Smith's data Compression Tension Figure 14. Model-predicted longitudinal coefficient vs. stress for [111] direction. Smith’s data [20] are included for comparison.

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42 50 60 70 80 90 100 -500-300-100100300500Stress / MPa||[110] / 10-11Pa-1Smith's data Compression Tension Figure 15. Model-predicted longitudinal coefficient vs. stress for [110] direction. Smith’s data [20] are included for comparison.

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43 relaxation time and quantization effect in th e inversion layer of pMOSFET as described in subsection 2.3.2. The predictions associ ated with [001] and [ 110] direction will be verified by the experiments presen ted later in Chapters 4 and 5. In comparison to the previous works [19, 22, 23], first, all of them only consider coefficient for [111] direction and do not e xplicitly discuss the stress dependence of coefficient. Second, they all use stressed effective mass in calculation of unstressed conductivity without making assumption or givi ng explanation. Third, they all assume constant scattering time. For example, Suzuki et al. [22] consid er stress-induced hole transfer between the heavyand light-hole bands and the effective mass shift for light hole due to the stress-induced coupling between the light-hole and split-off bands. However, they make an assumption that the conductivity change due to the hole transfer is given by T k E T k s DB lh hh lh hh B u lh hh lh hh 0 44 03 ( 59 ) where hh and lh are the mobility of heavy and light hole, E is the band splitting for [111] direction defined in Eq. (10), kB is the Boltzmann constant and T is the absolute temperature. The authors do not provide justif ication or explanation to the assumption. Using Eq. (29), our model predicts the conductiv ity change due to the hole transfer as T k E p p p p p pB lh hh lh hh lh lh hh hh lh hh 0, ( 60 ) which can be derived from semiconductors equations. The authors obtain longitudinal and transverse coefficients as 11310-11Pa-1 and -5610-11Pa-1 respectively, without specifying at what stress. Kanda [19] uses th e model and result from Suzuki et al. [22]. Kleimann et al. [23] consider hole transfer and effective mass shift for the light hole due

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44 to (i) coupling between the light-hole and spli t-off bands and (ii) incomplete decoupling between the heavyand light-hole bands. Howeve r, they postulate that the effective mass shift due to the incomplete decoupling does not affect the longit udinal heavy and light hole effective masses. For transverse effectiv e masses of the heavy and light holes, they introduce correction terms proporti onal to stress to represent the effective mass shift due to incomplete decoupling, 11 hhm and 21 lhm, ( 61 ) where 1 and 2 are two parameters. The correctio n terms shown in Eq. (61) are contradictory to the results from Hasegawa [28] and Hensel et al. [27], to which Kleimann et al. refer in their paper. The reason is because the incomplete decoupling effect should decrease as the stress increases and, at very high stress, the heavyand light-hole bands will decouple completely and the correction terms will disappear, in contradiction to Eq. (61). The authors obtained 8910-11Pa-1 for longitudinal coefficient for [111] direction. As for transverse coefficient, the authors fit the experimental value, -44.510-11Pa-1, and get values for 1 and 2, 2.410-9Pa-1 and -2.410-9Pa-1 respectively. The authors give both unstressed and stressed ef fective masses in their paper but used the stressed effective mass to calculate the unstressed conductivity without a model or explanation. The strain effect described in section 2.2 was derived fr om Kleiner-Roth 44 strain Hamiltonian [27, 32], which ne glects the spilt-off band. To estimate the error introduced by using the Kleiner-Roth 44 st rain Hamiltonian [27, 32], we also use the Bir-Pikus 66 strain Hamiltonian [55, 56], wh ich takes the split-off band in to account, to calculate the

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45 band splitting and effective masses for [001 ] direction and compare to the results presented in section 2.3. Figure 16 illustrates the band energy vs. stress calculated from 44 and 66 strain Hamiltonians for [001] direction. The 44 strain Hamiltonian overestimates the band splitting between th e heavyand light-hole bands by 41% at 500MPa uniaxial compression but underestimates 16% at 500MPa uniaxial tension. Note that the energy separation between the upper hol e band and split-off band is larger at high stress than zero stress, which implies that th e hole concentration in the split-off band is even smaller at high stress and neglecting th e split-off band in the model is justified. Comparisons of longitudinal and transverse heavy and light hole effective masses are shown in Figs. 17(a) and (b) respectively. Fo r the longitudinal light hole effective mass, the 44 strain Hamiltonian underestimates about 2% and 5% at 500MPa uniaxial compression and tension respectively, and the deviations are about 1% and 4% overestimations for the transverse light hole effective mass respectively. For the heavy hole, both the longitudinal and transverse effective masses ar e the same for the 44 and 66 strain Hamiltonians. Based on these compar isons, we conclude that using 44 strain Hamiltonian is a good approximation and the mode l is suitable for the stress less than 500MPa. 2.5 Summary The strain effects on valence band are explained in detail in this chapter. The constant energy surfaces of the heavyand light-hole bands, heavy and light hole effective masses, stress-induced band splitting, hole repopulation are explained and derived using Kleiner-Roth 44 strain Hamiltoni an. At high stress, th e influence from the split-off band is taken into account by adding an effective mass shift to the light hole

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46 -0.06 -0.04 -0.02 0 0.02 -500-2500250500Stress / MPaEnergy / eVLight Hole 6x6 Light Hole 4x4 Heavy Hole 6x6 & 4x4 Split-off 4x4 Split-off 6x6 Figure 16. The energies of the heavyand light-hole and split-off bands vs. stress for [001] direction calculated using 4 4 and 66 strain Hamiltonians.

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47 0.17 0.19 0.21 0.23 0.25 0.27 0.29 -500-2500250500 Stress / MPaLongitudinal Effective Mass / m0Light Hole 6x6 Light Hole 4x4 Heavy Hole 6x6 & 4x4(a) 0.17 0.19 0.21 0.23 0.25 0.27 0.29 -500-2500250500 Stress / MPaTransverse Effective Mass / m0Light Hole 6x6 Light Hole 4x4 Heavy Hole 6x6 & 4x4(b) Figure 17. Effective masses vs. stress for th e heavy and light holes for [001] direction calculated using 44 and 66 strain Hamiltonians. (a) Longitudinal. (b) Transverse.

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48 effective mass (for [001] and [111] directions) an d to both heavy and light hole effective masses (for [110] direction). The strain effects are used to model pi ezoresistance in ptype silicon. The longitudinal and transverse coefficients for three major cr ystal axes, [001], [111], and [110] directions, are calculate d and compared to the publishe d data. The model-predicted coefficients are stress-dependent and the tensile and compressive coefficients are different. The reasons are two folds: (i) th e stress-induced band sp litting causes the stress dependence of hole population an d (ii) light hole effective mass (for [001] and [111] directions) or both heavy and light hole effective masses (for [110] direction) are stressdependent due to the influence of the split-off band at high stress. Finally, the valid stress range that the model is applicable is estimated by comparing the band splitting and the heavy and light hole effective masses calculated using 44 and 66 strain Hamiltonians. The comparisons show that 44 strain Hamiltonian is a good approxima tion to 66 strain Hamiltonian at stress under 500MPa, and hence, the piezoresistance model is good with stress less th an 500MPa. Later in Chapters 4 and 5, the piezoresistance model will be verified by the experiments.

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49 CHAPTER 3 HOLE MOBILITY ENHANCEMENT IN BIAXIAL AND UNIAXIAL STRAINEDSILICON PMOSFET 3.1 Introduction The underlying mechanisms of mobility enhancement in uni axial and biaxial strained-Si pMOSFETs are the same as the piezore sistance effect in ptype silicon as seen from Eqs. (20) and (21). As shown in Fi g. 18, by uniaxial strained-Si pMOSFETs, we mean that an in-plane uniaxial compressive stress (uniaxial compressi on) is applied along the channel of pMOSFET, i.e., along [110] direction in the ( 001) plane that contains the channel. By biaxial strained-Si pMOSFETs, we mean that an in-plane biaxial tensile stress (biaxial tension) is applied to the cha nnel in the (001) plane, which is illustrated in Fig. 19. Biaxial stress, like uniaxial stress, can lift the degeneracy at the valence band edge and cause hole repopulation and mobility change. Uniaxial and biaxial strained-Si pMOSFETs are two important technologies used to enhance the hole mobility [1-14]. Thompson et al. [11, 12, 44] compared uni axial vs. biaxial in terms of device performance and process complexity and conc luded that uniaxial strained-Si pMOSFETs is preferable since comparable mobility enhancement is attained at smaller stress (500MPa compared to >1GPa) which is re tained at high effective field. In this chapter, we will use the strain effects on the valence band described in Chapter 2 to explain quantita tively how uniaxial compressive and biaxial tensile stresses change the hole mobility in pMOSFETs in the low stress regime (<500MPa). In section 3.2, the mobility enhancement will be calculated. An equivalent out-of-plane uniaxial compressive stress will be derived for biaxial-strained pMOSFET. The valence band

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50 Figure 18. Illustration of uniaxial strained -Si pMOSFET. The arrows represent the uniaxial compressive stress in the channel. The uniaxial compressive stress can be generated by process, for example, source/drain refilled with SiGe [1012] or highly compressive stressed SiN capping layer [13], or by four-point wafer bending, which will be described in Chapter 4.

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51 Figure 19. Illustration of biaxial strained-Si pMOSFET. The arrows represent the biaxial tensile stress in the channel. The biaxial tensile stress can be generated by process, for example, using thick relaxed Si1-xGex layer to stretch the Si channel [1-8], or by concentric-ring wafer bending, which will be described in Chapter 4.

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52 structure, inand out-of-plane heavy and light hole effective masses, band splitting, and hole transfer are explained and calculated using the model described in Chapter 2. Section 3.3 is the discussion. Fina lly, section 3.4 is the summary. 3.2 Mobility Enhancement in Strained-Silicon PMOSFET In this section, we will calculate a nd compare the mobility improvement in uniaxialand biaxial-strained pMOSFETs. Th e equations of constant energy surface and band splitting described in Chapter 2 were de rived by Hensel and Feher [27] in terms of uniaxial stress, for example, Eqs. (7) and (10) In order to use their equations to model biaxial-strained pMOSFETs, we will show that in-plane, (001), biaxial-tensile stress bi can be represented by an equivalent out-of-plane uniaxial compressive stress uni of the same magnitude along the [001] direction, which will create the same band splitting along the same direction as shown in Eq. (7). The band splitting for uniaxial stress in the [001] direction is given by [27], S D Eu 3 2 2 20, ( 62 ) where uD is the valence band deformation potential for [001] direction and the strain is give by xx zzs s S 12 11, ( 63 ) where is the uniaxial stress along [001] direction, and zz and xx are the uniaxial strain along the [001] and [ 100] directions respectively. The strain-stress relation for material with cubic symmetry is revi ewed in Appendix A. In Eq. (63), 11s and 12s are compliances of silicon [57] and

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53 12 11 12 11 12 11 112c c c c c c s and 12 11 12 11 12 122c c c c c s ( 64 ) where Pa c11 1110 657 1 and Pa c11 1210 639 0 are the normal and off-diagonal stiffnesses of silicon. For in-plane biaxial tensile strain, yy xx, and zz can be derived from the stress-strain equa tion with zero shear strains [58], 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 044 44 44 11 12 12 12 11 12 12 12 11 zz yy xx xy zx yz zz yy xxc c c c c c c c c c c c ( 65 ) where xx, yy, and zz are the uniaxial stresses, and xy, yz, and zx are the shear stresses. On the left hand side of Eq. (65) is the stress tensor, and the right hand side is the elastic stiffness tensor and strain tensor. With in-plane biaxial tensile strain, there is no out-ofplane stress, i.e., zz yy xx zzc c 11 120 Then, zz can be obtained as 11 122c czz and S in Eq. (63) becomes 12 11 11 122c c c c Sbi ( 66 ) where bi is the in-plane biaxial tensile stress in (001) plane and 1Ebi, ( 67 )

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54 where E is the Young’s modulus, is the Poisson ratio, and Pa s s E11 12 1110 805 1 1 1 is the biaxial modulus and invariant in the (001) plane [57]. For a uniaxial compressive stress uni 12 11 12 11c c s s Suni uni uni ( 68 ) Comparing Eqs. (66) and (68), if uni bi then the in-plane biaxial tensile and out-ofplane uniaxial compressive stresses will create the same strain S and thus the same band splitting along [001] directi on. Therefore, the in-pla ne biaxial tensile stress bi can be represented by an equivalent out-ofplane uniaxial compressive stress uni of the same magnitude. The equations and band parameters presente d in section 2.3 are used to calculate the effective mass, band splitting, and hole transfer between the heavyand light-hole bands. Table 3 gives the zeroth-order, ina nd out-of-plane heavy and light hole effective Table 3. Inand out-of-plane effective masse s of the heavy and li ght holes for uniaxial compression and biaxial tension. Uniaxial Compression Biaxial Tension Heavy Hole Light HoleHeavy Hole Light Hole In-Plane 0.54 0.15 0.21 0.26 Out-of-Plane 0.21 0.26 0.28 0.20 masses for uniaxial compression and biaxial te nsion. Figure 20 shows the in-plane heavy and light hole effective mass with mass corr ection vs. stress. The band splitting induced by both stress and quantum confinement vs. st ress is shown in Fig. 21. The mobility enhancement in uniaxialand biaxial-strained pMOSFETs are calculat ed using Eqs. (20), (21), and (58) and shown in Fig 22. Two publis hed experimental data points for biaxial-

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55 0.1 0.2 0.3 0.4 0.5 0.6 0100200300400500Stress / MPaIn-Plane Effective Mass / m0Uniaxial Heavy Hole Biaxial Light Hole Biaxial Heavy Hole Uniaxial Light Hole Figure 20. In-plane effective masses of the h eavy and light holes vs. stress for uniaxial compression and biaxial tension.

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56 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0100200300400500Stress / MPaBand Splitting / e V Biaxial Tension Uniaxial Compression Figure 21. Band splitting vs. stress for uniax ial compression and biaxial tension. Both contributions from stress and quantum confinement are included. Doping density is assumed 1017cm-3.

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57 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 1.E+071.E+081.E+09Stress / Pa/effThompson et al. 04 (Uniaxial Compression) Uniaxial Compression This Work Rim et al. 02 (Biaxial Tension) Biaxial Tension This Work Rim et al. 03 (Biaxial Tension) Oberhuber 98 (Biaxial Tension) Figure 22. Model-predicted mobility enhancement vs. stress for uniaxial compression and biaxial tension. Published theoretical [3 ] and experimental [5, 8, 12] works are also shown for comparison.

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58 strained pMOSFET [5, 8] and one for uniaxial-strained pM OSFET [12] are included for comparison. Also included is the model pr ediction for biaxial-st rained pMOSFET from Oberhuber et al. [3]. In comparison with their theoretical work [3], our model gives better prediction that shows mobility degradation instead of mobility enhancement at low biaxial tensile stress. At 500MPa, the mode l predicts 48% mobility improvement for uniaxial compression and 4% mobility degradat ion for biaxial tension, in good agreement with published data [5, 8, 12]. 3.3 Discussion The underlying mechanism of the mobility enhancement in uniax ialand biaxialstrained pMOSFETs is the hol e repopulation from the heavyto light-hole bands. For uniaxial compression, as shown in Figs. 8 a nd 20, since the heavyand light-hole bands are prolate and oblate ellipsoi ds, the heavy hole effective ma ss along the channel in (001) plane is larger than the lig ht hole and hole repopulation fr om the heavyto light-hole bands improves the mobility. For biaxial tension, as seen from Figs. 6 and 20, the effective mass along the channel in (001) plane for the heavy hole is actually smaller than the light hole and hole repopulation from th e heavyto light-hol e bands results in mobility degradation. In addition, as seen from Fig. 20, the difference in the in-plane effective masses of the heavy and light holes is larger for uniaxial compression than biaxial tension. This causes larger mobili ty improvement for uniaxial compression. The summary of the underlying mechanism of mob ility enhancement in uniaxialand biaxialstrained pMOSFETs is illustrated in Fig. 23. At high stress, the band splitting will beco me very large, and hole transfer will finally stop when all holes populate only one band (upper band with lower energy). At this point, the mobility enhancement will mainly come from the suppression of the

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59 Figure 23. The underlying mechanism of mob ility enhancement in uniaxialand biaxia l-strained pMOSFETs. The effective mass shown next to the constant energy surfaces are the zeroth-order in-plane effective masses.

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60 interband optical phonon scattering, as shown in Eq. (51). For biaxial tension, it implies that, at certain high stress, the mobility will stop decreasing and start increasing with even higher stress. The band splitting between the heavyand light-hole bands is ~25meV for 500MPa biaxial tensile stress and ~100meV for 2.2GPa, corresponding to x=0.28 in Si1xGex. When the band splitting becomes larger than the optical phonon energy, 63meV [42], the interband scattering between the h eavyand light-hole band is suppressed and hole mobility increases. This explains the published biaxial tension data at high Ge concentration [5, 8]. However, due to the limitation of 44 strain Hamiltonian, the analytical model cannot provide accurate prediction for hole mobility enhancement at such high stress. 3.4 Summary At low stress (<500MPa), the uniaxial-str ained pMOSFET is shown to have large mobility improvement due to hole repopulation from the heavy-hole band with larger inplane effective mass to the light-hole band with smaller one. For biaxial-strained pMOSFET, because the in-plane effective mass of the heavy hol e is smaller than the light hole, hole repopulation from the heavyto lig ht-hole bands degrades the mobility. Both predictions are in good agreement with the published data, 48% improvement for uniaxial-strained pMOSFET and 4% degrad ation for biaxial-strained pMOSFET at 500MPa. At large biaxial tensile stress, the suppressed interband scattering due to the large band splitting, greater than the optic al phonon energy, results in the mobility enhancement.

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61 CHAPTER 4 WAFER BENDING EXPERIMENT AND MOBILITY ENHANCEMENT EXTRACTION ON STRAINED-SILICON PMOSFETS 4.1 Introduction In this chapter, wafer bending experiments designed to test the models described in previous chapters are presented. The hole mobility and mobility enhancement vs. stress will be extracted and the piezoresistance coefficients of p-type silicon vs. stress will then be calculated. Concentric-ring and fou r-point bending apparatus are used to apply six kinds of mechanical stress to the ch annels of pMOSFETs, biaxial tensile and compressive and uniaxial longitudinal and tran sverse, tensile and compressive stress. The stress range used in this experiment is 50MPa to 300MPa. PMOSFETs from 90nm technology [11, 12, 59] with the channels or iented along [110] direction on (001) wafers are used in the experiments. In section 4.2, wafer bending experiments are presented. Firs t, the four-point bending apparatus used to apply uniaxial stress will be explai ned in detail and equations for calculating the uniaxial stress will be derived. For the concentric-ring bending jig used to apply biaxial stress, finite element analysis simulation is used due to nonlinear bending. Uncertainty analysis in the applie d stress will be pe rformed. Section 4.3 explains the methods to extract threshold vol tage, hole mobility, and vertical effective field. Uncertainty analysis for effective mobility will be performed. Section 4.4 is the summary.

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62 4.2 Wafer Bending Experime nts on pMOSFETs 4.2.1 Four-Point Bending for Applying Uniaxial Stress Uniaxial stress is applied to the channe l of a pMOSFET using four-point bending. Figure 24(a) and (b) are the pictures of the apparatus used to bend the substrate and the illustrations of calculating the uniaxial stress. As shown in Fig. 24, the upper and lower surfaces of the substrate will experience un iaxial compressive and tensile stress along [110] direction, respectively. The stress on both surfaces can be calculated using the following analysis [60] with the assumptions: (i) The substrate is simply supported. (ii) Four loads applied by four cylinders ar e approximated by fo ur point forces, P. As shown in Fig. 24(b), let the deflection at an y point on the upper surface be designated by y(x), where y(0)=0 and y(L)=0. The stress on the upper and lo wer surfaces at the center of the substrate are given by r EHxupper2 and r EHxlower2 ( 69 ) respectively, where Pa E1110 689 1 [57] is the Young’s modulus of crystalline silicon along the [110] direction on (001) substrate, H is the substrate thickness, r is the radius of curvature given by [60] z zEI Pa EI M r 1 ( 70 ) where Pa M is the moment for 2L x a and 123bH Iz is the moment of inertia for a substrate with rectangular cross section and width of b. Eq. (69) can then be expressed as [60]

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63 (a) (b) Figure 24. Apparatus used to apply uniaxial st ress and schematic of four-point bending. (a) The picture of jig. In this picture, uniaxial compressive and tensile stresses are generated on the upper and lower surf aces of the substrate respectively. (b) Schematic of four-point bending. The substrate is simply supported. Four loads applied by cylinders are appr oximated by four point forces, P. The deflection at any point on th e upper surface is designated by y(x).

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64 z z xupperI PaH I MH2 2 and z z xlowerI PaH I MH2 2 ( 71 ) For a x 0, the moment Px M and Px M dx y d EIz 2 2. ( 72 ) Solving Eq. (72) we get 2 1 36 1 C x C x P EI yz, ( 73 ) where C1 and C2 are integration constants. For 2 L x a the moment Pa a x P Px M and [60] Pa M dx y d EIz 2 2. ( 74 ) Solving Eq. (74) we obtain 4 3 22 1 C x C x Pa EI yz, ( 75 ) where C3 and C4 are integration constants. The f our integration constants can be determined from the boundary conditions [60]: (i) the slope dx dy determined from Eqs. (73) and (75) should be equal at a x (ii) the slope 0 dx dy at 2 L x i.e., at the center of the substrate, (iii) at a x y determined from Eqs. (73) and (75) should be equal, and (iv) at 0 x, 0y. With these four boundary conditions, Eqs. (73) and (75) become [60] x a L Pa x P EI yz2 6 13 a x 0, ( 76 )

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65 6 2 2 13 2Pa x PaL x Pa EI yz 2 L x a ( 77 ) Using Eqs. (76) and (77), the deflection at a x and 2 L x can be calculated [60] 3 2 22a L EI Pa yz a x, ( 78 ) 2 2 24 3 24 a L EI Pa yz L x ( 79 ) Measuring the deflection at a x zI P can be obtained as 3 2 22a L a Ey I Pa x z, ( 80 ) and then the stress on both surfaces of the substrate in Eq. (71) can then be calculated 3 2 2 2 a L a EHya x xupper and 3 2 2 2 a L a EHya x xlower. ( 81 ) The radius of curvature in Eq. (70) can also be obtained as a L a y EI Pa ra x z3 2 2 1. ( 82 ) Finite element analysis (FEA) using ABAQUS [61] is also performed to verify the assumptions used to obtain Eq. (81) and the results are shown in Fig. 25. In Fig. 25, two cases are simulated as shown in two insets The upper inset shows that the distance between the two top pins is larger than the two bottom ones. When the two top pins move downward, a tensile stress is generated on the upper surface. The lower inset is the

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66 -500 -400 -300 -200 -100 0 100 200 300 400 500 00.10.20.30.40.50.60.70.80.91Deflection / mmStress / MPa Uniaxial Tension (Simulation) Uniaxial Tension (Equation) Uniaxial Compression (Equation) Uniaxial Compression (Simulation) Maximum Deflection Used Figure 25. Stress at the center of the upper su rface of the substrate vs. the deflection of the top pins. The calculated stress values are from Eq. (81). Simulated stress values are obtained using finite element analysis with ABAQUS [61].

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67 opposite of the upper one and a compressive stress is generated on the top surface. The stress at the center of the t op surface is extracted from the simulation and compared with the stress calculated using Eq. (81). The calcula ted stresses using Eq. (81) agree well with the results of finite element analysis for the range of deflection used. Figure 26(a) is a plot of st ress vs. relative position y to the neutral axis along the cross section, AA’ in Fig. 26(b), extracted fr om simulation. The deflection is assumed as 0.91mm. Figure 26(b) is an illustration of be nding substrate, neutral axis, top and bottom planes, and cross section AA’ cut at the cent er of substrate used in simulations. The substrate thickness H is 0.77mm. As shown in Fig 26(a), the stress vs. position y curve is linear and symmetric about the origin. This resu lt verifies the validit y of Eq. (81) within the range of deflection 0.91mm. The finite element analysis simulations shown in Figs. 25 and 26 need further investigations due to no systematic studies of grid convergence. More detailed analysis can be done in future work. Regarding the effect of sample location on stress, as will be shown in Fig. 35, the stress variation at the pos ition between the two top pins on the upper surface is less than 0.01%. Therefore, th e effect of sample lo cation on stress can be neglected for the range of deflection used. 4.2.2 Concentric-Ring Bending fo r Applying Biaxial Stress Biaxial stress is applied to the channel of a pMOSFET using concentric-ring bending. Figures 27(a) and (b) show a picture of the apparatus used to bend the substrate and an illustration for simulating the biaxial stress. Unlike beams (uniaxial stress state), even deflections comparable to the plate th ickness produce large stresses in the middle plane and contribute to stre ss stiffening. Hence one should use large deflection (nonlinear

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68 -500 -400 -300 -200 -100 0 100 200 300 400 500 -400-300-200-1000100200300400Position y / mStress / MPa Stress at top plane 443MPa Stress at bottom plane -443MPa Stress at neutral axis 0 (a) (b) Figure 26. Stress vs. position and schematic of bending substrate. (a) Plot of stress vs. relative position y to the neutral axis along the cross section, AA’ in (b), extracted from simulation. The defl ection is assumed as 0.91mm. (b) Illustration of bending substrate, neutra l axis, top and bottom planes, and cross section AA’ cut at the center of substr ate. The substrate thickness is 0.77mm.

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69 (a) (b) Figure 27. Apparatus used to apply biaxia l stress and schematic of concentric-ring bending. (a) The picture of jig. In this picture, biaxial compressive and tensile stresses are generated on the upper and lower surfaces of the substrate respectively. (b) Schematic of concentric -ring bending. The plate (substrate) is simply supported. The deflection at any point on the upper surface is designated by w(r)

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70 analysis) to calculate deflections and stresses in a plate. The finite element analysis (FEA) using ABAQUS [61, 62] considerin g both the nonlinearity and orthotropic property of Si is used in this work to cal culate the biaxial stress from the measured deflections. The constitutive equation of Si is expressed as xy zx yz zz yy xx xy zx yz zz yy yz xx xz zz zy yy xx xy zz zx yy yx xx xy zx yz zz yy xxG G G E E v E v E v E E v E v E v E 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 ( 83 ) where Pa E E Ezz yy xx1110 302 1 are the three Young’s moduli [57], Pa G G Gzx yz xy1110 796 0 are the three shear moduli, 279 0 zx yz xyv v v are the three Poisson ratios, xx, yy, and zz are the normal stresses, xx, yy, and zz are the normal strains, xy, yz, and zx are the shear stresses, and xy, yz, and zx are the shear strains. The values of Young’s moduli, shear moduli, and Poi sson ratios are needed in the finite element analysis. The simulation assumed that the smaller ring is on the top and the larger one is on the bottom as illustrated in Fig. 27. The results are shown in Figs. 28 and 29 [62]. In Fig. 28, the stre sses along x and y axes, xx and yy respectively, at the center of the bottom, middle, and top planes vs. the displacement of the smaller ring are presented. As can be seen from Fig. 28, xx = yy, this result confirms that the stress at the center of the planes is biaxial. In a ddition, Figure 29 (a) shows the shear stress xy at

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71 -300 -200 -100 0 100 200 300 400 500 600 00.20.40.60.81Displacement of Small Ring (mm)Stress at Center (MPa) sigma(xx) sigma(yy)Bottom Middle Top Maximum Displacement Use d xxyy (a) (b) Figure 28. Stress vs. displacement and schematic of bending plate. (a) Finite element analysis simulation of the bending plate (substrate). The radial stresses, xx and yy, at the center of the top, middle, and bottom planes of the bending plate vs. the displacement of the smaller ring are shown. xx= yy. (b) Illustration of top, middle, and bottom planes of the plate.

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72 0 0.5 1 1.5 00.20.40.60.81 Displacement of Small Ring (mm)Shear Stress at Center (MPa)Bottom Plane Middle Plane Top Plane Maximum Displacement Used (a) 0 1000 2000 3000 4000 5000 6000 00.20.40.60.81Displacement of Small Ring (mm)Load (N) (b) Figure 29. Finite element analysis simulatio n of the bending plat e (substrate). (a) The shear stress at the centers of three pl anes, top, middle, and bottom and (b)the load required on the smaller ring as a function of displa cement are extracted from the simulations.

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73 the center of three planes, and Fig. 29(b) is the load applied to the smaller ring vs. the displacement of the smaller ring. On the bottom plane of the substrat e, the stress at the center is tensile as expected while, on th e top plane, the center stress appears as compressive first, then gradually decrease s and finally becomes tensile. This can be explained by the nonlinearity of bending plate with large defl ection. At small deflection (<< the thickness of substrate 0.77 mm), the stress on the top and bottom planes are of nearly the same magnitude but opposite sign, as shown in Fig. 28. There is no stress in the middle plane. However, at large deflection, the middle plane stretches and experiences tensile stresses, which implies that the whole substrate stretches. For the bottom plane, the total tensile stress will be the sum of the original stress and the additional tensile stress due to the substr ate stretching, while for the top plane, the compressive stress will be reduced by the addi tional tensile stress. As the displacement of smaller ring reaches about 0.89mm, the compressive stress on the top plane will be completely cancelled out by the tensile stress due to the substrate stretching, as can be seen from Fig. 28. According to Fig. 29(b), the corresponding load is about 4500N or 1000lb when displacement re aches 0.89mm. The maximum deflection achieved in this work is about 0.46mm, correspond ing to about 1100N or 250l b. According to Figs. 28(a) and 29(a), xy is about three orders of magnitude smaller than xx and yy at the center. Detailed analysis of the shear stress shows that xy has no effect on mobility enhancement at the center of the concentr ic ring due to symmetry of xy and yx. Thus, the mobility enhancement at the center is due to biaxial stress alone. The finite element analysis simulations shown in Figs. 28 and 29 need further investigations due to no systematic stud ies of grid convergence and no uncertainty

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74 analysis in sample location variation from exact center. More detailed analysis can be done in future work. 4.2.3 Uncertainty Analysis In this subsection, the uncertainty in the applied stress will be estimated. There are four major sources of uncertainty in applied uniaxial stress using the four-point bending jig shown in Fig. 24. One major source of un certainty is the starti ng point of bending. If the top plate does not lower enough to make the two top pins contact with the substrate and the grooves perfectly, the ac tual applied stress will be sm aller than the expected one, while if it lowers too much, additional stress wi ll be generated and ca use the actual stress to be higher than expected. To estimate th e uncertainty of the starting point, we use the approach as follows [63]: First, lower the top plate such that the two top pins, the substrate, and the grooves can be seen in co ntact. Tapping the two top pins, if the pins can move with slight friction, this is the starting point. Second, measure the distance between the top and bottom plates at four locations as indicated in Fig. 30. The average value is used as the distance between the two plates. Finally, repeat the procedure 10 times and calculate the uncertainty with 95% confidence with these 10 values. This value is the uncertainty of the star ting point in terms of deflecti on. The uncertainty of starting point is estimated as 0.07mm. The second source of uncertainty is th e micrometer for setting the displacement [63]. The resolution of the micrometer is 1/1000 inch (0.03mm), the uncertainty is one half of the resolution or about 0.02mm. Th e total uncertainty in deflection from the starting point and micrometer is mm073 0 02 0 07 02 2

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75 Figure 30. Uncertainty analysis of the starting point for the four-point and concentric-ring bending experiments. The distance betw een two plates are measured at the locations designated 1, 2, 3, and 4.

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76 The third source is the variation of the substrate thickness. The typical thickness of a 12 inches (300mm) wafer is 77520 m [64]. The uncertainty in wafer thickness is 0.02mm. The uncertainty in stress from the fi rst three sources can be calculated by differentiating Eq. (81) [65] H a L a Ey y a L a EH 3 2 2 2 3 2 2 2, ( 84 ) 2 2 2 H H y y ( 85 ) In Eq. (84), the variations in L and a are negligible because they are fixed by the grooves as shown in Fig. 24. In Eq. (85), 0007 0 775 202 2 H H is negligible compared with 02 0 57 0 073 02 2 y y where 0.57mm is the maximum deflection achieved in the experiment. The total uncertaint y in stress is estimated about 40MPa. The fourth uncertainty is from the substrate angle misalignment as shown in Fig. 31 [66]. During the experiment, the substrate is difficult to align because there is only a circular hole (smaller than the substrate) on the top plate, the view from the top and bottom is obstructed by the metal plate. The only markers that can be used for alignment are the patterns on the substrate. This will ca use an uncertainty in alignment visually estimated to be about 10, corresponding to about 30MPa uncertainty in stress [66]. The total uncertainty from all four sources is estimated as MPa50 30 402 2 The applied uniaxial stress can be calibrate d by extracting the radius of curvature of the substrate after bending. Let the elas tic curve for a beam after bending be y ( x ), then the radius of curvature r can be expressed as [67]

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77 Figure 31. Uncertainty analysis of the misali gnment of the substrate with respect to the pins for the four-point bending experiment.

PAGE 89

78 2 3 2 2 21 1 dx dy dx y d r ( 86 ) assuming the beam deflections occur only due to bending [67]. Substituting into Eq. (69), we get 2 3 2 2 21 2 dx dy dx y d EH. ( 87 ) Both Eqs. (69) and (87) are valid for eith er small or large radii of curvature [67]. Extracting the elastic curve y ( x ) and substituting into Eq. (87), the stress is obtained. The PHILTEC FiberOptic Displacement Measuremen t System [68] is used to extract the elastic curve, and the setup of experiment is shown in Fig. 32. The optical sensor sweeps across the substrate between the two top pins, and the distance between the sensor and the substrate is measured and recorded. To refe rence the original unstressed wafer surface, the before bending curve is measured first, and the after bending curves are measured subsequently. Subtracting the before bending cu rve from the after bending one, the elastic curve is measured and the result is shown in Fig. 33. A similar method was used by Uchida et al. [69] previously. The elastic curve and radius of curvature are extracted from the polished surface of a bare wafer instead of the device wafer due to poor reflectivity on the patterned and passivated device wafer. There is a passiva tion on top of the device wafer, which typically consists of a phosp horus-doped silicon dioxide laye r and then silicon nitride

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79 Figure 32. The experimental setup for calibra ting the uniaxial stress in the four-point bending experiment.

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80 y = -1E-06x2 + 0.0069x + 8.4543 R2 = 0.9975 y = -6E-07x2 + 0.0033x + 4.5056 R2 = 0.9929 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 01000200030004000500060007000Position x / mDisplacement y / m fitting curve fitting curve Figure 33. Extracted displacement vs. posit ion curves on the upper surface of the substrate in the four-point bendi ng experiment. The experimental displacement curves are LSF to 2nd order polynomials with R2=0.9975 and 0.9929 respectively.

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81 layer. The reflectivity of passivation is too low for optical sensor to operate accurately, therefore, the polished surface of bare wafer is used to extract the elastic curve and radius of curvature. The extracted elastic curves are least-square fit (LSF) to a second order polynomial with R2=0.9975 and 0.9929. Substituting the resulting LSF second order polynomial, 8.4543 + 0.0069x + x 10 = y 2 -6 and 4.5056 + 0.0033x + x 10 6 = y 2 -7 into Eq. (87), the applied stress is obtained. Fi gures 34 and 35 are the extracted radius of curvature and corresponding stress respectively. As can be seen from Figs. 34 and 35, the difference between the maximum and minimum of the radi us of curvature is about 0.01% and also about 0.01% for applied stress. Figure 36 shows both the calculated stress from displacement and extracted stress from the measured wafer curvature. The uncertainty analysis of the stress extracted from the ex perimental curvature data in Fig. 36 is described in Appendix C. For biaxial stress, there are two major sources of uncertainty in the applied biaxial stress, the starting point and the micrometer. Us ing the same procedure as for the uniaxial stress, the uncertainty of starting point is estimated as about 0.04mm, smaller than 0.07mm for the uniaxial stress, because the starting point is ea sier to be seen with a ring than pins and grooves. The micrometer has the same 0.02mm uncertainty as in uniaxial case. The total uncertainty is mm045 0 02 0 04 02 2 Since the finite element analysis simulation is used to predict the stress due to the nonlinear bending, hence no simple equation similar to Eq. (84) can be used. Instead, the uncertainty range in displacement is projected to the stress on the fi nite element analysis calculated stress vs. displacement curves in Fig. 28 to extract the uncertainty in stress. At each pre-set

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82 -4.44745E+05 -4.44740E+05 -4.44735E+05 -4.44730E+05 -4.44725E+05 -4.44720E+05 -4.44715E+05 -4.44710E+05 -4.44705E+05 -4.44700E+05 -4.44695E+05 -4.44690E+05 01000200030004000500060007000Position x / mRadius of Curvature / m Figure 34. Extracted radius of curvature vs. position on the upper surface of the substrate in the four-point bending experiment, corresponding to the upper curve in Fig. 33. The difference between the maximum and minimum radius of curvatures is only about 0.01%.

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83 -1.46230E+08 -1.46228E+08 -1.46226E+08 -1.46224E+08 -1.46222E+08 -1.46220E+08 -1.46218E+08 -1.46216E+08 -1.46214E+08 -1.46212E+08 01000200030004000500060007000Position x / mStress / Pa Figure 35. Extracted uniaxial stress vs. posi tion on the upper surface of the substrate in the four-point bending experiment corresponding to the Fig. 34. The difference between the maximum and mi nimum stresses is only about 0.01%.

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84 0 50 100 150 200 250 300 00.10.20.30.40.50.6Displacement / mmStress/ MPa Experimental stress data extracted from curvature Calculated value from displacement Figure 36. Calibration of the four-point bendi ng experiment. The extracted stress values are close to the calculated ones using Eq. (81) and within the uncertainty range at 95% confidence level.

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85 displacement point x we project the uncertainty range at x and find the corresponding stress interval from the curve as the uncerta inty range for the stress. For example, at a smaller ring displacement of 0.114mm, the uncertainty range of the displacement is mm 045 0 114 0 114 0 045 0 114 0 and the corresponding uncertainty range of the tensile stress on the bottom plane can be found by projection as MPa6 113 8 80 8 48 or MPa8 64 8 48 6 113 and the uncertainty range for the compressive stress on the top plane is MPa4 87 8 66 2 42 or MPa2 45 4 87 2 42 Figure 37 is the demonstration of the projection approach. Us ing this approach, the uncertainty in the smaller ring displacement can then be converte d to an uncertainty in stress on the top and bottom planes. Note that this approach is only as accurate as the finite element analysis calculated stress vs. displacement curve. 4.3 Extracting Threshold Voltage, Mob ility, and Vertical Effective Field In this section, the methods of extractin g effective mobility, effective vertical electric field, and coefficients will be describe d. The effective mobility will be extracted from the drain current in the linea r region (low drain bias) for a long-channel MOSFET. At low drain bias VDS, the linear drain current of an ideal MOSFET can be approximated as DS T GS ox eff DSV V V L W C I ( 88 ) and the effective mobility eff can then be expressed as DS T GS ox DS effV V V L W C I ( 89 )

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86 -300 -200 -100 0 100 200 300 400 500 600 00.20.40.60.81Displacement of Smaller Ring (mm)Stress at Center (MPa) Bottom Plane Top Plane Figure 37. Uncertainty analysis for the co ncentric-ring bending experiment. At each preset displacement point, the displacement with the uncertainty range at 95% confidence level is projected on the stress vs. displacement curve to get the stress value with the uncertainty range in stress.

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87 where Cox is the gate oxide capacitance, W and L are the channel width and length respectively, and VGS and VT are the gate bias voltage and threshold voltage respectively. The definition of threshold voltage is illust rated in Fig. 38(a). The linear region threshold voltage is extracted by drawing a tangent line to the IDS-VGS curve at the point where the slope is the largest and extending it to intercept the x axis. The gate voltage at the intercept is defined as the threshold vo ltage. Figure 38(b) is a snapshot of the IDS-VGS curve and threshold voltage extraction using the Agilent 4155C Semiconductor Parameter Analyzer. In Fig. 38(b), an additional cu rve proportional to the gradient of the IDS-VGS curve, GS DSV I is also shown to help to determin e the point with the largest slope on the IDS-VGS curve. The effective vertical electric field Eeff is expressed as [70] s inv b effQ Q E ( 90 ) where is a fitting parameter an d equal to 1/3 for holes, Qb is the bulk depletion charge, T GS ox invV V C Q is the inversion charge, and s is the dielectric constant of silicon. At the interface of gate oxide and silicon channe l, the electric displacement continuity gives S S ox oxE E ( 91 ) where Eox is the electric field in the oxide, ox is the dielectric constant of oxide, ES is the silicon surface field at the interface, and S b inv SQ Q E ( 92 ) With Eqs. (91) and (92), the bulk depletion charge Qb can be expressed as

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88 (a) (b) Figure 38. Illustration of extracting threshold vo ltage. (a)The tangent line with the largest slope intercepts x axis at -0.4V, therefore, the threshold voltage VT= -0.4V. (b)Snapshot of the IDS-VGS characteristic and thres hold voltage extraction from Agilent 4155C Semiconductor Parameter Analyzer.

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89 inv ox ox bQ E Q ( 93 ) and the effective vertical field Eeff can then be obtained as S t GS GS ox S inv ox ox effV V V C Q E E 1 1 ( 94 ) where Eox is approximated by VGS / tox, which is valid for VDS<< VGS [70], and Cox is the gate capacitance. The uncertainty in extracted effective mobility can be estimated using Eq. (89) by evaluating the sensitivity of eff for variations in IDS, Cox, L, W, VGS, VT, and VDS as shown in Eqs. (95) and (96) [65], 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 DS DS eff T T eff GS GS eff eff eff ox ox eff DS DS eff effV V V V V V W W L L C C I I ( 95 ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 DS DS T T T GS T GS GS T GS GS ox ox DS DS DS DS DS eff eff DS T T T eff eff T GS GS GS eff eff GS eff eff eff eff ox ox ox eff eff ox DS DS DS eff eff DS eff effV V V V V V V V V V V V W W L L C C I I V V V V V V V V V V V V W W W W L L L L C C C C I I I I ( 96 )

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90 For the Agilent 4155C Semiconductor Parameter Analyzer, the measurement accuracy are 0.1% and 0.02% for current and voltage measurement [71] respectively, therefore, 6 210 DS DSI I, 8 210 4 GS GSV V, and 8 210 4 DS DSV V. According to ITRS [72], the uncertainty in gate dimens ion for 90nm technology is 3.3nm. The gate length and width of pMOSFET used in experiment are 1m and 50m respectively, therefore, 5 2 210 1000 3 3 L L and 9 2 210 4 50000 3 3 W W. The variation in threshold voltage of all measured pMOSFETs is less than 5mV. The mobility enhancement is extracted at effective fiel d 0.7MV/cm, corresponding to gate voltage VGS about 0.8V. With the threshold voltage VT=0.3V, 4 210 8 2 T TV V, 56 22 T GS GSV V V, and 36 02 T GS TV V V. For uncertainty in Cox and doping density, it can be estimated from analyzing the uncertainty in threshold voltage. The threshold voltage VT is expressed as ox B a Si B fb TC qN V V 4 2 ( 97 ) where Vfb is the flat-band voltage, B is the difference between the Fermi potential and the intrinsic potential, Si is the permittivity of silicon, q is the electron charge, and Na is the doping density. There are two main f actors that affect the uncertainty in VT, Na and Cox. As shown previously, the uncertainty in threshold voltage is 4 210 8 2 T TV V, therefore, 4 210 8 2 ox oxC C and 4 210 8 2 A AN N. The total uncertainty in extracted mobility can be estimated as (eff / eff)2 =10-6 + 2.810-4 + 10-5 + 410-9 + 2.56410-8 + 0.362.810-4 + 410-8 = 3.9210-4. Therefore, 02 0 eff eff

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91 4.4 Summary The four-point and concentric-ring bendi ng apparatus are used to apply uniaxial and biaxial stresses to the pMOSFETs. An anal ytic equation is derived for calculating the uniaxial stress from the deflection, and finite element analysis is used to determine the biaxial stress from the deflection due to the nonlinearity of the bendi ng plate. Uncertainty analysis is also performed, and the calibra tion of the uniaxial stress is within the uncertainty range of the calculated value. Th e approaches used to extract the mobility, threshold voltage, and vertical effective fiel d are described. These approaches will be used to extract mobility enhancement and calculate the coefficients in Chapter 5. The uncertainty analysis in extracted effective mobility is also performed.

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92 CHAPTER 5 RESULTS AND DISCUSSIONS 5.1 Mobility Enhancement and Coefficient versus Stress The devices used in the experiment are from 90nm technology [11, 12, 59] with 1 and 2m channel length and 12 thin gate oxide thickness. An Agilent 4155C Semiconductor Parameter Analyzer is used to measure the IDS-VGS characteristics with gate voltage VGS swept from 0 to -1.2V and drain voltage VDS fixed at -50mV. The extracted effective mobility vs. effective fiel d of the pMOSFETs before and after bending with uniaxial longitudinal and transverse and biaxial compressive and tensile stresses are shown in Figs. 39, 40, and 41, respectivel y. One important observation from these experimental results is that before and after bending, the mobility vs. effective field curves are parallel with each other in all three figures. It implies that the mobility enhancement is independent of effective field fo r all six types of stresses at this stress level and within this range of effective field. Mobility enhancement 0 where 0 is the before bending mobility, is extracted at an effective field, 0.7MV/cm and plotted vs. stress in Fig. 42. This figure compares the experimentally extracted mob ility enhancement with the model prediction (Chapters 2 and 3) and published experiment al data [73, 74] and numerical simulation [74]. Each data point has 3 to 6 devices meas ured and the average value is plotted with 95% confidence level uncertainty. The uncertainty analysis will be given in Appendix C. Good agreement is obtained between experiments and analytical model predictions

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93 45 50 55 60 65 70 75 80 85 90 95 0.70.80.91Effective Field / (MV/cm)Mobility / (cm2/V.sec) before bending longitudinal compression 226MPa 20.4% 21.1% -13.3% -13.3% longitudinal tension 226MPa Universal Mobility Figure 39. Effective hole mobility vs. effective field before and after bending with uniaxial longitudinal tensile and co mpressive stresses at 226MPa. The magnitude of mobility change is larg er for longitudinal compression than tension. The curve of universal mobility is also included fo r comparison.

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94 45 50 55 60 65 70 75 80 85 90 95 0.70.80.91Effective Field / (MV/cm)Mobility / (cm2/V.cm) transverse compression 113MPa transverse tension 113MPa 4.9% 4.9% -3.5% -4.1% before bending Universal Mobility Figure 40. Effective hole mobility vs. effective field before and after bending with uniaxial transverse tensile and co mpressive stresses at 113MPa. The magnitude of mobility change is larger for transverse tension than compression. The curve of universal mob ility is also included for comparison.

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95 45 50 55 60 65 70 75 80 85 90 95 0.70.80.91Effective Field / (MV/cm)Mobility / (cm2/V.sec) biaxial tension 303MPa biaxial compression 134MPa 3.2% 3.1% -5.1% -5.3% before bending Universal Mobility Figure 41. Effective hole mobility vs. effective field before and after bending with biaxial tensile stress at 303MPa and compre ssive stress at 134MPa. The curve of universal mobility is also included for comparison.

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96 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 -500-2500250500Stress / MPaMobility Enhancement / (/ Uniaxial Longitudinal Compression Uniaxial Transverse Compression Biaxial Compression Uniaxial Transverse Tension Biaxial Tension Uniaxial Longitudinal Tension Gallon et al. 2004 Thompson et al. 2004 Wang et al. 2004 Symbol: experimental data Line: model prediction Figure 42. Mobility enhancement vs. stress for six kinds of stresses biaxial tensile and compressive and uniaxial longitudinal and transverse, tensile and compressive. The mobility enhancements are extracted at 0.7MV/cm. The solid lines are the model pr edictions: blue: this work, orange: Wang et al. [74]. The symbols are experimental data: blue circle: this wor k, green triangle: Thompson et al. [11], orange diamond: Wang et al. [74], and purple square: Gallon et al. [73].

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97 presented in Chapters 2 and 3. The experiment al results from both Gallon et al. [73] and Wang et al. [74] are consistent with this wo rk within the estimated uncertainty. Both of them [73, 74] used four-point bending to appl y uniaxial stress. Gallon et al. used beam bending like this work except the authors measured the displacement at the center of substrate and then calculated the stress at the center of substrate 2 24 3 12a L EHycenter ( 98 ) where ycenter is the displacement at the center of substrate. Wang et al. used a more accurate and realistic approach by considering the four-point bending as 3-dimensional plate bending instead of 2-dimensional beam bending. For their theoretical stu dy, Wang et al. [74] employed a quantum anisotropic transport model using 6-band stress depe ndent kp Hamiltonian and momentumdependent scattering model. The predictions given by their numerical model for uniaxial longitudinal and transverse stresses also su pport our analytical model prediction. These comparisons provide independent corroboration of the experimental data and analytical model developed in this study. The coefficient can be calculated from th e results of mobility enhancement vs. stress using Eqs. (20) and (21). Figure 43 is a plot of extracted coefficient vs. stress. Only longitudinal and transverse, tensile and compressive coefficients for [110] direction and transverse tensile and compressive coefficients for [001] direction are provided since they pertain to [110] cha nnel direction pMOSFET on (001) surface with uniaxial and biaxial stresses. Our model predictions and the results from Smith [20], Wang et al. [74], and Gallon et al. [7 3] are also included for comparison. Good

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98 -1.5E-09 -1.0E-09 -5.0E-10 0.0E+00 5.0E-10 1.0E-09 1.5E-09 -500-2500250500stress / MPa coefficientlongitudnal [110] transverse [110]transverse [001] Smith's transverse [001] Smith's longitudnal [110] Smith's transverse [110] Gallon et al. trannsverse [110] Gallon et al. longitudinal [110] Wang et al. transverse [110] Wang et al. longitudnal [110] Symbol: experimental data Line: model prediction Figure 43. coefficient vs. stress, includi ng longitudinal and transverse coefficients for [110] direction and transverse coefficient for [001] di rection. The solid lines are the model predictions: blue: this work, orange: Wang et al. [74]. The symbols are experimental data: blue ci rcle: this work, green triangle: Smith [20], orange diamond: Wang et al. [74], and purple square: Gallon et al. [73].

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99 agreements are obtained between the experime ntal data and model predictions presented in Chapter 2. However, compared with Smith’s piezoresistance data [20], good agreement is obtained only for the longitudinal coefficient for [110] direction. For the transverse coefficients for [110] and [001] dir ections, significant deviations exist between our work, model prediction and expe rimental data, and Smith’s data [20], however, they are within the estimated un certainty. The experimental and theoretical results from Wang et al. [74] and experiment al data from Gallon et al. [73] agree with this work. The stress dependence of coefficient has been verifi ed by experimental data from this work as well as Wang et al. [74] and Gallon et al. [73]. At low stress, the extracted coefficients are smaller than the model predictions, possibly due to the uncertainty in starting point, causing the actual st ress smaller than the calculated stress. 5.2 Discussion 5.2.1 Identifying the Main Factor Contribut ing to the Stress-Induced Drain Current Change As can be seen from Eq. (88), there are fi ve possible factors that contribute to the stress-induced linear drain current change, VT, Cox, W, L, and eff. Using the same approach as shown in Eqs. (95) and (96), th e stress-induced drain current change can be expressed as 2 2 2 2 2 2 2 2 2 2 DS DS T T T GS T GS GS T GS GS ox ox eff eff DS DSV V V V V V V V V V V V L L W W C C I I ( 99 ) Take Fig. 39 with a 226MPa longitudi nal compressive stress applied along the channel as an example, the measured be fore and after bending drain currents IDS show

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100 043 02 DS DSI I. For the extracted threshold voltage VT, the difference in VT before and after bending is 2mV and the extracted threshold voltage is about 0.3V. Therefore, 5 210 4 T TV V and 36 02 T GS TV V V as described in sub-section 4.3.1, thus the contribution from the stress-induced threshold voltage variation is 0.36410-5=1.410-5 and can be neglected. This result is consistent with the published works that the threshold voltage variation du e to stress is negligible [73, 75, 76]. For a 226MPa longitudinal stress ap plied along the [110] direction, the corresponding strain is about 0.0013. (the Young’s modulus for the [110] direction is 1.6891011Pa [57].) At the interface of the gate oxide and Si channe l, the strain parallel to the interface is continuous across the inte rface. Therefore, the silicon dioxide (SiO2) will experience the same strain, about 0.0013, along the [110] direction. The Poisson ratio for SiO2 is about 0.17 [77] so the corresponding outof-plane ([001] direction) strain due to the Poisson effect is only about 2.310-4. Since ox ox oxt C 8 2 210 2 5 ox ox ox oxt t C C. Therefore, the contribution from the stress-induced change in gate oxide capacitance can be neglect ed. This result is also consistent with the observation from Matsuda and Kanda [78]. Regarding the channel width W (along the 0 1 1 direction if the channel direction is along the [110] direction), for a 226MPa longitudinal stress, due to the small Poisson ratio of Si for the [110] and 0 1 1 directions, 0.064 [57], the stress-induced change in W is very small, 9 2 210 3 7 064 0 0013 0 W W. Therefore, the stress-induced change in W can also be neglected.

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101 Since the gate VGS and drain VDS voltages are fixed (within the uncertainty of machine 410-8) before and after bend ing, there are no stress-induced changes of VGS and VDS, i.e., 0 GS GSV V and 0 DS DSV V. For the effective channel length L, the situation is more complex because the metallurgical channel length is not necessarily equal to the effective channel length [50]. For the metallurgical channel length, the cha nge due to stress is about 0.0013 for 226MPa longitudinal stress. Therefore, the stress-indu ced change in metallurgical channel length is also negligible. To experimentally study the effective channel length as a function of stress, we first use the shift and ratio meth od [50, 79] to extract the effective channel length before and after bending. Figures 44 and 45 show the effective channel length ratio of 1m pMOSFET, defined as after be nding effective channel length Leff,bending divided by before bending Leff0, 0 eff bending effL L r, for longitudinal and transverse stresses applied, respectively. In Fig. 44, the percent decrea se of the effective channel length after bending with longitudinal compre ssive stress and the percent increase with longitudinal tensile stress are close to the percent mobility enhancement and degradation respectively. It implies that the change in effective channel length might be a key factor of the stress-induced drain current change. However, as can be seen from Fig. 45, the decrease and increase of the effective channe l length after bending with transverse tensile and compressive stresses, respectively, show a similar result as the longitudinal stresses. Considering the small Poisson ratio, 0.064 [57], th e result for transverse stress in Fig. 45 is contradictory to the conclusion for longitudi nal stress in Fig. 44. This contradiction can be resolved by examining the key assumpti on of the shift and ratio method that the T GSV V dependence of effective mobility eff is a common function for all devices

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102 Longitudinal Compression, Mobility Enhancement 8.8% 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 -0.03-0.024-0.018-0.012-0.00600.0060.0120.0180.0240.03Vg shift ( ) 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1< 2> 0.918 (a) Longitudinal Tension, Mobility Enhancement -8% 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 -0.03-0.024-0.018-0.012-0.00600.0060.0120.0180.0240.03Vg shift ( )0 0.02 0.04 0.06 0.08 0.1 0.12 0.14< 2> 1.073 (b) Figure 44. Average effective channel length ratio (blue diamond) and variance <2> (orange square) vs. gate voltage shift for longitudinal stress. (a) Compression. (b) Tension. The numbers in dicated on the figures are the ratios with the minimum variance.

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103 Transverse Compression, Mobility Enhancement -4.4% 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 -0.03-0.024-0.018-0.012-0.00600.0060.0120.0180.0240.03Vg shift ( ) 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04< 2> 1.049 (a) Transverse Tension, Mobility Enhancement 3.8% 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 -0.03-0.024-0.018-0.012-0.00600.0060.0120.0180.0240.03Vg shift ( ) 0 0.02 0.04 0.06 0.08 0.1 0.12< 2> 0.965 (b) Figure 45. Average effective channel length ratio (blue diamond) and variance <2> (orange square) vs. gate voltage shift for transverse stress. (a) Compression. (b) Tension. The numbers indicated on the figures are the ratios with the minimum variance.

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104 [50]. In the experiment, the threshold voltage of pMOSFETs is essentially constant, so (VGS VT) and hence the effective mobility eff are assumed to be constant, which is incorrect for stressed pMOSFET. Therefore, the shift and ratio method is not applicable for strained-Si pMOSFET. In fact, Scott et al. [15] studied the influence of trench isolation-induced stress on the drain current of nMOSFETs. They compared the drain currents in nMOSFETs with the same phys ical gate length but different width of source/drain region, which is defined as the distance between the gate and the trench isolation along the channel direction. Smaller width of source/drain regions will induce higher longitudinal compressive stress in th e channel of the device and vice versa. They concluded that the longitudinal compressive stress induced by trench isolation degrades the electron mobility and the drain current of nMOSFETs. The effective channel lengths are essentially the same for all nMOSFETs w ith different widths of source/drain regions because the drain induced barrier lowering (D IBL) is very small between devices with different widths of source/drain. If the varia tion of the effective channel length is really a key factor, the DIBL should be much higher. Ther efore, it is the stress-induced change in the mobility and not the effective channel length that affects the drain current of nMOSFETs under the stress caused by trench isolation. We can also prove the contri bution from the change in effective channel length is negligible by examining the definition of the effective channel length. The effective channel length Leff is defined as the portion controlled by the gate [50]. If the source/drain doping is graded, Leff is greater than the metallurgical channel length Lmet because part of the source/drain overlap is also controlled by the gate [50]. Figure 46 [80] is an illustration of the definition of Leff and

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105 Figure 46. Schematic diagram of doping concen tration gradient and current flow pattern near the metallurgical junction between the source/drain and body. The dashed lines are contours of constant doping concentration. The dark region is the accumulation layer and the arrows i ndicate the position where the current starts spreading into bulk source/dr ain region. (after Taur [50].)

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106 ov met effL L L 2, ( 100 ) where Lov is the part of the source/drain overlap controlled by the gate, and Lmet is the metallurgical channel length. In Fig. 46, the metallurgical junction is defined as when the body doping Nd is equal to the source/drain doping Na, d aN N The dotted lines represent the doping gradient. The junction de pth of the source/drai n is designated by xj, and xc is the thickness of the accumulation la yer. Due to the doping gradient, the doping density near the metallurgical junction of the source/drain is not as high as in the bulk region of the source/d rain, and hence the resistivity is also higher. When a MOSFET is turned on, an inversion layer is formed in th e channel region near the interface of Si and gate oxide, while an accumulation layer is formed in the source/drain overlap region Lov. The length of Lov is determined by the point where the carrier concentration of the accumulation layer is equal to the background dop ing concentration of the source/drain or the resistivity of the accumulation layer is eq ual to the background re sistivity [80]. When the channel current flows across the metallurgi cal junction, the current will flow through the accumulation layer instead of spreading in to the bulk region of source/drain because the background doping concentration is much lower and the resistivity is much higher than the accumulation layer near the meta llurgical junction. At the point where the background doping concentratio n or resistivity is equal to the accumulation layer, the current starts spreading [80]. The distance between this point and the metallurgical junction is Lov. Since this accumulation layer is al so induced and controlled by the gate similar to the inversion layer, it needs to be included in the effective channel length [50]. When applying stress to a MOSFET, not only the effective mobility or conductivity of the channel is increasing, but the conductivity of the source/drain is also increasing and

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107 hence the resistivity is decreasing. It implies that Lov is decreasing. The amount of Lov reduction is approximately proportional to the decrease of background resistivity. According to the Internatio nal Technology Roadmap for Semiconductors [72], for 90nm technology, the extension lateral abruptness is 4.1nm/decade. Assume the worst case that Lov is about 10nm based on source/drain doping density 51019cm-3 and channel doping 1017cm-3. Applying a longitudinal compressive stress 226MPa to a MOSFET along the channel direction [110], the effective mobility or conductivity will increase about 20% and thus the resistivity will drop about 20%. It means Lov will reduce approximately 20%, to about 8nm which is the worst case si nce the piezoresistance coefficient decreases at high doping concentration [81]. For a pMOSFET with an 1 m channel length, which are used in the experiment, the change in 2 Lov is about 0.004m and the change in the metallurgical channel length is about 0.0013m. The total change in the effective channel length is about 0.0053m and 5 210 8 2 L L. This small amount of change in effective channel length cannot explain th e much larger change in the linear drain current. Therefore, the contribution from the change in effective channel length can also be neglected. From Fig. 39, the mobility enhancement at effective field 0.7MV/cm is 20.4% and 042 02 t enhancemen eff eff Considering the uncertainty in extracted mobility 4 2 y uncertaint10 92 3 eff eff as described in sub-section 4.3.1, the stress-induced mobility enhancement is still close to 043 02 DS DSI I. Based on the above discussion, we conclude that the main factor contributing to the stress-induced change in the linear drain current is effective mobility change and affirm the validity of extracting

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108 mobility enhancement from Eq. (88). It is also shown that the shift and ratio method is not applicable for extracting the effectiv e channel length from strained-Si MOSFETs. 5.2.2 Internal Stress in the Channel The 90nm technology [11, 12, 59] used SiGe in source and drain to induce about 600MPa longitudinal compressive stress in the channel of a 45nm channel length pMOSFET as shown in Fig. 3 [14]. For longer channel pMOSFETs, the channel compressive stress from SiGe source/drain is reduced as shown by the stress simulation results using the FLOOPS-ISE process simulato r in Fig. 47 [82, 83]. In Fig. 47, the channel length is 1m; outside the source/drain is shallo w trench isolation (STI); and the source/drain is filled with Si0.83Ge0.17 as shown in Fig. 3. The longitudinal compressive stress at the center of the ch annel is reduced to about 85M Pa due to the longer channel length while it is about 200MPa near the source/drain metallurgi cal junction. When reducing the channel length to 45nm, the simulated channel compressive stress is increased to about 600MPa, consistent with the simulation result shown in Fig. 3. Therefore, the 1 m long channel length device was used for the mobility extraction to investigate the effect of applied external stress. 5.2.3 Stress-Induced Mobility Enhancement at High Temperature So far we only discussed the models and experiments at room temperature. However, the devices normally operate not at room temperature but at a much higher temperature, about 100C. In this subs ection, the model pred iction of mobility enhancement at 100C will be given and compar ed to the one at room temperature. The experimental data will also be provided.

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109 Figure 47. Simulation result of the internal stress distribution in a pMOSFET. The channel length is 1m and the source/drain is built with Si0.83Ge0.17. Outside the source/drain is shallow trench isol ation (STI). Near the Si channel and gate oxide interface, the stress is about -85MPa (compressive stress) at the center and about -200MPa (compressive st ress) near the source/drain junction.

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110 Figure 48 is a plot of mobility enhancement vs. stress at room temperature and 100C for both model prediction and experime ntal data: the blue lines and symbols are for room temperature, while the orange ones are for 100C. As can be seen from Fig. 48, the model prediction of the mobility degradation is smaller at 100C than at room temperature for the longitudinal tension, tr ansverse compression, and biaxial tension. However, the transverse tension and biaxial compression show slig htly higher mobility enhancement at 100C than at room temper ature. For the longitudinal compression, the mobility enhancement at 100C is slightly smaller than at room temperature. The reasons for these observations are attributed to two f actors: (i) smaller stress-induced hole transfer and (ii) interband optical phonon scattering becomes more important at higher temperature. As can be seen from Eq. (29), th e hole transfer is inversely proportional to the temperature. At 100C, the amount of hole transfer is smaller than at room temperature, thus the mobility change will al so be smaller than at room temperature. However, at high temperature, the influe nce of interband optical phonon scattering becomes more important. The suppression of interband optical phonon scattering due to the stress-induced band splitting will help more at higher temperature. This will result in even less mobility degradation for the longi tudinal tension, transv erse compression, and biaxial tension. Regarding the longitudinal compression, transverse tension, and biaxial compression, although the mobility enhancement is reduced at high temperature due to less hole transfer, the reduced interband op tical phonon scattering will compensate the reduction in mobility enhancement, for exam ple, longitudinal compression, or even cause slightly higher mobility enhancement, for example, transverse tension, and biaxial compression.

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111 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 -500-2500250500Stress / MPaMobility Enhancement / (/ Uniaxial Longitudinal Compression Uniaxial Transverse Compression Biaxial Compression Uniaxial Transverse Tension Biaxial Tension Uniaxial Longitudinal Tension Room Temperature 100C 100C 100C 100C Room Temperature Room Temperature Room Temperature Figure 48. Mobility enhancemen t vs. stress at room temperature and 100C for six kinds of stresses, biaxial tensile and compressive and uniaxial longitudinal and transverse, tensile and compressive. The solid lines are the model predictions: blue: at room temperature, orange: at 100C. The symbols are experimental data: blue circle: at room temperat ure, orange triangle: longitudinal compression at 100C, and orange diam ond: transverse compression at 100C.

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112 As for experiment, there are only two data po ints shown in Fig. 48 due to the difficulty of measurement. Most of the devices either failed (gate oxide breakdown) or became leaky (higher gate leakage current) at 100C befo re or after bending and these failures are permanent. Figures 49 and 50 show the before and after bending IDS-VGS characteristics at 100C under normal and leaky conditions, and Fig. 51 illustrates the IDS-VGS characteristics at 100C when the device fail s (gate oxide breakdown). As seen from Fig. 50, when the device becomes leaky, part of the drain current will become gate tunneling current and the extracted mobility enhancement will be unreliable. During the bending experiment at room temperature, the same failures have already been observed. The devices show normal char acteristics before bending and then become leaky or failed as the stress increases. At high temperature, the situation becomes worse. The devices show higher gate leakage current or failure after the temperature rises to 100C even before bending. Based on the observations from the experiments, the possible root cause for the failure may be dr awn as the following. The physical gate oxide thickness of the devices is 1.2nm, which has fi ve layers of silicon atoms. Excluding two of the silicon atoms forming two interfacial laye rs at the interfaces of (i) channel and gate oxide and (ii) polysilicon gate and gate oxide, there are only three layers of “bulk” SiO2, in the gate oxide [84, 85]. This ultra-thin ga te oxide is very fragile, and the combination of the external mechanical stress and high oxide electric field (more than 5MV/cm) may very likely damage the gate oxide and cause higher gate leakage cu rrent or even gate oxide breakdown. With high temperature, the situ ation deteriorates and the ultra-thin gate oxide becomes even more fragile. The possible solution to these problems is to use devices with thicker gate oxide. To our best knowledge, so far, there is no report about

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113 -250 -200 -150 -100 -50 0 -1.2-0.9-0.6-0.30VG / VCurrent / AIG Before Bending IG After Bending ID After Bending ID Before Bending Figure 49. Before and after bending drain current ID and gate current IG vs. gate voltage VG. The after bending gate current coinci des with the before bending one. The extracted mobility enhancement is reliable.

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114 -200 -150 -100 -50 0 -1.2-0.9-0.6-0.30VG / VCurrent / AIG Before Bending IG After Bending ID After Bending ID Before Bending Figure 50. Before and after bending drain current ID and gate current IG vs. gate voltage VG. The after bending gate current is much higher than the before bending one. It implies that part of the drain current ID becomes the gate tunneling current IG. Although the device is s till working, the gate oxide is leaky and the extracted mobility enhancement is unreliable. This is permanent damage to gate oxide.

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115 -200 -150 -100 -50 0 50 100 150 200 -1.2-0.9-0.6-0.30VG / VCurrent / AID Before Bending IG After Bending ID After Bending IG Before Bending Figure 51. Before and after bending drain current ID and gate current IG vs. gate voltage VG. The after bending gate current is ex tremely higher than the before bending one. In fact, the after bending drain cu rrent reverses the direction and provides the current for gate current like source curr ent. It implies that the gate oxide is breakdown and the device is no longer fu nctional. This is also permanent damage to gate oxide.

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116 the gate oxide failure due to applying exte rnal bending stress and high temperature during measurements. However, the devices used by ot her groups have thicker gate oxide than ours. For example, the gate oxide thicknesses in the devices used by Gallon et al. [73, 75] are 2 and 6.5nm, Zhao et al. [76] used devi ces with 8nm thick gate oxide, the devices used by Haugerud et al. [86] have 8.5 and 17nm thick gate oxide, and Uchida et al. [69] used devices with the gate oxide thickness 10 nm. Based on the above reviews, the ultrathin gate oxide may be the root cause of the failure and increasing the gate oxide thickness may improve the mechanical bending probing yield. Haugerud et al. [86] reported that the stress-induced mobility enhancement or degradation at temperature range from 296 to 367K does not change significantly. However, the uniaxial strain they applied to the substrate is only 0.044%, corresponding to uniaxial stress about 74MPa. With this small stress, our model prediction and experimental data also showed very little ch ange in mobility enhancement or degradation at 100C (373K), which is consis tent with their results [86]. 5.2.4 Stress-Induced Gate Leakage Current Change Applying stress to a pMOSFET, the gate leakage current will also change. Figure 52 shows stress-induced gate leakage current change vs. stress for biaxial and uniaxial longitudinal and transverse stresses. For all tensile stresses, the gate leakage current increases as the stress increases. However, fo r all compressive stresses, the gate leakage current decreases as the stress increases. These can be explained by hole repopulation and out-of-plane effective masses of heavy and light holes. The detailed explanation of underlying mechanism is given in Ref. [87].

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117 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 -400-300-200-1000100200300400Stress / MPa Jg/Jg transverse stress longitudinal stress biaxial stress Figure 52. Stress-induced gate leakage current change vs. stress. For all tensile stresses, the gate leakage current increases as the stress increases; for all compressive stresses, the gate leakage current de creases as the stress increases.

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118 5.3 Summary The hole mobility is extracted from the drain current in the linear region, and the mobility enhancements extracted at the effect ive field of 0.7MV/cm agree well with the model prediction for all six kinds of stresses. The coefficients are also calculated from the mobility enhancement vs. stress data a nd agree well with the model predictions. The main factor that contributes to the stre ss-induced drain current increase is identified as the effective mobility change. The contribution from the change of the effective channel length is proved to be negligible. The internal channel stress in pMOSFETs due to the SiGe source/drain is also determined by simulation for 1m pMOSFETs. The mobility enhancement at high temperature is also studied. The net effect of temperature on strain-induced mobility enhancement is attributed to two key factors: hole repopulation and suppression of interband optical phonon s cattering. The experimental data of stress-induced gate leakag e current change is also presented.

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119 CHAPTER 6 SUMMARY, CONTRIBUTIONS, AND RECOMMENDATIONS FOR FUTURE WORK 6.1 Summary The main purpose of this disse rtation is to create a simple, analytical model that can be easily understood and provide quick, accurate predictions for piezoresistance in p-type silicon and mobility enhancement in strained-Si pMOSFETs. In Chapter 2, starting from the valence ba nd theory and 44 stra in Hamiltonian, the stress-induced valence band alteration, heavy and light hole effective mass changes, and hole repopulation due to the stress-induced band splitting are discussed and calculated. An important result is obtained: the coefficients are stressdependent, different from Smith’s data, and the incr ease or decrease of coefficients depend on the type of stresses. The main reason is the stress-de pendent hole repopulation. Depending on the type of stresses, one of the two hole bands will rise above the other one, and the holes will transfer from the lower hole band to upper one, causing the stress dependence of coefficients, noting that hole energy is the negative of the electron energy. Based on the model developed in Chapter 2, mobility enhancements in biaxial and uniaxial strained-Si pMOSFETs are discusse d in Chapter 3. For uniaxial strained-Si pMOSFETs, good agreement with the published work is obtained. For biaxial strained-Si pMOSFETs, a contradictory result from the p ublished theoretical work is obtained: at low biaxial tensile stress, the hole mobility is actually degraded instead of improved. However, this result is supported by other published experimental works. The reason for

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120 this result can be attributed to hole transfer This result strongly support the analytical model and, especially, the concept of hole transfer. In Chapters 4 and 5, four-point and c oncentric-ring bending experiments for uniaxial and biaxial stresses respectively ar e performed and the resu lts support the model prediction. Compared with other published ex perimental works, good agreements are also obtained. The main factor contributing to th e stress-induced drain current increase is identified as the mobility change among many other factors. The internal channel stress due to the SiGe source/drain is also exam ined by simulation. The mobility enhancement vs. stress at high temperatur e is discussed. The model shows that hole transfer at high temperature will be reduced and cause less mo bility changes. The contribution from the suppression of interband optical phonon scat tering will become more important than at room temperature. Only limited experimental data are available due to the difficulty of experiments. Finally, the experimental data of stress-induced gate leakage current change is presented. 6.2 Contributions The contributions of this work are summarized as follows. First, the analytical model predicts that coefficient has stress dependence and the prediction is confirmed by the experiment. Th e main reason is due to stress-dependent hole repopulation. The original data by Smith [20] on piezoresistance of bulk silicon was obtained with applied stresses of 106 N/m2 (1 MPa) to 107 N/m2 (10 MPa). Within this range of stress, the stress dependence of coefficient is very small. However, at larger stress (>100MPa), the hole repopulation becomes important and coefficient shows

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121 stress dependence. Previous works [19, 22, 23] gave a prediction on coefficient, and yet, did not discuss the stress dependence. Second, the model predicts that, at low bi axial tensile stress (<500MPa), the hole mobility is actually degraded. This prediction is confirmed by the experimental data from this work and also by the previous works [5, 8]. As shown in Fig. 22, the model prediction from Oberhuber et al. [3] shows hole mobility enhancement at low biaxial stress, which is contradictor y to the experimental data. Third, as shown in Fig. 48, the theoreti cal study shows that, for a longitudinal compressive stress applied along the channel [110] direction, which is the most important approach to increase hole mobility [10-13], the mobility enhancement is not significantly affected by high temperature (100C), alth ough the hole mobility decreases as the temperature increases. This theoretical result is consistent with the work from Haugerud et al. [86]. 6.3 Recommendations for Future Work Rapid scaling of CMOS technology has driven the channel length into the nanometer regime. Recently, strained silicon technology has become an extensive research subject because it can enhance the carrier mobility and continue device performance improvement as discussed in prev iously chapters. However, as the channel length continues to shrink, the importance of mobility is not clear and improving the mobility using strained silicon technology may or may not be helpful. In addition, we will still have to face the ultimate limit, zer o channel length, and then finally, CMOS technology will probably stop improving. Many excellent works studying the performance limits of the CMOS devices [30, 88-93] have been published. The question

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122 is: is it possible to break the performan ce limit? Based on this work, the answer is probably yes and will be recommended as the future work. In a nano-pMOSFET, hole transport may be ballistic and the concept of mobility may not be valid. However, as discussed in Chapter 3, the main mechanism contributing to mobility enhancement is hole repopulation, i.e., holes transfer from the lower hole band with heavier in-plane effective mass to the higher one with light er in-plane effective mass. The hole transfer and effective mass c oncepts are derived from the valence band theory and strain Hamilton and should still be valid in nano-scale hole transport. For nano-device operation, the drain current is found to be [30] DS T GS B T ox DSV V V q T k r v WC I 2 1, q T k VB DS ( 101 ) or T GS T ox DSV V r r v WC I 1 1, q T k VB DS, ( 102 ) where vT is the source injection velocity [90] given by *2m T k vL B T, ( 103 ) TL is the lattice temperature, kB is the Boltzmann constant, and m* is the effective mass. In Eq. (103), the silicon is assumed to be nondegenerate and the degeneracy factor approaches one [90]. In Eqs. (101) and (102), r is the channel back-scattering coefficient, 1 0 r and [30] l l l r 1 1, ( 104 )

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123 where is the mean free path, and l is the distance it takes for the carrier to drift ~kBT/q down the channel from the top of the source barrier. When r approaches zero, the carrier approaches the ballistic limit, and Eq. (102) becomes *2m T k V V WC V V WC IL B T GS ox T T GS ox DS q T k VB DS. ( 105 ) Equation (105) sets an upper limit of the device performance due to the thermal injection velocity from the source. To break it, one possible way is to increase the thermal injection velocity by changing the effective mass m*. In Eq. (103), the effective mass in the source injection velocity is the average effective mass of the heavy and light holes, neglecting the split-off hole. Applying stress w ill cause hole transfer and change the hole repopulation and average hole effective mass, the same mechanism described in Chapter 2. By applying the right stress, the hole re population will reduce the average effective mass, increase the source injection velocity in Eq. (105) and break the performance limit. The same idea can also be applied to nMOSFETs because the main mechanism of the stress-induced electron mobility enhancement is also the conduction band structure alteration and electron repopulation [21, 23, 94, 95]. The by-product of this future work is to verify which model is valid in nano-scale devices. As can be seen from Eq. (19), the mobility enhancement has a different formula from the source injection velocity enhancement in terms of the effective mass dependence. By fitting both models to the experimental data, one will be able to verify which model will hold in the nanometer regime, mobility or ballistic transport.

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124 APPENDIX A STRESS-STRAIN RELATION In this appendix, we will briefly review the stress-strain relation and define stiffness cij and compliance sij coefficients of material. We start from the definitions of stress and strain [77]. Stress is define d as the force per unit area. Stress acting perpendicularly on a surface is called uni axial or normal stress and labeled as Stress acting in parallel on a surface is ca lled shear stress and labeled as Figure 53 illustrates the definitions of the two kinds of stresses. In static equilibr ium, there are no net forces or torques, therefore, in Fig. 53, yx xy zx xz and zy yz Strain is defined as the change in length per unit length. Uniaxial or normal strain, corresponding to uniaxial stress, is defined as [77] dx du x x u x x ux x x x ( 106 ) Shear strain, corresponding to sh ear stress, is defined as [77] dx du dy du x u y uy x y x xy. ( 107 ) Figure 54 illustrates the definitions of uniaxial and shear strains. With the definitions of stress and strain, the stress-strain relation can be expressed as [77]

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125 Figure 53. Definitions of uniaxial stresses, x, y, and z, and shear stresses, xy, yx, xz, zx, yz, andzy. In static equilibr ium, it requires that yx xy zx xz and zy yz (Adapted from Ref. [18])

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126 Figure 54. Definitions of strain. (a) Uniaxial strain. (b) Shear strain. (Adapted from Ref. [77])

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127 xy zx yz z y x xy zx yz z y xc c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c 66 56 46 36 26 16 56 55 45 35 25 15 46 45 44 34 24 14 36 35 34 33 23 13 26 25 24 23 22 12 16 15 14 13 12 11, ( 108 ) where cij, i, j =1..6, are the matrix elements of the matrix C and called stiffness coefficients. The stress-strain relation can also be expressed as xy zx yz z y x xy zx yz z y xs s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s 66 56 46 36 26 16 56 55 45 35 25 15 46 45 44 34 24 14 36 35 34 33 23 13 26 25 24 23 22 12 16 15 14 13 12 11, ( 109 ) where sij, i,j,=1..6, are the matrix elements of the matrix S and called compliance coefficients. As can be seen from Eqs. (108) and (109), the matrix S is the inverse of the matrix C. For a cubic material such as single-cryst al silicon, there are only three independent matrix elements in the matrix C and S and the matrixes are simplified to [77] 44 44 44 11 12 12 12 11 12 12 12 110 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c c c c c c c c c c c c C, ( 110 )

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128 44 44 44 11 12 12 12 11 12 12 12 110 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 s s s s s s s s s s s s S. ( 111 ) The matrix elements sij of the simplified matrix S can be expressed in terms of the matrix elements cij of the simplified matrix C [57], 12 11 12 11 12 11 112c c c c c c s ( 112 ) 12 11 12 11 12 122c c c c c s ( 113 ) 44 441c s ( 114 ) For silicon, the values of stiffness cij and compliance sij coefficients [57] are listed in Table 4. Table 4. Stiffness cij, in units of 1011Pa, and compliance sij, in units of 10-11Pa-1, coefficients of silicon [57]. c11 c12 c44 s11 s12 s44 1.657 0.639 0.7956 0.768 -0.214 1.26

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129 APPENDIX B PIEZORESISTANCE COEFFICIEN T AND COORDINATE TRANSFORM In this appendix, we will define the piezoresistance coefficients. For a cubic crystal such as sili con, the piezoresistance coefficients are defined as [18] 3 2 1 3 2 1 44 44 44 11 12 12 12 11 12 12 12 11 6 5 4 3 2 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ( 115 ) where x 1, y 2, and z 3 are uniaxial stresses and yz 1, zx 2, and xy 3 are shear stresses as defined in Fig. 53. In order to obtain the coefficient along an arbitrary direction, the c oordinate transform of the coefficient is needed. One coordinate system 3 2 1, ,x x x can be transformed into another coordinate system 3 2 1, ,x x x by coordinate transformation: 3 2 1 33 32 31 23 22 21 13 12 11 3 2 1x x x a a a a a a a a a x x x, ( 116 ) where aij, i,j=1, 2, 3 are the direction cosines, the cosine of the angle between 'ix and jx. Along an arbitrary direction, two coefficients are defined: longitudinal and transverse coefficients. The longitudinal coefficient is defined when the uniaxial stress, electric field and electric current are all in the same direction as shown in Fig. 1. The transverse coefficient is defined when the electric field is parallel to the electric current but normal

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130 to the direction of the uniaxial stress as also shown in Fig. 1. Using Eqs. (115) and (116) and some mathematical manipulations, we can calculate the longitudinal coefficient, l and transverse coefficient, t along an arbitrar y direction [18]: 2 13 2 12 2 13 2 11 2 12 2 11 11 12 44 112a a a a a al ( 117 ) 2 23 2 13 2 22 2 12 2 21 2 11 11 12 44 122a a a a a at ( 118 ) where 11a, 12a, 13a and 21a, 22a, 23a are the direction cosines between the longitudinal and transverse directions and the axes of the coordinate system, x, y, and z respectively. For the longitudinal direction along [001] di rection and the transverse direction along [100] direction, 011 a, 012 a, 113 a, and 121 a, 022 a, 023 a. Substituting these values into Eqs. (117) and (118), we can get 11 ] 001 [ l and 12 ] 001 [ t. For the longitudinal direction along [111] direc tion and the transverse direction along ] 0 1 1 [ direction, 3 111 a, 3 112 a, 3 113 a, and 2 121 a, 2 122 a, 023 a, and the longitudinal and transverse coefficients are obtained as 44 12 11 ] 111 [2 2 3 1 l and 44 12 11 ] 111 [2 3 1 t. For the longitudinal direction along [110] direction, th ere are two transverse directions ] 0 1 1 [ and [001], labeled as 2 and 3 respectively, and the direction cosines are 2 111 a, 2 112 a, 013 a, 2 121 a, 2 122 a, 023 a, and 031 a, 032 a, 133 a. Substituting the values of a’s into Eqs. (117) and (118), we obtain 44 12 11 ] 110 [2 1 l, 44 12 11 ] 110 [ 22 1 t, and 12 ] 110 [ 3 t. In Table 5, we summarize the above calculation results [18].

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131 Table 5. Longitudinal and transverse coefficients for [001], [ 111], and [110] directions [18]. Longitudinal Direction l or || Transverse Direction t or [001] 11 [100] 12 [111] 44 12 112 2 3 1 ] 0 1 1 [ 44 12 112 3 1 ] 0 1 1 [ 44 12 112 1 [110] 44 12 112 1 [001] 12

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132 APPENDIX C UNCERTAINTY ANALYSIS C.1 Systematic and Random Errors The difference between the measured value an d the true value is defined as the total measurement error (). As shown in Fig. 55(a) [65], the total error is the sum of the systematic (or bias) error and the random (or precision) error. The systematic error () is the fixed or constant component of the total er ror, for example, instrument resolution, and is sometimes referred to as the bias [65]. The system error can be reduced by calibration. The random error () is the uncertainty that changes with each measurement and sometimes referred to as the repeatability, re peatability error, or precision error [65]. Figure 55(b) shows the k and k+1 measurements after making many measurements. The systematic error is the same for each measurement because it is a fixed error. However, the random error is different for each measurement. Therefore, the total error is different for each measurement and i i [65]. If the number of measurements approaching infinity, the data would become as shown in Fig. 55(c). The systematic error is given by the difference between the mean () and the true value X. The distribution of measurement around the mean is caused by the random error. Figure 56 [96] is an illustration of error types in gun shots at a target. In Fig. 56(a), the systematic error is large and the random error is small while Fi g. 56(b) shows the oppo site. In reality, the true value is often unknown, therefore, the ex act systematic and random errors can not be specified in the measurement.

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133 Figure 55. Measurement errors in X. (a) Si ngle measurement. (b) Two measurements. (c) Infinite number of measurement. (Adapted from Ref. [65]).

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134 Figure 56. Random and bias errors in gun sh ots. (a) Large bias error and small random error. (b) Small bias error and large ra ndom error. (Adapted from Ref. [96]).

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135 To make a statement about the true valu e X based on the measurements, the best way to say is that the true value X lies within Xbest UX with C% confidence [65]. Xbest is assumed as the mean value of the N measurements and UX is the uncertainty in X with the combination of systematic and random uncertainties with C% confidence. For example, 90% confidence estimate of UX means that about 90 times out of 100, Xtrue would be in Xbest UX. C.2 Uncertainty Analysis in Figures 36 and 42 In Fig. 36, there are four data points, th e first point has nine measurements, the second one has eight measurements, the third one has three measurements, and the fourth one also has three measurements. The uncertainty is performed as follows. Take the first point as an example. First, the standard deviation S of nine measurements is calculated. Using t distribution with 9–1=8 degree of free dom, the uncertainty of nine measurements is expressed as p=tS, where t=2.306 with 8 degree of freedom and 95% confidence level [65], the standard deviation S is 7MPa. The uncertainty of nine measurements is then 2.306716MPa. The rest of data points use the same procedure to calculate the uncertainty in stress. Table 6 lists all the experimental data used in Fig. 36. Table 6. Experimental data used in Fig. 36. Displacement (mm) 0.11 0.23 0.34 0.46 69 152 201 256 66 136 179 246 73 145 182 245 72 146 66 116 62 119 57 134 59 124 Stress (MPa) 53 Average (MPa) 64 134 187 249 Standard Deviation (MPa) 7 13 12 6 95% Uncertainty t (MPa) 16 32 51 26

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136 In Fig. 42, each data point has three to si x devices measured. The error bar of stress is obtained from sub-section 4.2.3. The error ba r of mobility enhancemen t is calculated as follows. The uncertainty in extracted effective mobility described in sub-section 4.3.1 is 02 0 eff eff The extracted mobility enhancem ent is calculated as shown in Figs. 39, 40, and 41. At each data point, based on the number of devices measured, the average value of the measurements and th e standard deviation are calculated. The uncertainty p is then calculated using t di stribution, p=tS, as described above. This uncertainty p is random uncerta inty. The total uncertainty U is obtained using root-sumsquare method 2 2p U [65]. The mobility enhancement experimental data / and total uncertainty U for uniaxial longitudinal and transverse and biaxial stresses are listed in Tables 7, 8, and 9 respectively. Table 7. Mobility enhancement experime ntal data and uncertainty for uniaxial longitudinal stresses. Longitudinal Stress (MPa) -226 -170 -113 -57 57 113 170 226 283 Uncertainty (MPa) 50 50 50 50 50 50 50 50 50 0.198 0.134 0.090 0.040 -0.028 -0.065 -0.098 -0.140 -0.170 0.204 0.145 0.067 0.049 -0.024 -0.054 -0.092 -0.133 -0.166 0.214 0.153 0.101 0.044 -0.029 -0.059 -0.103 -0.146 -0.179 0.226 0.162 0.078 0.054 -0.034 -0.084 -0.129 -0.149 -0.185 0.100 Mobility Enhancement / 0.110 Average 0.211 0.149 0.091 0.047 -0.029 -0.065 -0.105 -0.142 -0.175 Standard Deviation 0.012 0.012 0.016 0.006 0.004 0.013 0.016 0.007 0.009 Random Uncertainty (p) 95% t 0.040 0.038 0.041 0.020 0.013 0.043 0.051 0.023 0.027 Systematic Uncertainty ( ) 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 Total Uncertainty (U) 0.044 0.043 0.046 0.028 0.024 0.047 0.055 0.031 0.034

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137 Table 8. Mobility enhancement experimental data and uncertainty for uniaxial transverse stresses. Transverse Stress (MPa) -113 -57 57 113 170 226 283 Uncertainty (MPa) 50 50 50 50 50 50 50 -0.044 -0.016 0.0140.0330.057 0.082 0.111 -0.045 -0.009 0.0120.0310.053 0.079 0.103 -0.035 -0.020 0.0130.0350.058 0.087 0.107 -0.038 0.0140.0350.056 0.087 0.111 Mobility Enhancement / -0.021 Average -0.037 -0.015 0.0130.0330.056 0.084 0.108 Standard Deviation 0.010 0.006 0.0010.0020.002 0.004 0.004 Random Uncertainty (p) 95% t 0.027 0.024 0.0040.0050.006 0.013 0.012 Systematic Uncertainty ( ) 0.020 0.020 0.0200.0200.020 0.020 0.020 Total Uncertainty (U) 0.033 0.032 0.0200.0210.021 0.024 0.023 Table 9. Mobility enhancement experimental data and uncertainty for biaxial stresses. Biaxial Stress (MPa) -133 -111 -67 81 162 236 303 Uncertainty ( ) -6 -15 -25 34 31 29 26 Uncertainty ( + ) -3 -11 -21 33 30 28 25 0.0290.0220.007-0.015 -0.019 -0.046 -0.061 0.0320.0210.005-0.007 -0.016 -0.063 -0.074 0.0290.0200.005-0.024 -0.024 -0.041 -0.055 Mobility Enhancement / 0.005 -0.014 Average 0.0300.0210.006-0.010 -0.018 -0.050 -0.063 Standard Deviation 0.0020.0010.0010.012 0.004 0.012 0.010 Random Uncertainty (p) 95% t 0.0090.0060.0050.039 0.014 0.050 0.042 Systematic Uncertainty ( ) 0.0200.0200.0200.020 0.020 0.020 0.020 Total Uncertainty (U) 0.0220.0210.0210.044 0.025 0.054 0.047

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PAGE 157

146 BIOGRAPHICAL SKETCH Kehuey Wu was born in Kaohsiung, Taiwan, in 1969. He received his bachelor’s degree in electrophysics from the National Chiao Tung University, Taiwan, in 1992 and his Master of Science degree in electrical a nd computer engineering from the University of Florida in 1999, where he is currently pursuing a Ph.D. degree focusing his research on strain effect on valence band of silicon, incl uding piezoresistance in p-type silicon and mobility enhancement in strained silicon pMOSFET.


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Title: Strain Effect on the Valence Band of Silicon: Piezoresistance in P-Type Silicon and Mobility Enhancement in Strained Silicon pMOSFET
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Title: Strain Effect on the Valence Band of Silicon: Piezoresistance in P-Type Silicon and Mobility Enhancement in Strained Silicon pMOSFET
Physical Description: Mixed Material
Copyright Date: 2008

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STRAIN EFFECTS ON THE VALENCE BAND OF SILICON: PIEZORESISTANCE
IN P-TYPE SILICON AND MOBILITY ENHANCEMENT IN STRAINED SILICON
PMOSFET















By

KEHUEY WU


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2005

































Copyright 2005

by

Kehuey Wu
















TABLE OF CONTENTS

Page

L IS T O F T A B L E S ................................................. ................................. ..................... v

LIST OF FIGURES .................................................. ............................ vi

A B S T R A C T ........................................................................................................ ........ .. x

CHAPTER

1 INTRODUCTION .................... ............ ..............................

1.1 Relation of Strained Silicon CMOS to Piezoresistive Sensor and
P iezoresistivity .................................................................. .. ............
1.2 M otiv atio n ......................................................... ................. ...... .... .......... 4
1.3 Valence Band Structure- A Preview .............. ....................................8...
1.4 Focus and Organization of Dissertation................................ ................ 10

2 STRAIN EFFECTS ON THE VALENCE BAND AND PIEZORESISTANCE
M O D E L ................................................................................................................ 12

2 .1 Introduction ............... .. ............................ .............. .............. 12
2.2 Review of Valence Band Theory and Explanations of Strain Effects on
V alence B and ................................................................... ..... ...... ........... 13
2.3 Modeling of Piezoresistance in p-Type Silicon....................................23
2.3.1 Calculations of Hole Transfer and Effective Mass.....................23
2.3.2 Calculation of Relaxation Time and Quantization-Induced Band
Splitting .................................................................................. 30
2 .4 R results and D iscu ssion ......................................................... ................ 37
2 .5 S u m m ary .................................................................................................... 4 5

3 HOLE MOBILITY ENHANCEMENT IN BIAXIAL AND UNIAXIAL
STR A IN ED -SIL ICON PM O SFET ........................................................................49

3.1 Introduction ........................................... .............. ... ............... 49
3.2 Mobility Enhancement in Strained-Silicon PMOSFET..........................52
3.3 D iscu ssion ......................................................................................... 58
3 .4 S u m m ary .................................................................................................... 6 0









4 WAFER BENDING EXPERIMENT AND MOBILITY ENHANCEMENT
EXTRACTION ON STRAINED-SILICON PMOSFETS...............................61

4 .1 In tro d u ctio n .................... .. ...... ........ ....... ................. ..................... 6 1
4.2 Wafer Bending Experiments on pMOSFETs ......................................62
4.2.1 Four-Point Bending for Applying Uniaxial Stress......................62
4.2.2 Concentric-Ring Bending for Applying Biaxial Stress ..............67
4.2.3 U uncertainty A nalysis................................................. ................ 74
4.3 Extracting Threshold Voltage, Mobility, and Vertical Effective Field .....85
4 .4 S u m m ary .................................................................................................... 9 1

5 RESU LTS AND D ISCU SSION S ..................................................... ................ 92

5.1 Mobility Enhancement and 7t Coefficient versus Stress.........................92
5.2 D iscu ssion ......................... .. ..................... .. ... ... ............. 99
5.2.1 Identifying the Main Factor Contributing to the Stress-Induced
D rain C current C change ............................................... ................ 99
5.2.2 Internal Stress in the Channel ....................................................108
5.2.3 Stress-Induced Mobility Enhancement at High Temperature...... 108
5.2.4 Stress-Induced Gate Leakage Current Change ..........................116
5.3 Sum m ary .......................................................................................... 118

6 SUMMARY, CONTRIBUTIONS, AND RECOMMENDATIONS FOR
FU TU R E W O R K ..... .................................................................. ............... 119

6.1 Sum m ary .......................................................................................... 119
6.2 Contributions ..................................................... .... ............... ........ ..... 120
6.3 Recommendations for Future Work...... ....................................... 121

APPENDIX

A STRESS-STRA IN RELA TION ...........................................................................124

B PIEZORESISTANCE COEFFICIENT AND COORDINATE TRANSFORM. 129

C U N CER TA IN TY AN A LY SIS ............................................................................132

LIST O F R EFEREN CE S .. .................................................................... ............... 138

BIOGRAPH ICAL SKETCH .................. .............................................................. 146















LIST OF TABLES


Table Page

1 Values of the inverse mass band parameters and deformation potentials used in
th e calcu latio n s ......................................................................................................... 2 9

2 Calculated zeroth-order longitudinal ( || ) and transverse ( L ) stressed effective
masses of the heavy and light holes (in units of mo) for [001], [111], and [110]
direction s. ............. ................................................ ..................................... . 3 0

3 In- and out-of-plane effective masses of the heavy and light holes for uniaxial
com pression and biaxial tension ......................................................... ................ 54

4 Stiffness c,, in units of 101Pa, and compliance s,, in units of 10-1Pa-1,
coefficients of silicon. ............. ................ .............................................. 128

5 Longitudinal and transverse coefficientss for [001], [111], and [110]
d ire ctio n s. ............................................................................................................... 1 3 1

6 Experim ental data used in Fig. 36 ....... ...... ..... ..................... 135

7 Mobility enhancement experimental data and uncertainty for uniaxial
longitudinal stresses. ............. ................ .............................................. 136

8 Mobility enhancement experimental data and uncertainty for uniaxial transverse
store s se s ................................................................................................................ ... 1 3 7

9 Mobility enhancement experimental data and uncertainty for biaxial stresses......137















LIST OF FIGURES


Figure Page

1 Definitions of longitudinal and transverse directions for defining zcoefficients...... 3

2 Schematic diagram of the biaxial strained-Si MOSFET on relaxed Sil-xGex
lay er .......................................................................................... ............. ...... 5

3 Schematic diagram of the uniaxial strained-Si pMOSFET with the source and
drain refilled with SiGe and physical gate length 45nm ................. ..................... 6

4 Strain effect on the valence band of silicon .......................................... ...............9...

5 E-k diagram and constant energy surfaces of the heavy- and light-hole and
split-off bands near the band edge, k=0, for unstressed silicon ............................ 15

6 E-k diagram and constant energy surfaces of the heavy- and light-hole and
split-off bands near the band edge, k=0, for stressed silicon with a uniaxial
compressive stress applied along [001] direction. .............................. ................ 18

7 E-k diagram and constant energy surfaces of the heavy- and light-hole and
split-off bands near the band edge, k=0, for stressed silicon with a uniaxial
com pressive stress applied along [111] direction. .............................. ................ 19

8 E-k diagram and constant energy surfaces of the heavy- and light-hole and
split-off bands near the band edge, k=0, for stressed silicon with a uniaxial
com pressive stress applied along [110] direction. .............................. ................ 21

9 Top view (observed from [110] direction) of the constant energy surfaces with
a uniaxial compressive stress applied along [110] direction................................22

10 Stress-induced band splitting vs. stress for [001], [111], and [110] directions........31

11 The scattering times due to acoustic and optical phonons and surface roughness
sc atte rin g s ............................................................................................................ .. 3 6

12 Calculated effective masses of heavy and light holes vs. stress using 6x6 strain
H am ilto n ian .............................................................................................................. 3 9

13 Model-predicted longitudinal z coefficient vs. stress for [001] direction .............40









14 Model-predicted longitudinal z coefficient vs. stress for [111] direction .............41

15 Model-predicted longitudinal z coefficient vs. stress for [110] direction .............42

16 The energies of the heavy- and light-hole and split-off bands vs. stress for [001]
direction calculated using 4x4 and 6x6 strain Hamiltonians ..............................46

17 Effective masses vs. stress for the heavy and light holes for [001] direction
calculated using 4x4 and 6x6 strain Hamiltonians.. ...........................................47

18 Illustration of uniaxial strained-Si pM OSFET ................................... ................ 50

19 Illustration of biaxial strained-Si pM OSFET ..................................... ................ 51

20 In-plane effective masses of the heavy and light holes vs. stress for uniaxial
com pression and biaxial tension ......................................................... ................ 55

21 Band splitting vs. stress for uniaxial compression and biaxial tension.................56

22 Model-predicted mobility enhancement vs. stress for uniaxial compression and
b iax ial ten sio n ......................................................................................................... 5 7

23 The underlying mechanism of mobility enhancement in uniaxial- and biaxial-
strained pM OSFETs .. ................................................................................ 59

24 Apparatus used to apply uniaxial stress and schematic of four-point bending ........63

25 Stress at the center of the upper surface of the substrate vs. the deflection of the
to p p in s ................................................................. .............................................. ... 6 6

26 Stress vs. position and schematic of bending substrate.......................................68

27 Apparatus used to apply biaxial stress and schematic of concentric-ring
b e n d in g ................................................................................................................ .... 6 9

28 Stress vs. displacement and schematic of bending plate................ ................71

29 Finite element analysis simulation of the bending plate (substrate). .....................72

30 Uncertainty analysis of the starting point for the four-point and concentric-ring
bending experim ents.. ............ ................ ................................................ 75

31 Uncertainty analysis of the misalignment of the substrate with respect to the
pins for the four-point bending experim ent......................................... ................ 77

32 The experimental setup for calibrating the uniaxial stress in the four-point
bending experim ent. ................ .............. .............................................. 79









33 Extracted displacement vs. position curves on the upper surface of the substrate
in the four-point bending experim ent.................................................. ............... 80

34 Extracted radius of curvature vs. position on the upper surface of the substrate
in the four-point bending experim ent.................................................. ............... 82

35 Extracted uniaxial stress vs. position on the upper surface of the substrate in the
four-point bending experim ent............................................................ ............... 83

36 Calibration of the four-point bending experiment .............................................. 84

37 Uncertainty analysis for the concentric-ring bending experiment ...........................86

38 Illustration of extracting threshold voltage.. ....................................... ................ 88

39 Effective hole mobility vs. effective field before and after bending with uniaxial
longitudinal tensile and compressive stresses at 226MPa ..................................... 93

40 Effective hole mobility vs. effective field before and after bending with uniaxial
transverse tensile and compressive stresses at 113MPa......................................94

41 Effective hole mobility vs. effective field before and after bending with biaxial
tensile stress at 303MPa and compressive stress at 134MPa ...............................95

42 M obility enhancem ent vs. stress ......................................................... ................ 96

43 ircoeffi cient v s. stress ................................................................... ................ 98

44 Average effective channel length ratio and variance vs. gate voltage
shift 6 for longitudinal stress....................................................... ............... 102

45 Average effective channel length ratio and variance vs. gate voltage
shift 6 for transverse stress .................................................. 103

46 Schematic diagram of doping concentration gradient and current flow pattern
near the metallurgical junction between the source/drain and body ................... 105

47 Simulation result of the internal stress distribution in a pMOSFET..................... 109

48 Mobility enhancement vs. stress at room temperature and 1000C.........................111

49 Before and after bending drain current ID and gate current IG vs. gate voltage
VG. The after bending gate current coincides with the before bending one.........113

50 Before and after bending drain current ID and gate current IG vs. gate voltage
VG. The after bending gate current is much higher than the before bending
o n e ......................................... ................................................ ........... 1 14









51 Before and after bending drain current ID and gate current IG vs. gate voltage
VG. The after bending gate current is extremely higher than the before bending
o n e .......................................... ............................................... ........... 1 1 5

52 Stress-induced gate leakage current change vs. stress. ............... ...................117

53 Definitions ofuniaxial stresses, -x, Zy, and Zz, and shear stresses, ixy, iKy, Kxz,
K -s, K-y a n d K -i y ........................................................................................................ 12 5

54 D efinitions of strain .. ................................................................... .............. 126

55 M easurem ent errors in X ..................................... ........................ ............... 133

56 R andom and bias errors in gun shots.. ................................................................. 134















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

STRAIN EFFECT ON THE VALENCE BAND OF SILICON: PIEZORESISTANCE IN
P-TYPE SILICON AND MOBILITY ENHANCEMENT IN STRAINED-SILICON
PMOSFET

By

Kehuey Wu

December 2005

Chair: Toshikazu Nishida
Cochair: Scott E. Thompson
Major Department: Electrical and Computer Engineering

This dissertation explores strain effects on the valence band of silicon to explain

and model piezoresistance effects in p-type silicon and mobility enhancements in

strained-Si pMOSFETs. The strain effects are manifested as changes in the valence band

when applying a stress, including band structure alteration, heavy and light hole effective

mass changes, band splitting, and hole repopulation. Using the 4x4 k-p strain

Hamiltonian, the stressed effective masses of the heavy and light holes, band splitting,

and hole repopulation are used to analytically model the conductivity and effective

mobility changes and the piezoresistance r coefficients. The model predictions agree well

with the experiments and other published works. Mobility enhancements and 7t

coefficients are extracted from four-point and concentric-ring wafer bending experiments

used to apply external stresses to pMOSFET devices. The theoretical results show that

the piezoresistance r coefficient is stress-dependent in agreement with the measured n









coefficients. The analytical model predictions for mobility enhancements in uniaxial and

biaxial strained-Si pMOSFETs are consistent with experiments as well as published

experimental data and numerical simulations. In addition, for biaxial tensile stress, the

model correctly predicts mobility degradation at low biaxial tensile stress.

The main factor contributing to the stress-induced linear drain current increase is

identified as mobility enhancement. The contribution from the change in effective

channel length is shown to be negligible. The temperature dependence of stress-induced

mobility enhancement is also considered in the model. At high temperature, the hole

repopulation is smaller than at room temperature, causing smaller mobility change

whereas stress-induced band splitting suppresses the interband optical phonon scattering

which reduces the mobility degradation.














CHAPTER 1
INTRODUCTION

Aggressive scaling of complementary-metal-oxide-semiconductor (CMOS)

technology has driven the performance improvement of very large scale integrated

(VLSI) circuits for years. However, as CMOS technology advances into the nanometer

regime, scaling down the channel length of CMOS devices is becoming less effective for

performance improvement mainly due to mobility degradation resulting from the high

channel doping density, and hence high vertical effective field, in the channel. Strained-

Si CMOS provides a very promising approach for mobility enhancement and has been

extensively investigated recently [1-14].

In addition, as CMOS technology advances into the deep submicron regime,

process-induced stresses, for example, shallow trench isolation [15], contact etch stop

nitride layer [16], and source and drain silicide [17], etc., may affect device performance

and reliability.

1.1 Relation of Strained Silicon CMOS to Piezoresistive Sensor and Piezoresistivity

Silicon has been widely used in mechanical stress and pressure sensors for a long

time due to its high sensitivity, good linearity and excellent mechanical properties [18,

19]. The strain effects responsible for the transduction physics of micromachined

piezoresistive sensors is closely related to mobility enhancement in strained silicon

CMOS. The strain effect on the valence band of silicon can be used to explain and

quantify the piezoresistance effect in p-type strained silicon as well as the hole mobility









enhancement in strained-Si pMOSFETs. Details of the strain-stress relation in a material

with cubic symmetry such as silicon are discussed in Appendix A.

The piezoresistance effect in strained silicon, first discovered by Smith [20] fifty

years ago, is the stress-induced resistance change. A coefficient, T, used to characterize

the piezoresistance is defined as [18]

1 Rx -Ro 1 AR
Z R0 Z RO


where Ro and R. are the unstressed and stressed resistances and is the uniaxial stress.

Since R = pA /L where p is the resistivity, A is the cross-sectional area, and L is the

length, the resistance change may be due to a combination of resistivity change and

geometry change. However, in semiconductors, the contribution from the geometrical

change may be neglected because it is 50 times smaller than the resistivity change [19].

Hence, the r coefficient may be expressed in terms of resistivity change or conversely in

terms of conductivity change as follows:

1 AR 1 Ap 1 Ac
7r=- =- ---, (2)
X RO Z Po X oo0


where p and c are the resistivity and conductivity respectively. This conductivity

change is directly related to mobility change since acrqup where q is the electronic

charge, u/ is the hole mobility, and p is the valence hole concentration. Two types of

uniaxial stresses are defined in Fig. 1 in order to distinguish two kinds of z coefficients,

longitudinal and transverse [18]. Further details on the piezoresistance coefficient and

coordinate transformations for an arbitrary direction are given in Appendix B.








Stressed

X t Longitudinal


Unstressed


+


xJ


x
A--


Vx




Transverse


Svx
V.j


Figure 1. Definitions of longitudinal and transverse directions for defining f coefficients
(adapted from Ref. [18]). The strain-stress relation is discussed in more detail
in Appendix A.









Longitudinal means that the uniaxial stress, electric field, and electric current are all in

the same direction. Transverse means that the electric field is parallel to the electric

current but normal to the uniaxial stress.

Recently, biaxial strained-Si pMOSFETs using a thick relaxed Sil-xGex layer to

stretch the Si channel has been studied extensively because high biaxial tensile stress can

increase the hole mobility [1-8]. Figure 2 [9] is a schematic diagram of the biaxial

strained-Si MOSFET using relaxed Sil-xGex. Alternatively, applying uniaxial

compressive stress along the channel can also enhance the hole mobility [10-12]. Two

approaches have been used to apply uniaxial compressive stress; one method employs

source and drain refilled with SiGe [10-12], another uses highly compressive stress SiN

layer [13]. Figure 3 [14] shows a pMOSFET with the source and drain refilled with SiGe

and a physical gate length of 45nm. Although both biaxial tensile and uniaxial

compressive stresses can improve the hole mobility, the efficacies of the two stresses are

different. For uniaxial compressive stress, about 50% hole mobility enhancement can be

achieved with about 500MPa [11, 12]; however, the biaxial tensile stress needs more than

1GPa to be able to increase the hole mobility. In fact, at low biaxial tensile stress, the

hole mobility is actually degraded [5, 8], which is contradictory to the theoretical

prediction made by Oberhuber et al. [3].

1.2 Motivation

The piezoresistance effect in n-type strained silicon has been well explained by the

many-valley model [21], while, in p-type strained silicon, the piezoresistance effect has

not yet been fully understood and characterized; most of the previous theoretical









* s 6tr ne S *




Relaxed SiGe .."l

e e Substrate
0 0 0 0 0"


Figure 2. Schematic diagram of the biaxial strained-Si MOSFET on relaxed Sil-xGex
layer (adapted from Ref [9]).
















0.0 -

pm

0.1-


I Si 200 MPa
FLOOP (FLorida Object Onented Process Simulator)


Figure 3. Schematic diagram of the uniaxial strained-Si pMOSFET with the source and
drain refilled with SiGe and physical gate length 45nm (adapted from Ref.
[14]).


45 wr









works [19, 22, 23] only show the longitudinal and transverse '7 coefficients along [111]

direction.

There is also a need for extended measurement of the piezoresistance coefficients

in silicon at higher stresses. The original data by Smith on piezoresistance of bulk silicon

were obtained with applied stresses of 106 N/m2 (1 MPa) to 107 N/m2 (10 MPa) using

loads attached to the free end of a clamped crystalline silicon or germanium sample with

electrodes attached on parallel or perpendicular faces of the sample [20]. However, much

higher fixed stresses on the order of 100 to 1000 MPa in the surface is required to

significantly improve the performance of transistor devices (n-channel metal-oxide-

semiconductor field-effect transistors (nMOSFET) and p-channel metal-oxide-

semiconductor field-effect transistors (pMOSFET)) [5, 24, 25].

There is a need for a better fundamental understanding of the effect of these high

stresses and corresponding strains on the carrier transport properties in advanced

MOSFETs and the influence of quantum confinement in nanostructures as well as

temperature. Since the piezoresistance effect is precisely the conductivity enhancement

obtained in strained-Si CMOS, understanding strain effects in semiconductors is vital for

continuing performance enhancement in advanced CMOS technologies. An accurate

model is needed to estimate the impact of process-induced stresses on the device

performance. Such a model can also be used in process and device simulations to

estimate the overall impact from various process-induced stresses.

Recently, due to the advance of silicon IC technology, the mass production of

high precision sensors and the integration of mechanical sensors and electronic circuits

(system on chip SOC) are now possible [18]. To design high precision sensors or SOC, a









more accurate model and better understanding of the piezoresistance effect in silicon are

needed.

1.3 Valence Band Structure-A Preview

The underlying physics of piezoresistance in p-type silicon and mobility

enhancement in strained-Si pMOSFETs can be explained by the strain effect on the

valence band of silicon. The valence band of silicon consists of three bands, heavy- and

light-hole and split-off bands. The heavy- and light-hole bands are degenerate at the band

edge and the split-off band is 44meV below the band edge [26, 27]. Applying stress to

silicon will lift the degeneracy and alter the valence band structure. As a result, the

effective masses of the heavy and light holes will change and holes will repopulate

between the heavy- and light-hole bands. To the first order approximation, the

contribution from the split-off band can be neglected [27-30] since it is 44meV below the

band edge. These stress-induced changes in the valence band are collectively called the

strain effect. Figure 4 is the illustration of the strain effects in silicon. Both the unstressed

and stressed silicon (with a uniaxial stress applied along the [111] direction) are shown in

the figure. For unstressed silicon on the left hand side of the figure, the E-k diagram

shows the degeneracy of the heavy- and light-hole bands at the band edge and the split-

off band is 44meV below the band edge. The energy surfaces of three hole bands are

shown next to the E-k diagram. On the right hand side of the figure are the E-k diagram

and the constant energy surfaces of the three hole bands for stressed silicon, assuming a

uniaxial compressive stress is applied along the [111] direction. As can be seen from the

E-k diagram, the degeneracy at the band edge is lifted and the light-hole band rises above

the heavy-hole band with a band splitting AE, resulting in hole repopulation from the










Unstressed
4 [001]


1IKeavy Hole
Band

Light Hole
Band

' Split-off
SBand


E(k)


Stressed


t [111]


[111]
P k


, Light Hole
Band


Hole


tUniaxial Stress
11[1111

I Heavy Hole
Band


*-- Split-off
Band


Figure 4. Strain effect on the valence band of silicon. On the left hand side are the E-k diagram and the constant energy surfaces of the
heavy- and light-hole and split-off bands for unstressed silicon. The degeneracy of the heavy- and light-hole bands at the
band edge is also shown. The E-k diagram and the constant energy surfaces of the three hole bands are shown on the right
hand side. The degeneracy is lifted and the light-hole band rises above the heavy-hole band with a band splitting AE,
causing the hole repopulation from the heavy- to light-hole bands. The shapes of the constant energy surfaces of the heavy-
and light-hole bands are altered and effective masses of the heavy and light holes are also changed.


E(k)









heavy- to light-hole bands. In the meantime, the band structures or the shapes of the

constant energy surfaces of the heavy- and light-hole bands are altered, causing the

changes in the effective masses of the heavy and light holes. Detailed discussion of the

strain effects will be presented in Chapter 2.

1.4 Focus and Organization of Dissertation

This dissertation will mainly focus on creating a simple, analytical model that can

be easily understood and provide quick, accurate predictions for the piezoresistance in p-

type silicon and mobility enhancement in strained-Si pMOSFETs. Most of the theoretical

works on strain effects on metal-oxide-semiconductor field-effect transistors (MOSFETs)

have employed pseudo-potential [2, 31] or k-p full band numerical simulations [3, 7]. A

simple, analytical model can provide a quick check for numerical simulations as well as

provide physical insight.

We develop a simple, analytic model using 4x4 k-p strain Hamiltonian [27, 28, 32].

The split-off band will be neglected [27-30] but the influence from it is added into the

model as a correction. In Chapter 2, the strain effects on the valence band of silicon will

be explained and quantified using 4x4 strain Hamiltonian. The stressed effective masses

of the heavy and light holes, band splitting between the heavy- and light-hole bands, and

the amount of hole repopulation will be calculated. Then, the longitudinal and transverse

'7 coefficients along three major crystal axes, [001], [111], and [110] directions will be

calculated. The result of the strain effects calculated using 4x4 strain Hamiltonian will be

compared with the result using 6x6 Hamiltonian to estimate the valid stress range that the

model is applicable. Chapter 3 is the calculation and comparison of mobility

enhancements in uniaxial- and biaxial-strained Si pMOSFETs using the results of the









strain effects developed in Chapter 2. The reason for the mobility degradation at low

biaxial tensile stress will be explained in detail. In Chapter 4, experiments designed to

test the validity of the models are presented. The four-point and concentric-ring wafer

bending experiments are used to apply the uniaxial and biaxial stresses respectively. The

approaches used to extract the threshold voltage, mobility, and vertical effective field are

described. In Chapter 5, the hole mobility and the mobility enhancement will be extracted

from the drain current in the linear region. The mobility enhancement vs. stress will be

plotted and compared with the model. The 7t coefficients will then be calculated from the

mobility enhancement vs. stress. The main factor contributing to the linear drain current

increase will be identified from analyzing the variables in the linear drain current

equation. The internal channel stress will be estimated using the process simulator

FLOOPS-ISE. The mobility enhancement at high temperature will be discussed and

compared with the mobility enhancement at room temperature. And, finally, Chapter 6 is

the summary and the recommendation for future work.














CHAPTER 2
STRAIN EFFECTS ON THE VALENCE BAND AND PIEZORESISTANCE MODEL

2.1 Introduction

In this chapter, the strain effects on the valence band will be explained in detail.

The valence band theory will be reviewed first. The equations of the constant energy

surfaces, effective masses, band splitting, and hole repopulation, which are derived from

Kleiner-Roth 4x4 strain Hamiltonian [27, 32], will be presented and explained. The

equations will then be used to model the piezoresistance in p-type silicon.

The piezoresistance in n- and p-type silicon was discovered by Smith 50 years ago

[20]. The n-type silicon piezoresistance can be well explained by the many-valley model

[21, 23]. Recently, piezoresistance in p-type silicon has been modeled in terms of stress-

induced conductivity change due to two key mechanisms [19, 22, 23]: (i) the difference

in the stressed effective masses of the heavy and light holes and (ii) hole repopulation

between the heavy- and light-hole bands due to the stress-induced band splitting.

However, previous works [19, 22, 23] only focus on the piezoresistance along the [111]

direction. In this chapter, we will extend the previous works to model the piezoresistance

along three major crystal axes, [001], [110], and [111] directions.

In section 2.2, we review the valence band theory and explain the strain effect on

the valence band using Kleiner-Roth 4x4 strain Hamiltonian [27, 32]. The valence band

structure, band splitting, and hole repopulation between the heavy- and light-hole bands

will be explained. In section 2.3, a model of piezoresistance in p-type silicon is presented.

The longitudinal and transverse conductivity effective masses of the heavy and light









holes, the magnitude of the band splitting and hole repopulation, and the corrections due

to the influence of the split-off band are calculated. The relaxation times due to the

acoustic and optical phonon scatterings are calculated. Since the model is employed to

estimate mobility enhancement on pMOSFETs, surface roughness scattering is also taken

into account. In addition, the quantization effect in the inversion layer of a pMOSFET is

also considered. Section 2.4 is the result and discussion. Comparisons of the model with

the previous works will be made. The valid stress range in which the model is applicable

is estimated by comparing the strain effects calculated using the 4x4 with the 6x6 strain

Hamiltonians, which includes the split-off band. Finally, section 2.5 is the summary.

2.2 Review of Valence Band Theory and Explanations of Strain Effects on Valence
Band

Single-crystal silicon is a cubic crystal. Without strain or spin-orbit interaction, the

valence band at the band edge, k=0, is a sixfold degenerate p multiple due to cubic

symmetry [27]. The sixfold p multiple is composed of three bands, and each band is

twofold degenerate due to spin. The spin-orbit interaction lifts the degeneracy at the band

edge, and the sixfold p multiple is decomposed into a fourfold p3/2 multiple, J=3/2

state, and a twofold pl/2 multiple, J=1/2 state, with splitting energy A=44meV between

the two p multiplets [26, 27]. The p3/2 state consists of two twofold degenerate bands

designated as heavy- and light-hole band. The pl/2 state is a twofold degenerate band

called spin-orbit split-off band. Near the band edge, k=0, the constant energy surface for

the p3/2 state can be determined by k-p perturbation [27, 33], approximated as









E(k)= Ak2 + B2k4 +C2 (k k2y k2+ k +kk ), (3)


and the pl/2 state is given by


E(k)= Ak2 A, (4)


where A, B, and C are the inverse mass band parameters determined by cyclotron

resonance experiments [27, 33, 34]. The upper and lower signs in Eq. (3) represent the

heavy- and light-hole band respectively. Figure 5 shows the E-k diagram and the constant

energy surfaces of the heavy- and light-hole and split-off bands near the band edge for

unstressed silicon. As seen from the E-k diagram in Fig. 5, the heavy- and light-hole

bands are degenerate at the band edge and the split-off band is below the band edge with

a splitting energy A=44meV. Also seen from Fig. 5, the constant energy surfaces of the

heavy- and light-hole bands are distorted, usually called "warped" or fluted," due to the

coupling between them. As for the split-off band, it is decoupled from the heavy- and

light-hole bands and has a spherical constant energy surface.

Applying stress to silicon will break the cubic symmetry and lift the degeneracy of

the fourfold p3/2 multiple at the band edge [27, 28]. If a uniaxial stress is applied along

an axis with higher rotational symmetry, for example, [001] direction with four-fold

rotational symmetry or [111] direction with three-fold rotational symmetry, the p3/2 state

will be decoupled into two ellipsoids. For the [001] direction, the constant energy

surfaces of the heavy- and light-hole bands become [27, 28]


E (k)=(A+ B)k2 +(A-B)k2 + (5)








Unstressed
S[001]


V IS ea am am a

A=44meV
me. .. .


S[01o]

Heavy Hole
Band


Light Hole
Band


-'


"a Split-off
Band


Figure 5. E-k diagram and constant energy surfaces of the heavy- and light-hole and
split-off bands near the band edge, k=0, for unstressed silicon. The heavy- and
light-hole bands are degenerate at the band edge, as shown in the E-k
diagram. The constant energy surfaces of the heavy- and light-hole bands are
distorted or warped. The split-off band is 44meV below the band edge and has
a spherical constant energy surface.


E(k)









E (k)= (A- B)ki +(A+B)k2 ,, (6)


where E3/2 (k) and E1/2 (k) are the heavy- and light-hole bands respectively,

k2 = k + k2 k2 k The band splitting AEoo1 between the heavy- and light-hole bands

is expressed as


AEooi = 2E0 = 2. 2-D (s, -S12), (7)
3

where Eo is the energy shift for the heavy- and light-hole bands for the [001] direction,

- is the uniaxial stress, D, is the valence band deformation potentials for [001]

direction, sll and s12 are compliance coefficients of silicon. The definition of compliance

coefficients is given in Appendix A. Along the [111] direction, the heavy- and light-hole

bands become [27, 28]


E (k)=(A+lN)k2 +(A -N)k2 +E, (8)
2 6 3


E1 (k)=(A -N)k26 +(A+ N)k ( 9 )
2 6 3 (9)

where ki=k2 +k2, kl =k2, ki and k2 are along the [110] and [112] directions

respectively, k3 is along the [111] direction, and N2 = 9B2 +3C2 is an inverse mass band

parameter. The band splitting AE1,1 between the heavy- and light-hole bands is expressed

as


AE, = 2. =2D s44 (10)









where Es is the energy shift for the heavy- and light-hole bands for the [111] direction,

s44 is a compliance coefficient of silicon, D' is the valence band deformation potential

for the [111] direction. Figures 6 and 7 show the E-k diagrams and constant energy

surfaces of the heavy- and light-hole and split-off bands for stressed silicon with uniaxial

compressive stresses applied along the [001] and [111] directions respectively. As seen

from Figs. 6 and 7, the degeneracy of the heavy- and light-hole bands is lifted with a

band splitting energy AE and the heavy- and light-hole bands become prolate and oblate

ellipsoids respectively with axial symmetry about the stress direction [27, 28]. When a

uniaxial compressive stress is applied, the light-hole band will rise above the heavy-hole

band and holes will transfer from the heavy- to light-hole bands because the energy of the

light-hole band is lower than the heavy-hole band and vice versa. Note that the energy

axis E of the E-k diagrams represents the electron energy. The hole energy is the negative

of the electron energy and in the opposite direction of the electron energy axis.

Therefore, by valence band "rising" or "falling" it means that the hole energy in the

valence band is decreasing or increasing respectively.

When applying a uniaxial stress along the two-fold rotational symmetry axis, [110]

direction, the situation is more complex. The energy surfaces of the heavy- and light-hole

bands still are ellipsoids yet have three unequal principal axes and the constant energy

surface of the heavy- and light- hole band become [27, 28]

E (k) = h2 k2 + 2 k + h2 2 l (2 E '3V2 1/2
E 2m 22 2m3 +_ k (11)
2m1 2m2 2m3 2


where







Stressed [001]
4 [001]


k[001]




Hole
/ Transfer

cI4 m--------


U Light Hole
P Band

IUniaxial Stress
II [001]

I Heavy Hole
Band


Split-off
i emO Band


Figure 6. E-k diagram and constant energy surfaces of the heavy- and light-hole and split-
off bands near the band edge, k=0, for stressed silicon with a uniaxial
compressive stress applied along [001] direction. The degeneracy of the
heavy- and light-hole bands is lifted with a band splitting energy AE and the
light-hole band rise above the heavy-hole band. The heavy- and light-hole
bands become prolate and oblate ellipsoids respectively with axial symmetry
about the stress direction.


E(k)


AE
*
-








Stressed [111]
E(k)
k 1[111]

k [111] Light Hole
}E'7\ "...-*- Band


SHole Uniaxial Stress
Transfer II [111]


'War Heavy Hole
Band



-- e-- A Split-off
SBand

Figure 7. E-k diagram and constant energy surfaces of the heavy- and light-hole and split-
off bands near the band edge, k=0, for stressed silicon with a uniaxial
compressive stress applied along [111] direction. The degeneracy of the
heavy- and light-hole bands is lifted with a band splitting energy AE and the
light-hole band rise above the heavy-hole band. The heavy- and light-hole
bands become prolate and oblate ellipsoids respectively with axial symmetry
about the stress direction.









h2 B N
=AT -, 172 (12)
2m1 2 2


h2
2m2= A+B (13)
2m2


h2 B N
= AT-1- 2, (14)
2m3 2 2

and


Y '72 ?2
17, = and 772 = Y-- (15)
1+332 1+3 32


with /f=c'0/ The upper signs in Eqs. (11) (14) belong to the heavy-hole band

E3/2(k) and the lower signs belong to the light-hole band E1/2(k). In Eq. (11), k3 is the

longitudinal direction along the [110] direction; k, and k2 are two transverse directions

along the [1 10] and [001] directions respectively. The band splitting AE110 between the

heavy- and light-hole bands is expressed as


AE110 = +2 3E)1/2 (16)


In Eq. (11), (2 + 3 2)1 /2 is the energy shift for the heavy- and light-hole bands for the

[110] direction [27]. Figure 8 shows the E-k diagram and constant energy surfaces of the

heavy- and light-hole and split-off bands for stressed silicon with a uniaxial compressive

stress applied along the [110] direction. Figure 9(a) and (b) are the top views of the

constant energy surfaces of the light- and heavy-hole bands respectively as observed from

the [110] direction. As can be seen from Figs. 8 and 9, the constant energy surfaces of the








Stressed [110]
+ [110]


tUniaxial Stress
II [110]


k [110]


[110]


Hole
Transfer


Vrft f t


Light Hole
Band

h [001]


Heavy Hole
Band


S'"" Split-off
Band


Figure 8. E-k diagram and constant energy surfaces of the heavy- and light-hole and split-
off bands near the band edge, k=0, for stressed silicon with a uniaxial
compressive stress applied along [110] direction. The degeneracy of the
heavy- and light-hole bands is lifted with a band splitting energy AE and the
light-hole band rise above the heavy-hole band. The heavy- and light-hole
bands still are ellipsoids yet have three unequal principal axes.


E(k)


AEi










0.06




0.02
OD



k2
0
[001]



-0-04


-0.j6 -0.04 -0.02



0.06

0.04

0.02

k2 0
[001]



-0.04


.0. 6 -0.04 -0.02


0 0.02 0.04 0.06
kI [110]



(b)


0 0.02 0.04 0.06
k1 [1 TO]


Figure 9. Top view (observed from [110] direction) of the constant energy surfaces with a
uniaxial compressive stress applied along [110] direction. (a) Light-hole band.
(b) Heavy-hole band.









heavy- and light-hole bands are ellipsoids with three unequal principal axes and the light-

hole band rises above the heavy-hole band.

2.3 Modeling of Piezoresistance in p-Type Silicon

2.3.1 Calculations of Hole Transfer and Effective Mass

From Eq. (2) and the illustrations in Fig. 1, the longitudinal and transverse r

coefficients can be defined as [23]


1 Ap 1 A 1 -o (17)
Z% Po Zi -o Z% -o

1 Ap 1 A 1 -o0 (18)

Xt Po Xt Co Xt Co

where Z, and -, are the longitudinal and transverse uniaxial stresses respectively, Ca

and co are the stressed and unstressed conductivity respectively, and

S Phh Ph q (19)
mhh mlh


where u/ff is the effective carrier mobility, q is the electron charge, C is the hole

relaxation time, mhh and mlh are the heavy and light hole conductivity effective mass

respectively. For silicon, the resistance change due to the geometrical change is 50 times

smaller than the resistivity change [19], therefore, in Eqs. (17) and (18), the contribution

from the geometrical change is neglected. Using Eq. (19), Eqs. (17) and (18) can then be

expressed as


1 AI 1 CZ -C o 1 A/ 1 -eff eff (20)
X1 /0 X1 0 X eff X1 'eff









1 A 1 z O 1 A/ 1 zef -,f (21)
Xt Co Xt Co Xt ejf Xt eff


where /ueff and /ff are the stressed and unstressed effective mobility, respectively. In

Eq. (19), p=Phh + Plh is the total hole concentration, Phh and Plh are the heavy and

light hole concentrations respectively. In order to simplify the model, the contribution

from the split-off band is neglected [27-30] because the split-off band is 44meV below

the valence band edge [35]. For a non-degenerate p-type silicon, Phh and P/h are given

by [36],


2'nmhkBT EF E, EF -E,
Phh = 2( h -) exp( F )= NVh exp(- ), (22)
h kBT kBT


2nkBT 3/2 E E E -E
P/h = 2( ) exp( F ) = Nv exp(- F v), (23)
h kBT kBT


where m*h =0.49n0 and m*n =0.16m0 [37, 38] are the density-of-state effective mass of

the heavy and light hole respectively, mn is the free electron mass, kB is the Boltzmann

constant, Tis the absolute temperature, EF is the Fermi level, and E is the energy at the

valence band edge. For the unstressed case, the valence band is degenerate at the band

edge, k=0, the heavy-hole band energy Evh and the light-hole band energy Ejv are equal

and Evh = Evi = E,. Using Eqs. (22) and (23), the heavy and light hole concentrations can

then be calculated from the doping density p [23]:

*3/2 *3/2
Phh hand Ph rnh (24)
Phh 3/2 + *3/2 +*3/2 *3/2
inhh + lIh inhh + lIh









When stress is applied, energy splitting of the heavy- and light-hole bands occur and

holes repopulate between the heavy- and light-hole bands. The concentration changes in

the heavy-hole band, Aphh, and the light-hole band, Aplh, can be obtained by

differentiating Eqs. (22) and (23) [39]


Aphh Nh exp( EF Evh)( )(AEF Ah Phh (A Avh), (25)
kBT kBT kBT

A =N ( -EE 1 p-
Aph exp(- v)(- )(AF E) (AEF AE), (26)
kBT kBT kBT

with

Aphh + Ah= 0, (27)


AEvh AE1 AE, (28)


where AEh and AE, are stress-induced energy shifts of the heavy- and light-hole bands.

Using Eqs. (25) (28), we can get [23]

Aphh = Phh AE and Ap F Plh AE (29)
^ph-kBT 1 I+ (m* /m* ]3/2 and Aplh = k i I = -3/2 P
kTh(mh rh kTl+(mlh hh )

In Eq. (29), the upper and lower signs are for uniaxial tensile and compressive stress

respectively.

The conductivity effective masses of the heavy and light holes, mhh and mlh, can be

derived from the E-k dispersion relations described in section 2.2, Eqs. (5), (6), (8), (9),

and (11),










m= -- (30)
d 2E
dk2

For a uniaxial stress applied along [001] direction, if the inverse mass band parameters, A

and B, are given in units of h2/2mo where mo is the free electron mass, using Eqs. (5),

(6), and (30), the longitudinal ( ||) and transverse ( _-) effective masses of the heavy and

light holes can be obtained as [27, 28]


mllhh[001] A B and ml|Ih[00o] =A (31)



mhh[001] and ih[001] 1- B/2'


where the units of effective masses are normalized by mo. Along the [111] direction, if N

is also given in units of h2/2mo, then using Eqs. (8), (9), and (30), the effective masses

are obtained as [27, 28]

1 1
ihh[ I] A-N/3 and mih[ll A'1] (33)



mlhh[111] =- and milh[111]- / (34)
A+N/6 A-N/6


For [110] direction, the situation is more complex due to the ellipsoids with three unequal

axes. We define ml, m2, and m3 as the effective masses corresponding to k1, k2, and k3 as

defined in Eq. (11). The longitudinal effective masses of the heavy and light holes can be

obtained using Eqs. (11), (14), and (30) [27, 28]









1 1
m3hh[110] =- B N and m3lh[110] B N (35)
A--A 2 A+-17,+ 72
2 2 2 2

and the two transverse effective masses of the heavy and light holes can be obtained from

Eqs. (11) (13) and (30)

1 1
mIhh[110] = B N and mllh[ll0] B N (36)
A I + 2 A +-2 2
2 2 2 2

1 1
2hh[11] and m21h[110] (37
A + B]71 A-B]71


where q7 and 72 are defined in Eq. (15).

At high stress, the influence from the split-off band is no longer negligible and the

4x4 strain Hamiltonian is subject to a correction [27, 28]. The correction can be

expressed in terms of effective mass shift added to the stressed effective masses obtained

in Eqs. (31)-(37), which are zeroth-order, stress-independent stressed effective masses

mi0. The experimentally measured effective mass mi can be expressed by an empirical

formula [28]


-=, -+A -=--+ (38)
inm m m) mo


where a is a parameter [27, 28]. For a special case that a uniaxial stress is applied along a

higher rotational symmetry axis, the [001] direction (four fold) or the [111] direction

(three fold), the correction and the effective mass shift only affects the light hole and the

heavy hole will not be affected. The longitudinal and transverse effective mass shifts for

the light hole for [001] and [111] directions are given by [27, 28],









A 1 = az +4B and A -- =aZ= -2B ', (39)
11001] A m1[001] A


A 1 = a +4 -N and AK- = Z =T2 2 (40)
-m11=111 3 A mi[ ] 3 A

In Eqs. (39) and (40), the upper and lower signs are for uniaxial tensile and compressive

stress respectively. For [110] direction, due to the lower rotational symmetry (two fold),

both the heavy and light holes will experience effective mass shifts [27, 28]. For the

heavy hole, the effective mass shifts, A(l/mlhh), A(/m2hh), and A(l/m3hh), are expressed

as [28]

S1 B 3'2 2 N 2 42
,lO L T3 a o L o (41)




m2hh[110] A( + 3 /2 (45)


A -136 0 0 0 (43)
m3hh[110] 3 X 2A _( +3 )1/2 0 2A ( +3 )1/2 (

For light holes, the three effective mass shifts are given by

1 B 3 2-_2 N 2 o6
A =\a--= 0 0+e -C_ +e (44)
m1lh[110] 2A _(e2+3 2 '1/2 0 2A _(2 +3, '1/2 0

1 1B 3^'2 _62
A--- =a2 0 -- 2 CO (45)
m21h[110] A (6 +3 '2 /









-1 B 3E -0 N 2, o
A -- I[- a = T~ -- +o]+" N2 +g (46)



In calculation of the stressed effective mass and band splitting for the three major

crystal axes, the values of the inverse mass band parameters and deformation potentials

we use are listed in Table 1. The values of A, B, and N are given in units of h2 2mo [27].

Table 1. Values of the inverse mass band parameters and deformation potentials used in
the calculations.
Parameters Symbol Units Values

Inverse Mass Band Parameters A h2/2mo -4.28 Ref.[27]

Inverse Mass Band Parameters B h /2mo -0.75 Ref.[27]

Inverse Mass Band Parameters N h /2mo -9.36 Ref.[27]

Deformation Potentials D eV 3.4 Ref.[40]

Deformation Potentials D eV 4.4 Ref.[40]


The calculated effective mass and band splitting will then be substituted into the

piezoresistance model to predict cr coefficients. Table 2 lists the zeroth-order stressed

effective mass m* calculated using Eqs. (31) (37) and Fig. 10 shows the stress-induced

band splitting vs. stress for [001], [111], and [110] directions calculated using Eqs. (7),

(10), and (16). In Table 2, for the [001] and [111] directions, mr| is the longitudinal

effective mass, i.e., the effective mass calculated using Eq. (30) and the direction of k is

parallel to the direction of the stress; m-L is the transverse effective mass, i.e., the effective

mass calculated using Eq. (30) and the direction of k is perpendicular to the direction of

the stress. For the [110] direction, mi, m2, and m3 are the effective masses with the









Table 2. Calculated zeroth-order longitudinal ( || ) and transverse ( ) stressed effective
masses of the heavy and light holes (in units of mo) for [001], [111], and [110]
directions.


[001]

heavy 0.28
mro-
light 0.20

heavy 0.21
m-L
light 0.26



[111]

heavy 0.86
mnr-
light 0.14

heavy 0.17
m -L-----
light 0.37



[110]

heavy 0.16
mi [11 0]
light 0.44

m2 [001] heavy 0.21
mH2 [001] -------
light 0.26

heavy 0.54
m3 [110]0.15
light 0.15


direction of k along the directions of ki, k2, and k3 respectively and the direction of k3 is

along the stress direction [110].

2.3.2 Calculation of Relaxation Time and Quantization-Induced Band Splitting

In a lowly doped surface inversion channel, the relaxation time Z in Eq. (19) is due

mainly to three scattering mechanisms: acoustic and optical phonon and surface










0.025

[110]

0.02 -
> [001]

0.015 -


U) [111]
'o 0.01 -



0.005 -



0 ----
0 100 200 300 400 500
Stress / MPa

Figure 10. Stress-induced band splitting vs. stress for [001], [111], and [110] directions.









roughness scattering. Including surface roughness scattering is necessary because the

model will be verified by experiments on hole inversion channel in pMOSFETs. The

relaxation time can be calculated using approximate analytical equations based on certain

assumptions. To calculate the acoustic and optical phonon scattering times, the following

assumptions have been made: (i) the heavy- and light-hole bands are assumed to be

parabolic as shown in Eqs. (5), (6), (8), (9), and (11) [41], (ii) the silicon is non-

degenerate [41], (iii) the acoustic phonon scattering is elastic and the optical phonon

scattering is inelastic and the corresponding scattering times depend only on hole energy

[41, 42], and (iv) all holes are scattered isotropicly [41, 42]. For the acoustic phonon

scattering, the scattering time raz is expressed as [41-43]

1 E mmi) 2 k,T (
=/, (47)
z-,,) z h puI

where e is the energy, E0, = 5.3eV [42] is the acoustic deformation potential constant of

the valence band, m, and m, are longitudinal and transverse effective mass, p is the

density of silicon, and u, is the longitudinal sound velocity. The total acoustic phonon

scattering time is given by [42]

1 1 1
+r (48)
Tac ,total ) Tachh ( Tac, )48


where rTa,hh (s) and raclh(s) are the acoustic phonon scattering time in the heavy- and

light-hole bands respectively. For non-degenerate silicon, the average scattering time for

the acoustic phonon can be obtained as [41]










3K ) = jrQT=fa k eT x kBT kBT

Since acoustic phonon energy is very small compared to the carrier thermal energy [41],

the acoustic phonon scattering mainly occurs in the intraband scattering and the stress-

induced band splitting will not affect its scattering rate [44, 45].

For optical phonon scattering, without stress, the scattering time rop, is given by

[41-43]

1 1 Dn 2(m2,exp(N )Re [( (50)
To ) 2 h2k p T T

where D =6.6xlOeV/cm [42] is the optical deformation potential constant of the

valence band, 0 = 735K is the Debye temperature and kB = hco0 = 63meV is the

optical phonon energy [41], Nq is the Bose-Einstein phonon distribution [42] and

Nq = [exp(ho/kBT)- 1 = [exp(0/T)-l11 [41]. The "Re" in Eq. (50) means Re(A)= A

if A is real; Re(A)= 0 if A is a complex number [41]. Applying stress, the stress-induced

band splitting will change the scattering rate [44, 45],

1 1 D2 (mmI12Nq nO [kB2 R k)
S+ 2 2kBp L k+ -AE) +expy Re[(- k AE) ( 51)
To,(s) h 2 fkp T IfL

where AE is the band splitting energy. Using Eqs. (48) and (49) with rac replaced by ropu

and op, in Eqs. (50) and (51) respectively, the average optical phonon scattering time

can be obtained.









For the surface roughness scattering, the scattering time rz,. is derived based on the

assumption that the surface roughness causes potential fluctuation to the carrier transport

and resulting in carrier scattering. The roughness of the surface can be characterized by

the power spectrum density S(q) [46]

S(q) = tL2A2 exp(- (Lq)4 /4), ( 52 )

where A = 2.7A [46] is the r.m.s value of the roughness asperities and L = 10.3A [46] is

the roughness correlation length, q = 2k sin(0/2), and k is the crystal momentum. Using

the simple parabolic band approximation, k can be expressed as k = 2ms/h. Then, the

surface roughness scattering time r,, is obtained as [46-48]


1 _e effm 2, ( -cos(O))(q)dO, (53)
T"2(') 2. 0 ( 3o

where EeLf is the surface effective field (normal to the channel), m is the conductivity

effective mass along the channel. Considering the effect of stress on the scattering time,

for a 500MPa stress, the corresponding strain is only about 0.3%, thus the L and A in Eq.

(52) is essentially unchanged. In addition, surface roughness scattering is independent of

the stress-induced band splitting. Therefore, the surface roughness scattering will not be

affected by stress. Using Eqs. (48), (49), (51), and (52) with 1rc replaced by ri, the

average scattering time due to surface roughness scattering can be obtained. Then the

total scattering time c in Eq. (19) can be calculated:

1 1 1 1
actotal) optotal srtotal








Figure 11 shows the calculation results of the scattering times. It is assumed that a

longitudinal tensile stress is applied along the [110] direction on a (001) wafer. In

calculation of the surface roughness scattering, the vertical effective field is chosen at

0.7MV/cm because the mobility enhancement in the pMOSFET will be extracted at

0.7MV/cm. As seen from Fig. 11, the surface roughness scattering is the most significant

compared to the acoustic and optical phonon scatterings. The optical phonon scattering

time increases as the stress increases, while the surface roughness and the acoustic

phonon scatterings are independent of stress.

Since the mobility enhancement and piezoresistance will be extracted using

pMOSFETs, in addition to the surface roughness scattering, we also must consider the

quantization effect in the channel due to surface effective field. Like stress, the surface

electric field can lift the degeneracy at the valence band edge and cause band splitting

due to the difference in the effective mass of the heavy and light hole. This band splitting

must be considered in addition to the stress-induced band splitting. We use the triangular

potential approximation [49, 50] to estimate the field-induced band splitting. The

quantized energy subbands for the heavy- and light-hole bands can be approximated by

[49, 50]


E =j3hq ( and E = 3hq j+ 3, j=0,1,2,.... (55)
EJhh 4 ajh j4m 4 0 .,


respectively, where ( is the surface electric field, mhh and mlh are the heavy and light

hole effective mass normal to the surface, and h is the Planck's constant. For our

application, only non-degenerate silicon is considered, therefore, we only take into
















----""" Optical Phonon Scattering




Acoustic Phonon Scattering



Total Surface Roughness Scattering

'p p p p


Stress / MPa


Figure 11 The scattering times due to acoustic and optical phonons and surface roughness
scatterings. It is assumed that a longitudinal tensile stress is applied along
[110] direction on a (001) wafer.









account the first subband, i.e., j=0. Then the field-induced band splitting can be

approximated as

AE = Elhh E11h. (56)


The field-induced band splitting will then be added to the stress-induced band splitting to

calculate the hole repopulation and the total band splitting is

AE,, = AEtrz + AE (57)


2.4 Results and Discussion

In this section, ir coefficients will be calculated and compared with the published

data [18, 20]. Later in Chapter 5, the calculated ir coefficients along [110] and [001]

directions will be compared with the experimental results. Using the definitions of zr

coefficient and conductivity in Eqs. (17) (19), hole concentrations in Eq. (24), hole

repopulation in Eq. (29), the zeroth-order stressed effective masses in Eqs. (31) (37),

the effective mass shifts in Eqs. (39) (46), the scattering time in Eq. (54), and the

quantization effect in Eq. (56), the stressed and unstressed conductivities, cr and co,

can be calculated from


7 =q2Phh +APhh + P/h+ lh and co =q2r Phh + P h (58)
Smhh mh ) mhh% mh)


and the longitudinal and transverse r coefficients can be obtained from Eqs. (20) and

(21). To calculate the unstressed conductivity co in Eq. (58), the stressed conductivity

effective masses are used instead of unstressed ones. This is because, for extrinsic silicon,

due to the lattice mismatch between the silicon and the dopant atom, there exists a small









but not insignificant lattice stress, estimated about 60kPa [51-53]. This small lattice stress

can lift the degeneracy at the valence band edge and change the shapes of constant energy

surfaces of the heavy- and light-hole bands and the effective masses of the heavy and

light holes. Figure 12 is a plot of calculated effective masses of holes in top and bottom

bands vs. stress using 6x6 strain Hamiltonian [54], assuming a uniaxial compressive

stress is applied along the [110] direction. On the left hand side of the figure, for silicon,

the top band represents the heavy-hole band and the bottom band represents the light-hole

band and they are degenerate at the band edge. As uniaxial stress increases to about

60kPa, on the right hand side of the figure, the degeneracy at the band edge is lifted and

the top band now represents the light-hole band and the bottom band represents the

heavy-hole band. The stressed effective masses of heavy and light holes saturate. As a

result, the stressed effective masses should be used in calculation of unstressed

conductivity due to the presence of small dopant-induced residual stress. However, this

small lattice stress, 60kPa, only causes very small band splitting, as can be seen from Fig.

10, thus the hole population is essentially unchanged.

Figures 13, 14, and 15 show the model-predicted longitudinal r coefficient vs.

stress for [001], [111], and [110] direction. The published data from Smith [18, 20] are

also included for comparison. Note that Smith's data were extracted with 1 to 10MPa

uniaxial tensile stress. One important observation from Figs. 13, 14, and 15 is that the r

coefficients for uniaxial tensile and compressive stresses are different and stress-

dependent. The main reasons are two folds: (i) the stress-induced hole repopulation

between the heavy- and light-hole band and (ii) the correction to the hole effective mass

is stress-dependent as shown in Eq. (38). The discontinuities at zero stress are due to
















E \
E
S0.4-

II

S0.3 *
LU V

0.2 \
oMMU-No U.__)__


.01 0.1 1 10 100
Stress / kPa

Figure 12. Calculated effective masses of heavy and light holes vs. stress using 6x6 strain
Hamiltonian [54], assuming a uniaxial compressive stress is applied along the
[110] direction. The solid line represents the effective masses of holes in the
top band and the dashed line represents the bottom band. On the left hand side
of the figure, for the unstressed silicon, the top band represents the heavy-hole
band and the bottom band represents the light-hole band. On the right hand
side of the figure, for the stressed silicon, the top band represents the light-
hole band and the bottom band represents the heavy-hole band. After about
60kPa stress, the degeneracy is lifted and the light-hole band rises above the
heavy-hole band and the stressed effective masses of heavy and light holes
saturate.

















Compression


-300


Tension






Smith's data


-100


Stress / M Pa


Figure 13. Model-predicted longitudinal z coefficient vs. stress for [001] direction.
Smith's data [20] are included for comparison.


-500


500






41



100



140
a-


Tension


100 -
Smith's data



-500 -300 -100 100 300 500
Stress / MPa


Figure 14. Model-predicted longitudinal f coefficient vs. stress for [111] direction.
Smith's data [20] are included for comparison.






42





190



SCompression
Tension
70


60 Smith's data



-500 -300 -100 100 300 500
Stress / MPa


Figure 15. Model-predicted longitudinal f coefficient vs. stress for [110] direction.
Smith's data [20] are included for comparison.









relaxation time and quantization effect in the inversion layer of pMOSFET as described

in subsection 2.3.2. The predictions associated with [001] and [110] direction will be

verified by the experiments presented later in Chapters 4 and 5.

In comparison to the previous works [19, 22, 23], first, all of them only consider r

coefficient for [111] direction and do not explicitly discuss the stress dependence of r

coefficient. Second, they all use stressed effective mass in calculation of unstressed

conductivity without making assumption or giving explanation. Third, they all assume

constant scattering time. For example, Suzuki et al. [22] consider stress-induced hole

transfer between the heavy- and light-hole bands and the effective mass shift for light

hole due to the stress-induced coupling between the light-hole and split-off bands.

However, they make an assumption that the conductivity change due to the hole transfer

is given by


Ac = lhh llh D S44- = Ck -hh -- lh (5
phh +/lh 3kBT phh + lh kBT

where phh and k/h are the mobility of heavy and light hole, AE is the band splitting for

[111] direction defined in Eq. (10), kB is the Boltzmann constant and T is the absolute

temperature. The authors do not provide justification or explanation to the assumption.

Using Eq. (29), our model predicts the conductivity change due to the hole transfer as

A =co P/hh -lh PhhPh AE (60)
Phhhh +Plhlh Phh +Plh kBT

which can be derived from semiconductors equations. The authors obtain longitudinal

and transverse r coefficients as 113 x 10-Pa-1 and -56x 10-1Pa-1 respectively, without

specifying at what stress. Kanda [19] uses the model and result from Suzuki et al. [22].

Kleimann et al. [23] consider hole transfer and effective mass shift for the light hole due









to (i) coupling between the light-hole and split-off bands and (ii) incomplete decoupling

between the heavy- and light-hole bands. However, they postulate that the effective mass

shift due to the incomplete decoupling does not affect the longitudinal heavy and light

hole effective masses. For transverse effective masses of the heavy and light holes, they

introduce correction terms proportional to stress to represent the effective mass shift due

to incomplete decoupling,


A 1 7Z and A = ~z, (61)
mhhi MIhi

where /1, and /2 are two parameters. The correction terms shown in Eq. (61) are

contradictory to the results from Hasegawa [28] and Hensel et al. [27], to which

Kleimann et al. refer in their paper. The reason is because the incomplete decoupling

effect should decrease as the stress increases and, at very high stress, the heavy- and

light-hole bands will decouple completely and the correction terms will disappear, in

contradiction to Eq. (61). The authors obtained 89x 10-11Pa-1 for longitudinal coefficientt

for [111] direction. As for transverse r coefficient, the authors fit the experimental value,

-44.5x10-11Pa-1, and get values for /3 and /2, 2.4x10-9Pa-1 and -2.4x10-9Pa-1 respectively.

The authors give both unstressed and stressed effective masses in their paper but used the

stressed effective mass to calculate the unstressed conductivity without a model or

explanation.

The strain effect described in section 2.2 was derived from Kleiner-Roth 4x4 strain

Hamiltonian [27, 32], which neglects the spilt-off band. To estimate the error introduced

by using the Kleiner-Roth 4x4 strain Hamiltonian [27, 32], we also use the Bir-Pikus 6x6

strain Hamiltonian [55, 56], which takes the split-off band into account, to calculate the









band splitting and effective masses for [001] direction and compare to the results

presented in section 2.3. Figure 16 illustrates the band energy vs. stress calculated from

4x4 and 6x6 strain Hamiltonians for [001] direction. The 4x4 strain Hamiltonian

overestimates the band splitting between the heavy- and light-hole bands by 41% at

500MPa uniaxial compression but underestimates 16% at 500MPa uniaxial tension. Note

that the energy separation between the upper hole band and split-off band is larger at high

stress than zero stress, which implies that the hole concentration in the split-off band is

even smaller at high stress and neglecting the split-off band in the model is justified.

Comparisons of longitudinal and transverse, heavy and light hole effective masses are

shown in Figs. 17(a) and (b) respectively. For the longitudinal light hole effective mass,

the 4x4 strain Hamiltonian underestimates about 2% and 5% at 500MPa uniaxial

compression and tension respectively, and the deviations are about 1% and 4%

overestimations for the transverse light hole effective mass respectively. For the heavy

hole, both the longitudinal and transverse effective masses are the same for the 4x4 and

6x6 strain Hamiltonians. Based on these comparisons, we conclude that using 4x4 strain

Hamiltonian is a good approximation and the model is suitable for the stress less than

500MPa.

2.5 Summary

The strain effects on valence band are explained in detail in this chapter. The

constant energy surfaces of the heavy- and light-hole bands, heavy and light hole

effective masses, stress-induced band splitting, hole repopulation are explained and

derived using Kleiner-Roth 4x4 strain Hamiltonian. At high stress, the influence from the

split-off band is taken into account by adding an effective mass shift to the light hole










0.02
Light Hole
6x6



0




S' Light Hole Heavy Hole
-0.02 44 6x6 & 4x4
I-
LU

Split-off
4x4 \
-0.04-



\ Split-off
6x6
-0.06 I1 .I I
-500 -250 0 250 500

Stress / MPa


Figure 16. The energies of the heavy- and light-hole and split-off bands vs. stress for
[001] direction calculated using 4x4 and 6x6 strain Hamiltonians.






47


E 0.29

0.27
S .2 Heavy Hole 6x6 & 4x4
0.25 Light Hole 6x6

| 0.23 (a)

2 0.21

.A 0.19 Light Hole 4x4

3 0.17 i
-500 -250 0 250 500
Stress / MPa



o 0.29
Light Hole 4x4
0.27 (b)

w 0.25
.> Light Hole 6x6
w 0.23

| 0.21 1
> Heavy Hole 6x6 & 4x4
0.19

0.17 I i I -
-500 -250 0 250 500
Stress / MPa


Figure 17. Effective masses vs. stress for the heavy and light holes for [001] direction
calculated using 4x4 and 6x6 strain Hamiltonians. (a) Longitudinal. (b)
Transverse.









effective mass (for [001] and [111] directions) and to both heavy and light hole effective

masses (for [110] direction).

The strain effects are used to model piezoresistance in p-type silicon. The

longitudinal and transverse z coefficients for three major crystal axes, [001], [111], and

[110] directions, are calculated and compared to the published data. The model-predicted

z coefficients are stress-dependent and the tensile and compressive r coefficients are

different. The reasons are two folds: (i) the stress-induced band splitting causes the stress

dependence of hole population and (ii) light hole effective mass (for [001] and [111]

directions) or both heavy and light hole effective masses (for [110] direction) are stress-

dependent due to the influence of the split-off band at high stress.

Finally, the valid stress range that the model is applicable is estimated by

comparing the band splitting and the heavy and light hole effective masses calculated

using 4x4 and 6x6 strain Hamiltonians. The comparisons show that 4x4 strain

Hamiltonian is a good approximation to 6x6 strain Hamiltonian at stress under 500MPa,

and hence, the piezoresistance model is good with stress less than 500MPa. Later in

Chapters 4 and 5, the piezoresistance model will be verified by the experiments.













CHAPTER 3
HOLE MOBILITY ENHANCEMENT IN BIAXIAL AND UNIAXIAL STRAINED-
SILICON PMOSFET

3.1 Introduction

The underlying mechanisms of mobility enhancement in uniaxial and biaxial

strained-Si pMOSFETs are the same as the piezoresistance effect in p-type silicon as seen

from Eqs. (20) and (21). As shown in Fig. 18, by uniaxial strained-Si pMOSFETs, we

mean that an in-plane uniaxial compressive stress (uniaxial compression) is applied along

the channel of pMOSFET, i.e., along [110] direction in the (001) plane that contains the

channel. By biaxial strained-Si pMOSFETs, we mean that an in-plane biaxial tensile

stress (biaxial tension) is applied to the channel in the (001) plane, which is illustrated in

Fig. 19. Biaxial stress, like uniaxial stress, can lift the degeneracy at the valence band

edge and cause hole repopulation and mobility change. Uniaxial and biaxial strained-Si

pMOSFETs are two important technologies used to enhance the hole mobility [1-14].

Thompson et al. [11, 12, 44] compared uniaxial vs. biaxial in terms of device

performance and process complexity and concluded that uniaxial strained-Si pMOSFETs

is preferable since comparable mobility enhancement is attained at smaller stress

(500MPa compared to >IGPa) which is retained at high effective field.

In this chapter, we will use the strain effects on the valence band described in

Chapter 2 to explain quantitatively how uniaxial compressive and biaxial tensile stresses

change the hole mobility in pMOSFETs in the low stress regime (<500MPa). In section

3.2, the mobility enhancement will be calculated. An equivalent out-of-plane uniaxial

compressive stress will be derived for biaxial-strained pMOSFET. The valence band










[001] [110]
[[1 10]

















Figure 18. Illustration of uniaxial strained-Si pMOSFET. The arrows represent the
uniaxial compressive stress in the channel. The uniaxial compressive stress
can be generated by process, for example, source/drain refilled with SiGe [10-
12] or highly compressive stressed SiN capping layer [13], or by four-point
wafer bending, which will be described in Chapter 4.








[001] [110]
[[11 0]


















Figure 19. Illustration of biaxial strained-Si pMOSFET. The arrows represent the biaxial
tensile stress in the channel. The biaxial tensile stress can be generated by
process, for example, using thick relaxed Sil-xGex layer to stretch the Si
channel [1-8], or by concentric-ring wafer bending, which will be described in
Chapter 4.










structure, in- and out-of-plane heavy and light hole effective masses, band splitting, and

hole transfer are explained and calculated using the model described in Chapter 2.

Section 3.3 is the discussion. Finally, section 3.4 is the summary.

3.2 Mobility Enhancement in Strained-Silicon PMOSFET

In this section, we will calculate and compare the mobility improvement in

uniaxial- and biaxial-strained pMOSFETs. The equations of constant energy surface and

band splitting described in Chapter 2 were derived by Hensel and Feher [27] in terms of

uniaxial stress, for example, Eqs. (7) and (10). In order to use their equations to model

biaxial-strained pMOSFETs, we will show that in-plane, (001), biaxial-tensile stress %b,

can be represented by an equivalent out-of-plane uniaxial compressive stress -Zun, of the

same magnitude along the [001] direction, which will create the same band splitting

along the same direction as shown in Eq. (7). The band splitting for uniaxial stress in the

[001] direction is given by [27],


AE= 20 22 DS, (62)


where D. is the valence band deformation potential for [001] direction and the strain is

give by

S = (s11- s2) zz (63)


where is the uniaxial stress along [001] direction, and E= and sE are the uniaxial

strain along the [001] and [100] directions respectively. The strain-stress relation for

material with cubic symmetry is reviewed in Appendix A. In Eq. (63), sl and s12 are

compliances of silicon [57] and






53


C11 +C12 and s12 1 (64)
11 (1 _C12)(C11 +2c12) 12 (cl -c12 )(c1 +2cl2)'

where cl =1.657x1011"Pa and c12 =0.639 x1011Pa are the normal and off-diagonal

stiffnesses of silicon. For in-plane biaxial tensile strain, E = Syy = E, and E= can be

derived from the stress-strain equation with zero shear strains [58],


xx C 1 C12 C12 0 0 0 Ex
y Cl2 C11 C12 0 0 0 yy
cz 2 C12 c1 1 0 0 0 E z
= 65)
Ty, 0 0 0 C44 0 0 0
Zr 0 0 0 0 c44 0 0
_Ty 0 0 0 0 0 c44 0


where c, Xyy, and Xzz are the uniaxial stresses, and Txy, yz, and rzx are the shear stresses.

On the left hand side of Eq. (65) is the stress tensor, and the right hand side is the elastic

stiffness tensor and strain tensor. With in-plane biaxial tensile strain, there is no out-of-

plane stress, i.e., _zz =0 = c12( xx+ yy+11zz Then, E= can be obtained as

E= = -(2c12/c 11) and Sin Eq. (63) becomes


S=-2 c _bE (66)
11 ) (11 _12) (66)


where Xb, is the in-plane biaxial tensile stress in (001) plane and


E









where E is the Young's modulus, v is the Poisson ratio, and

E/(1-v)= 1/(s11 +s12)= 1.805 x10"Pa is the biaxial modulus and invariant in the (001)

plane [57]. For a uniaxial compressive stress u ....


S s(- -) s(-j)= I um (68)
(c 11- c 2)

Comparing Eqs. (66) and (68), if b, = un, then the in-plane biaxial tensile and out-of-

plane uniaxial compressive stresses will create the same strain S and thus the same band

splitting along [001] direction. Therefore, the in-plane biaxial tensile stress %b, can be

represented by an equivalent out-of-plane uniaxial compressive stress un, of the same

magnitude.

The equations and band parameters presented in section 2.3 are used to calculate

the effective mass, band splitting, and hole transfer between the heavy- and light-hole

bands. Table 3 gives the zeroth-order, in- and out-of-plane heavy and light hole effective

Table 3. In- and out-of-plane effective masses of the heavy and light holes for uniaxial
compression and biaxial tension.
Uniaxial Compression Biaxial Tension
Heavy Hole Light Hole Heavy Hole Light Hole
In-Plane 0.54 0.15 0.21 0.26
Out-of-Plane 0.21 0.26 0.28 0.20

masses for uniaxial compression and biaxial tension. Figure 20 shows the in-plane heavy

and light hole effective mass with mass correction vs. stress. The band splitting induced

by both stress and quantum confinement vs. stress is shown in Fig. 21. The mobility

enhancement in uniaxial- and biaxial-strained pMOSFETs are calculated using Eqs. (20),

(21), and (58) and shown in Fig 22. Two published experimental data points for biaxial-















0.5


0.4


0.3


Uniaxial Heavy Hole










Biaxial Heavy Hole





*I I I I ,


0 100


200
Stress


300
/MPa


400


500


Figure 20. In-plane effective masses of the heavy and light holes vs. stress for uniaxial
compression and biaxial tension.










0.035


0.03 Biaxial Tension


> 0.025 -


E 0.02 -
Uniaxial
"F Compression
) 0.015 -

m 0.01 -


0.005 -


0 -
0 100 200 300 400 500
Stress / MPa


Figure 21. Band splitting vs. stress for uniaxial compression and biaxial tension. Both
contributions from stress and quantum confinement are included. Doping
density is assumed 10"cm-3.











0.6


Thompson et al. 04
(Uniaxial Compression) "


0.4
Uniaxial
Compression
0.3 This Work


0.2
wZ Oberhuber 98
3_ 01 .1(Biaxial Tension)
S Biaxial
,,Tension
0 RThis Work

Rim et al. 02 (Biaxial Tension)- I
-0.1
Rim et al. 03 (Biaxial Tension) -----

-0.2 f .
1.E+07 1.E+08 1.E+09
Stress / Pa


Figure 22. Model-predicted mobility enhancement vs. stress for uniaxial compression and
biaxial tension. Published theoretical [3] and experimental [5, 8, 12] works are
also shown for comparison.









strained pMOSFET [5, 8] and one for uniaxial-strained pMOSFET [12] are included for

comparison. Also included is the model prediction for biaxial-strained pMOSFET from

Oberhuber et al. [3]. In comparison with their theoretical work [3], our model gives better

prediction that shows mobility degradation instead of mobility enhancement at low

biaxial tensile stress. At 500MPa, the model predicts 48% mobility improvement for

uniaxial compression and 4% mobility degradation for biaxial tension, in good agreement

with published data [5, 8, 12].

3.3 Discussion

The underlying mechanism of the mobility enhancement in uniaxial- and biaxial-

strained pMOSFETs is the hole repopulation from the heavy- to light-hole bands. For

uniaxial compression, as shown in Figs. 8 and 20, since the heavy- and light-hole bands

are prolate and oblate ellipsoids, the heavy hole effective mass along the channel in (001)

plane is larger than the light hole and hole repopulation from the heavy- to light-hole

bands improves the mobility. For biaxial tension, as seen from Figs. 6 and 20, the

effective mass along the channel in (001) plane for the heavy hole is actually smaller than

the light hole and hole repopulation from the heavy- to light-hole bands results in

mobility degradation. In addition, as seen from Fig. 20, the difference in the in-plane

effective masses of the heavy and light holes is larger for uniaxial compression than

biaxial tension. This causes larger mobility improvement for uniaxial compression. The

summary of the underlying mechanism of mobility enhancement in uniaxial- and biaxial-

strained pMOSFETs is illustrated in Fig. 23.

At high stress, the band splitting will become very large, and hole transfer will

finally stop when all holes populate only one band (upper band with lower energy). At

this point, the mobility enhancement will mainly come from the suppression of the










Uniaxial
Compression
[110]
4E(k) t


[110]

k [110]


Biaxial
Tension


Light Hole
0.15mo

low [001]


S
S
S
I
9
a
S


Hole
Transfer


Heavy Hole
0.52mo


AEbi (
*


6
g
6
6
ieee


(


LE(k)



N"


[110]


k [110]


[001]
Light Hole
0.26mo




\Hole
: Transfer


Heavy Hole
0.21 mo


Figure 23. The underlying mechanism of mobility enhancement in uniaxial- and biaxial-strained pMOSFETs. The effective mass
shown next to the constant energy surfaces are the zeroth-order in-plane effective masses.


AEuni!
ma m
+
I
l
I
l
l
Iy


L_ M









interband optical phonon scattering, as shown in Eq. (51). For biaxial tension, it implies

that, at certain high stress, the mobility will stop decreasing and start increasing with even

higher stress. The band splitting between the heavy- and light-hole bands is ~25meV for

500MPa biaxial tensile stress and ~100meV for 2.2GPa, corresponding to x=0.28 in Sil-

xGex. When the band splitting becomes larger than the optical phonon energy, 63meV

[42], the interband scattering between the heavy- and light-hole band is suppressed and

hole mobility increases. This explains the published biaxial tension data at high Ge

concentration [5, 8]. However, due to the limitation of 4x4 strain Hamiltonian, the

analytical model cannot provide accurate prediction for hole mobility enhancement at

such high stress.

3.4 Summary

At low stress (<500MPa), the uniaxial-strained pMOSFET is shown to have large

mobility improvement due to hole repopulation from the heavy-hole band with larger in-

plane effective mass to the light-hole band with smaller one. For biaxial-strained

pMOSFET, because the in-plane effective mass of the heavy hole is smaller than the light

hole, hole repopulation from the heavy- to light-hole bands degrades the mobility. Both

predictions are in good agreement with the published data, 48% improvement for

uniaxial-strained pMOSFET and 4% degradation for biaxial-strained pMOSFET at

500MPa. At large biaxial tensile stress, the suppressed interband scattering due to the

large band splitting, greater than the optical phonon energy, results in the mobility

enhancement.














CHAPTER 4
WAFER BENDING EXPERIMENT AND MOBILITY ENHANCEMENT
EXTRACTION ON STRAINED-SILICON PMOSFETS

4.1 Introduction

In this chapter, wafer bending experiments designed to test the models described in

previous chapters are presented. The hole mobility and mobility enhancement vs. stress

will be extracted and the piezoresistance r coefficients of p-type silicon vs. stress will

then be calculated. Concentric-ring and four-point bending apparatus are used to apply

six kinds of mechanical stress to the channels of pMOSFETs, biaxial tensile and

compressive and uniaxial longitudinal and transverse, tensile and compressive stress. The

stress range used in this experiment is 50MPa to 300MPa. PMOSFETs from 90nm

technology [11, 12, 59] with the channels oriented along [110] direction on (001) wafers

are used in the experiments.

In section 4.2, wafer bending experiments are presented. First, the four-point

bending apparatus used to apply uniaxial stress will be explained in detail and equations

for calculating the uniaxial stress will be derived. For the concentric-ring bending jig

used to apply biaxial stress, finite element analysis simulation is used due to nonlinear

bending. Uncertainty analysis in the applied stress will be performed. Section 4.3

explains the methods to extract threshold voltage, hole mobility, and vertical effective

field. Uncertainty analysis for effective mobility will be performed. Section 4.4 is the

summary.









4.2 Wafer Bending Experiments on pMOSFETs

4.2.1 Four-Point Bending for Applying Uniaxial Stress

Uniaxial stress is applied to the channel of a pMOSFET using four-point bending.

Figure 24(a) and (b) are the pictures of the apparatus used to bend the substrate and the

illustrations of calculating the uniaxial stress. As shown in Fig. 24, the upper and lower

surfaces of the substrate will experience uniaxial compressive and tensile stress along

[110] direction, respectively. The stress on both surfaces can be calculated using the

following analysis [60] with the assumptions: (i) The substrate is simply supported. (ii)

Four loads applied by four cylinders are approximated by four point forces, P. As shown

in Fig. 24(b), let the deflection at any point on the upper surface be designated by y(x),

where y(0)=0 and y(L)=0. The stress on the upper and lower surfaces at the center of the

substrate are given by

EH EH
pr and cr- .- ( 69)
upper 2r ower 2r

respectively, where E = 1.689 x1011Pa [57] is the Young's modulus of crystalline silicon

along the [110] direction on (001) substrate, H is the substrate thickness, r is the radius of

curvature given by [60]

1 M Pa (70)
r El, El'

where M = Pa is the moment for a < x < L/2, and I = bH3/12 is the moment of inertia

for a substrate with rectangular cross section and width of b. Eq. (69) can then be

expressed as [60]





63




















(a)

iP Pi

SL -*
; ;;t :: -^ ---] ) -- --+ --- -- w%
I ----- ---------



(b)
Figure 24. Apparatus used to apply uniaxial stress and schematic of four-point bending.
(a) The picture of jig. In this picture, uniaxial compressive and tensile stresses
are generated on the upper and lower surfaces of the substrate respectively. (b)
Schematic of four-point bending. The substrate is simply supported. Four
loads applied by cylinders are approximated by four point forces, P. The
deflection at any point on the upper surface is designated by y(x).









MH PaH MH PaH (71)
pr and =c.=.. (71)
xupper 21 21 ower 21 21

For 0 < x _< a, the moment M = Px and


El, = -M =-Px. (72)
dx

Solving Eq. (72) we get



Y EI 6 C2 (73)


where C1 and C2 are integration constants. For a < x < L/2, the moment

M = Px P(x a) = Pa and [60]


d~y
EI d = -M = -Pa. (74)
dx

Solving Eq. (74) we obtain


Y = --Pa2 +C3x+C4 (75)
=EI\ 2

where C3 and C4 are integration constants. The four integration constants can be

determined from the boundary conditions [60]: (i) the slope dy/dx determined from Eqs.

(73) and (75) should be equal at x = a, (ii) the slope dy/dx =0 at x = L/2, i.e., at the

center of the substrate, (iii) at x = a, y determined from Eqs. (73) and (75) should be

equal, and (iv) at x = 0, y = 0. With these four boundary conditions, Eqs. (73) and (75)

become [60]

1 P C3 Pa(L a)
El x 6 x 0 EI 6 2









1 fPa 2 PaL Pa3
El 2 2 62. (77


Using Eqs. (76) and (77), the deflection at x = a and x = L/2 can be calculated [60]


Pa2 L 2a
y1=C = (78)



Y L (3L2 4a2). (79)
2 24EI,


Measuring the deflection at x = a, P/I, can be obtained as


P Ey^
S 2a, (80)
I L 2a
a ---
2 3


and then the stress on both surfaces of the substrate in Eq. (71) can then be calculated


EHy EHy^
Cupper EL 2a and oxIow-er EHyL 2 (81)
2aL a 2a2a 2a
2 3 2 3

The radius of curvature in Eq. (70) can also be obtained as

1 Pa yx(82)
r El aL 2
a a\---a\
2 3

Finite element analysis (FEA) using ABAQUS [61] is also performed to verify the

assumptions used to obtain Eq. (81) and the results are shown in Fig. 25. In Fig. 25, two

cases are simulated as shown in two insets. The upper inset shows that the distance

between the two top pins is larger than the two bottom ones. When the two top pins move

downward, a tensile stress is generated on the upper surface. The lower inset is the










500
Uniaxial
400 p i Tension
0 P0 (Equation)
300 -p p| t Uniaxial

200 Tension
(Simulation)

U) 0 ..-----------x---.. .----
S)0 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
-100
O ^ Maximum Uniaxial
-200 Deflection Used Compression
S(Simulation)
-300 p p, i /
Uniaxial
-400 Compression '
I (Equation)
-500

Deflection / mm


Figure 25. Stress at the center of the upper surface of the substrate vs. the deflection of
the top pins. The calculated stress values are from Eq. (81). Simulated stress
values are obtained using finite element analysis with ABAQUS [61].









opposite of the upper one and a compressive stress is generated on the top surface. The

stress at the center of the top surface is extracted from the simulation and compared with

the stress calculated using Eq. (81). The calculated stresses using Eq. (81) agree well with

the results of finite element analysis for the range of deflection used.

Figure 26(a) is a plot of stress vs. relative position y to the neutral axis along the

cross section, AA' in Fig. 26(b), extracted from simulation. The deflection is assumed as

0.91mm. Figure 26(b) is an illustration of bending substrate, neutral axis, top and bottom

planes, and cross section AA' cut at the center of substrate used in simulations. The

substrate thickness H is 0.77mm. As shown in Fig 26(a), the stress vs. position y curve is

linear and symmetric about the origin. This result verifies the validity of Eq. (81) within

the range of deflection 0.91mm.

The finite element analysis simulations shown in Figs. 25 and 26 need further

investigations due to no systematic studies of grid convergence. More detailed analysis

can be done in future work. Regarding the effect of sample location on stress, as will be

shown in Fig. 35, the stress variation at the position between the two top pins on the

upper surface is less than 0.01%. Therefore, the effect of sample location on stress can be

neglected for the range of deflection used.

4.2.2 Concentric-Ring Bending for Applying Biaxial Stress

Biaxial stress is applied to the channel of a pMOSFET using concentric-ring

bending. Figures 27(a) and (b) show a picture of the apparatus used to bend the substrate

and an illustration for simulating the biaxial stress. Unlike beams (uniaxial stress state),

even deflections comparable to the plate thickness produce large stresses in the middle

plane and contribute to stress stiffening. Hence one should use large deflection (nonlinear
















o100


o -4)0 -300 -200 -10 100 200 300 400
-100
--200
SStress at neutral axis 0

I Stress at bottom -400
plane -443MPa
-500

Position y / gm

(a)


Top Neutral
Plane A Axis








Bottom
Plane

(b)

Figure 26. Stress vs. position and schematic of bending substrate. (a) Plot of stress vs.
relative position to the neutral axis along the cross section, AA' in (b),
extracted from simulation. The deflection is assumed as 0.91mm. (b)
Illustration of bending substrate, neutral axis, top and bottom planes, and cross
section AA' cut at the center of substrate. The substrate thickness is 0.77mm.







































w(r) Top Ring
SI4Ll b Load
.... ......... ............. ...

Bottom Ring a
Support


(b)

Figure 27. Apparatus used to apply biaxial stress and schematic of concentric-ring
bending. (a) The picture of jig. In this picture, biaxial compressive and tensile
stresses are generated on the upper and lower surfaces of the substrate
respectively. (b) Schematic of concentric-ring bending. The plate (substrate) is
simply supported. The deflection at any point on the upper surface is
designated by w(r).






70


analysis) to calculate deflections and stresses in a plate. The finite element analysis

(FEA) using ABAQUS [61, 62] considering both the nonlinearity and orthotropic

property of Si is used in this work to calculate the biaxial stress from the measured

deflections. The constitutive equation of Si is expressed as

1 Vx x 0 0 0

1 v1 0 0 0
E, E, E


1 0 0 0
zz E zz E E (83)
7yz 0 0 0 1 0 0 'z


_y_ 0 0 0 0 1 0 _
Gzx
0 0 0 0 0 1
G.

where Ex= Eyy = E =1.302x101 Pa are the three Young's moduli [57],

Gy = = Gz = 0.796 x 1011Pa are the three shear moduli, vy = vy = v = 0.279 are

the three Poisson ratios, Zxx, Xyy, and ,Zz are the normal stresses, sx, syy, and Ez are the

normal strains, Txy, zy>, and z, are the shear stresses, and ",, 7y), and y/x are the shear

strains. The values of Young's moduli, shear moduli, and Poisson ratios are needed in the

finite element analysis. The simulation assumed that the smaller ring is on the top and the

larger one is on the bottom as illustrated in Fig. 27. The results are shown in Figs. 28 and

29 [62]. In Fig. 28, the stresses along x and y axes, Oxx and oyy respectively, at the center

of the bottom, middle, and top planes vs. the displacement of the smaller ring are

presented. As can be seen from Fig. 28, Oxx = Oyy, this result confirms that the stress at

the center of the planes is biaxial. In addition, Figure 29 (a) shows the shear stress Txy at










600

500 F xx
X CyYY Bottom
400 -

300 Middle

200

100--

0

-100 0.2 0.4 0.6
-100 -

-200 Maximum
Displacement
-300 Used
Displacement of Small Ring (mm)

(a)

Top Plane


H




Bottom Middle

Plane Plane

(b)

Figure 28. Stress vs. displacement and schematic of bending plate. (a) Finite element
analysis simulation of the bending plate (substrate). The radial stresses, Oxx
and yyy, at the center of the top, middle, and bottom planes of the bending
plate vs. the displacement of the smaller ring are shown. oxx= yy. (b)
Illustration of top, middle, and bottom planes of the plate.







72



1.5

Middle Plane


1 Bottom Plane


SMaximum
Displacement Used
V0.5



0 I Top Plane


0 0.2 0.4 0.6 0.8 1
Displacement of Small Ring (mm)


(a)

6000


5000


4000


3000

2000


1000



0 0.2 0.4 0.6 0.8 1
Displacement of Small Ring (mm)


(b)

Figure 29. Finite element analysis simulation of the bending plate (substrate). (a) The
shear stress at the centers of three planes, top, middle, and bottom and (b)the
load required on the smaller ring as a function of displacement are extracted
from the simulations.









the center of three planes, and Fig. 29(b) is the load applied to the smaller ring vs. the

displacement of the smaller ring. On the bottom plane of the substrate, the stress at the

center is tensile as expected while, on the top plane, the center stress appears as

compressive first, then gradually decreases and finally becomes tensile. This can be

explained by the nonlinearity of bending plate with large deflection. At small deflection

(<< the thickness of substrate 0.77 mm), the stress on the top and bottom planes are of

nearly the same magnitude but opposite sign, as shown in Fig. 28. There is no stress in

the middle plane. However, at large deflection, the middle plane stretches and

experiences tensile stresses, which implies that the whole substrate stretches. For the

bottom plane, the total tensile stress will be the sum of the original stress and the

additional tensile stress due to the substrate stretching, while for the top plane, the

compressive stress will be reduced by the additional tensile stress. As the displacement of

smaller ring reaches about 0.89mm, the compressive stress on the top plane will be

completely cancelled out by the tensile stress due to the substrate stretching, as can be

seen from Fig. 28. According to Fig. 29(b), the corresponding load is about 4500N or

10001b when displacement reaches 0.89mm. The maximum deflection achieved in this

work is about 0.46mm, corresponding to about 1100N or 2501b. According to Figs. 28(a)

and 29(a), cxy is about three orders of magnitude smaller than axx and ayy at the center.

Detailed analysis of the shear stress shows that cTxy has no effect on mobility enhancement

at the center of the concentric ring due to symmetry of Txy and Tyx. Thus, the mobility

enhancement at the center is due to biaxial stress alone.

The finite element analysis simulations shown in Figs. 28 and 29 need further

investigations due to no systematic studies of grid convergence and no uncertainty









analysis in sample location variation from exact center. More detailed analysis can be

done in future work.

4.2.3 Uncertainty Analysis

In this subsection, the uncertainty in the applied stress will be estimated. There are

four major sources of uncertainty in applied uniaxial stress using the four-point bending

jig shown in Fig. 24. One major source of uncertainty is the starting point of bending. If

the top plate does not lower enough to make the two top pins contact with the substrate

and the grooves perfectly, the actual applied stress will be smaller than the expected one,

while if it lowers too much, additional stress will be generated and cause the actual stress

to be higher than expected. To estimate the uncertainty of the starting point, we use the

approach as follows [63]: First, lower the top plate such that the two top pins, the

substrate, and the grooves can be seen in contact. Tapping the two top pins, if the pins

can move with slight friction, this is the starting point. Second, measure the distance

between the top and bottom plates at four locations as indicated in Fig. 30. The average

value is used as the distance between the two plates. Finally, repeat the procedure 10

times and calculate the uncertainty with 95% confidence with these 10 values. This value

is the uncertainty of the starting point in terms of deflection. The uncertainty of starting

point is estimated as 0.07mm.

The second source of uncertainty is the micrometer for setting the displacement

[63]. The resolution of the micrometer is 1/1000 inch (0.03mm), the uncertainty is one

half of the resolution or about 0.02mm. The total uncertainty in deflection from the

starting point and micrometer is 0.072 + 0.022 0.073mm.








/I
I



( I Microneler

2 /


Micrometer 0 :

4 )

I


Figure 30. Uncertainty analysis of the starting point for the four-point and concentric-ring
bending experiments. The distance between two plates are measured at the
locations designated 1, 2, 3, and 4.








The third source is the variation of the substrate thickness. The typical thickness of

a 12 inches (300mm) wafer is 77520|tm [64]. The uncertainty in wafer thickness is

0.02mm. The uncertainty in stress from the first three sources can be calculated by

differentiating Eq. (81) [65]

Ac = EH Ay Ey AH, (84)
2aL 2a aL 2af
Q2 3 2 3


= +(A- (85)


In Eq. (84), the variations in L and a are negligible because they are fixed by the grooves

as shown in Fig. 24. In Eq. (85), (AH/H)2 = (20/775)2 0.0007 is negligible compared

with (Ay/y)2 = (0.073/0.57)2 0.02, where 0.57mm is the maximum deflection achieved

in the experiment. The total uncertainty in stress is estimated about 40MPa.

The fourth uncertainty is from the substrate angle misalignment as shown in Fig. 31

[66]. During the experiment, the substrate is difficult to align because there is only a

circular hole (smaller than the substrate) on the top plate, the view from the top and

bottom is obstructed by the metal plate. The only markers that can be used for alignment

are the patterns on the substrate. This will cause an uncertainty in alignment visually

estimated to be about 100, corresponding to about 30MPa uncertainty in stress [66]. The

total uncertainty from all four sources is estimated as -V402 + 302 = 50MPa.

The applied uniaxial stress can be calibrated by extracting the radius of curvature of

the substrate after bending. Let the elastic curve for a beam after bending be y(x), then the

radius of curvature r can be expressed as [67]






























Figure 31. Uncertainty analysis of the misalignment of the substrate with respect to the
pins for the four-point bending experiment.









d2y
1 dx (86)
r



assuming the beam deflections occur only due to bending [67]. Substituting into Eq. (69),

we get

d2y
EH dx
2=- 3 (87)

1+(

Both Eqs. (69) and (87) are valid for either small or large radii of curvature [67].

Extracting the elastic curve y(x) and substituting into Eq. (87), the stress is obtained. The

PHILTEC FiberOptic Displacement Measurement System [68] is used to extract the

elastic curve, and the setup of experiment is shown in Fig. 32. The optical sensor sweeps

across the substrate between the two top pins, and the distance between the sensor and the

substrate is measured and recorded. To reference the original unstressed wafer surface,

the before bending curve is measured first, and the after bending curves are measured

subsequently. Subtracting the before bending curve from the after bending one, the elastic

curve is measured and the result is shown in Fig. 33. A similar method was used by

Uchida et al. [69] previously.

The elastic curve and radius of curvature are extracted from the polished surface of

a bare wafer instead of the device wafer due to poor reflectivity on the patterned and

passivated device wafer. There is a passivation on top of the device wafer, which

typically consists of a phosphorus-doped silicon dioxide layer and then silicon nitride







79





















a 6



p d o






Figure 32. The experimental setup for calibrating the uniaxial stress in the four-point
bending experiment.











20
19
18
17
16
15
14
E 13
12
11
10


8
5 7
6
5
4
3
2
1
0


0 1000


2000


3000


4000


5000


6000


7000


Position x / p.m


Figure 33. Extracted displacement vs. position curves on the upper surface of the
substrate in the four-point bending experiment. The experimental
displacement curves are LSF to 2nd order polynomials with R2=0.9975 and
0.9929 respectively.


__, fitting curve

X2 +
--y = -1E-06x + 0.0069x + 8.4543
--- R2 = 0.9975










,-- -fitting curve

y = -6E-07x2 + 0.0033x + 4.5056
R2 = 0.9929









layer. The reflectivity of passivation is too low for optical sensor to operate accurately,

therefore, the polished surface of bare wafer is used to extract the elastic curve and radius

of curvature.

The extracted elastic curves are least-square fit (LSF) to a second order polynomial

with R2=0.9975 and 0.9929. Substituting the resulting LSF second order polynomial,

y = -10-6x2 +0.0069x+8.4543 and y= 6 x10x2 + 0.0033x + 4.5056 into Eq. (87), the

applied stress is obtained. Figures 34 and 35 are the extracted radius of curvature and

corresponding stress respectively. As can be seen from Figs. 34 and 35, the difference

between the maximum and minimum of the radius of curvature is about 0.01% and also

about 0.01% for applied stress. Figure 36 shows both the calculated stress from

displacement and extracted stress from the measured wafer curvature. The uncertainty

analysis of the stress extracted from the experimental curvature data in Fig. 36 is

described in Appendix C.

For biaxial stress, there are two major sources of uncertainty in the applied biaxial

stress, the starting point and the micrometer. Using the same procedure as for the uniaxial

stress, the uncertainty of starting point is estimated as about 0.04mm, smaller than

0.07mm for the uniaxial stress, because the starting point is easier to be seen with a ring

than pins and grooves. The micrometer has the same 0.02mm uncertainty as in uniaxial

case. The total uncertainty is 0.042 +0.022 & 0.045mm. Since the finite element

analysis simulation is used to predict the stress due to the nonlinear bending, hence no

simple equation similar to Eq. (84) can be used. Instead, the uncertainty range in

displacement is projected to the stress on the finite element analysis calculated stress vs.

displacement curves in Fig. 28 to extract the uncertainty in stress. At each pre-set










-4.44690E+05 -
) 1000 2000 3000 4000 5000 6000 7000
-4.44695E+05


-4.44700E+05

-4.44705E+05
-4.44710E+05



-4.44715E+05

-4.44720E+05


-4.44725E+05

-4.44730E+05

-4.44735E+05

-4.44740E+05

-4.44745E+05
Position x / gm


Figure 34. Extracted radius of curvature vs. position on the upper surface of the substrate
in the four-point bending experiment, corresponding to the upper curve in Fig.
33. The difference between the maximum and minimum radius of curvatures
is only about 0.01%.










-1.46212E+08 -
) 1000 2000 3000 4000 5000 6000 7000

-1.46214E+08


-1.46216E+08


-1.46218E+08


0.. -1.46220E+08


-1.46222E+08


-1.46224E+08


-1.46226E+08


-1.46228E+08


-1.46230E+08
Position x / pm


Figure 35. Extracted uniaxial stress vs. position on the upper surface of the substrate in
the four-point bending experiment, corresponding to the Fig. 34. The
difference between the maximum and minimum stresses is only about 0.01%.










300
A Experimental stress
data extracted from
250 curvature
Calculated value from
S200 displacement

a 150 -

S100 -

50 -

0 -= T i
0 0.1 0.2 0.3 0.4 0.5 0.6
Displacement I mm

Figure 36. Calibration of the four-point bending experiment. The extracted stress values
are close to the calculated ones using Eq. (81) and within the uncertainty
range at 95% confidence level.









displacement point x, we project the uncertainty range at x and find the corresponding

stress interval from the curve as the uncertainty range for the stress. For example, at a

smaller ring displacement of 0.114mm, the uncertainty range of the displacement is

0.114 0.045 < 0.114 < 0.114 + 0.045mm and the corresponding uncertainty range of the

tensile stress on the bottom plane can be found by projection as 48.8 < 80.8 < 113.6MPa

or 113.6 -48.8 = 64.8MPa and the uncertainty range for the compressive stress on the

top plane is 42.2 > -66.8 > -87.4MPa or 42.2 (- 87.4) = 45.2MPa. Figure 37 is the

demonstration of the projection approach. Using this approach, the uncertainty in the

smaller ring displacement can then be converted to an uncertainty in stress on the top and

bottom planes. Note that this approach is only as accurate as the finite element analysis

calculated stress vs. displacement curve.

4.3 Extracting Threshold Voltage, Mobility, and Vertical Effective Field

In this section, the methods of extracting effective mobility, effective vertical

electric field, and zf coefficients will be described. The effective mobility will be

extracted from the drain current in the linear region (low drain bias) for a long-channel

MOSFET. At low drain bias VDS, the linear drain current of an ideal MOSFET can be

approximated as


IDS effC GS -V,)VDS (88)


and the effective mobility /eff can then be expressed as


S= ID (89)
Cox (VGS -V )VDS











600

500

400 Bottom Plane
400 -

300

200


.100 L



-100L

-200
Top Plane
-300

Displacement of Smaller Ring (mm)


Figure 37. Uncertainty analysis for the concentric-ring bending experiment. At each pre-
set displacement point, the displacement with the uncertainty range at 95%
confidence level is projected on the stress vs. displacement curve to get the
stress value with the uncertainty range in stress.









where Cox is the gate oxide capacitance, W and L are the channel width and length

respectively, and VGs and VT are the gate bias voltage and threshold voltage respectively.

The definition of threshold voltage is illustrated in Fig. 38(a). The linear region threshold

voltage is extracted by drawing a tangent line to the IDs-VGS curve at the point where the

slope is the largest and extending it to intercept the x axis. The gate voltage at the

intercept is defined as the threshold voltage. Figure 38(b) is a snapshot of the IDs-VGS

curve and threshold voltage extraction using the Agilent 4155C Semiconductor Parameter

Analyzer. In Fig. 38(b), an additional curve proportional to the gradient of the IDs-VGS

curve, JIDS/IVGs, is also shown to help to determine the point with the largest slope on

the IDS-VGS curve.

The effective vertical electric field Egffis expressed as [70]

Eff Qb + 17Q., (90)



where 7 is a fitting parameter and equal to 1/3 for holes, Qb is the bulk depletion charge,

Q,,, = Cox (VGs V ) is the inversion charge, and Es is the dielectric constant of silicon. At

the interface of gate oxide and silicon channel, the electric displacement continuity gives

Eocox = Escs, (91)


where Eox is the electric field in the oxide, Eox is the dielectric constant of oxide, Es is the

silicon surface field at the interface, and


Es =Qm Qb (92)
With Eqs. (91) and (92), the bulk depletion charge b can be expressed as


With Eqs. (91) and (92), the bulk depletion charge Qb can be expressed as













VGS /V VT=-0.4V


.2 -0.9 -0.6 -0.3


slope cc pfu


S-2.E-04






-4.E-04


(a)

R* fgllent 4155C GRAPH PLOT W Dan 8 9:47:51 1998 PRGE I


MARKE
(R)


ID


20. Ou
/d.lj


-582.00000000f V -61.314100000un
VTH -380.8S613339mV
MOBMRX 80.526* Qm2/Vs


80.52* em2,-Vs )
(cm2./Vss)


4 -- s -
l;2.S:iOuB A L4 7V2/Vs
-r d 6ra-
3: 5.8166u T "L_ 12 .34 56








-A-------
ttz_/ z \5_ _t
\_^_ _


/^^~


2-sU. 11


-1 .20


Vb CV)


C


100.


MOB


8.00






0.00
. 00


(b)


Figure 38. Illustration of extracting threshold voltage. (a)The tangent line with the largest
slope intercepts x axis at -0.4V, therefore, the threshold voltage VT= -0.4V.
(b)Snapshot of the IDS-VGS characteristic and threshold voltage extraction from
Agilent 4155C Semiconductor Parameter Analyzer.


R(


1UU.N .-diV







Qb = Eoxox ..nv, (93)
and the effective vertical field Eeff can then be obtained as

Ef = Eox ( 1)Q Cox [VGS (1 )(VGS -V,)] (94)

where Eox is approximated by VGS / tox, which is valid for VDS< gate capacitance.
The uncertainty in extracted effective mobility can be estimated using Eq. (89) by
evaluating the sensitivity of ueff for variations in IDS, Cox, L, W, VGS, VT, and VDS as
shown in Eqs. (95) and (96) [65],

A/ f) aD= S DS)2+ 2f(A2 2 +Af (AL)2 ffT 2(A)2 (9

+(AVGS 2 A)2VDS


2 2 2 / 2
Aff DS ef AIDS + Cox aeff ACo + L f, f A2
/e'ff e'ffa IDS IDS ) /eff OCox Cox fef OL L
WK J Kwj2 aVG AVS VJQ AV
eUff OW W ff 9VGKS VGS e f 0 VT VT
LVDS ff AVDS (96)
e qf VDS VDS
AIDS + AC0x +(AL2 (AW V GS )2AVGs)
IDS Cx L W VGS VT Y VGS
22
VT 2 AVT AVDs
VGS T VT )YVDS )