UFDC Home  myUFDC Home  Help 



Full Text  
STRAIN EFFECTS ON THE VALENCE BAND OF SILICON: PIEZORESISTANCE IN PTYPE 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 pType Silicon....................................23 2.3.1 Calculations of Hole Transfer and Effective Mass.....................23 2.3.2 Calculation of Relaxation Time and QuantizationInduced 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 StrainedSilicon 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 STRAINEDSILICON PMOSFETS...............................61 4 .1 In tro d u ctio n .................... .. ...... ........ ....... ................. ..................... 6 1 4.2 Wafer Bending Experiments on pMOSFETs ......................................62 4.2.1 FourPoint Bending for Applying Uniaxial Stress......................62 4.2.2 ConcentricRing 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 StressInduced D rain C current C change ............................................... ................ 99 5.2.2 Internal Stress in the Channel ....................................................108 5.2.3 StressInduced Mobility Enhancement at High Temperature...... 108 5.2.4 StressInduced 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 STRESSSTRA 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 zerothorder 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 outofplane 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 101Pa1, 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 strainedSi MOSFET on relaxed SilxGex lay er .......................................................................................... ............. ...... 5 3 Schematic diagram of the uniaxial strainedSi 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 Ek diagram and constant energy surfaces of the heavy and lighthole and splitoff bands near the band edge, k=0, for unstressed silicon ............................ 15 6 Ek diagram and constant energy surfaces of the heavy and lighthole and splitoff bands near the band edge, k=0, for stressed silicon with a uniaxial compressive stress applied along [001] direction. .............................. ................ 18 7 Ek diagram and constant energy surfaces of the heavy and lighthole and splitoff bands near the band edge, k=0, for stressed silicon with a uniaxial com pressive stress applied along [111] direction. .............................. ................ 19 8 Ek diagram and constant energy surfaces of the heavy and lighthole and splitoff 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 Stressinduced 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 Modelpredicted longitudinal z coefficient vs. stress for [001] direction .............40 14 Modelpredicted longitudinal z coefficient vs. stress for [111] direction .............41 15 Modelpredicted longitudinal z coefficient vs. stress for [110] direction .............42 16 The energies of the heavy and lighthole and splitoff 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 strainedSi pM OSFET ................................... ................ 50 19 Illustration of biaxial strainedSi pM OSFET ..................................... ................ 51 20 Inplane 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 Modelpredicted 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 fourpoint 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 concentricring 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 fourpoint and concentricring bending experim ents.. ............ ................ ................................................ 75 31 Uncertainty analysis of the misalignment of the substrate with respect to the pins for the fourpoint bending experim ent......................................... ................ 77 32 The experimental setup for calibrating the uniaxial stress in the fourpoint bending experim ent. ................ .............. .............................................. 79 33 Extracted displacement vs. position curves on the upper surface of the substrate in the fourpoint bending experim ent.................................................. ............... 80 34 Extracted radius of curvature vs. position on the upper surface of the substrate in the fourpoint bending experim ent.................................................. ............... 82 35 Extracted uniaxial stress vs. position on the upper surface of the substrate in the fourpoint bending experim ent............................................................ ............... 83 36 Calibration of the fourpoint bending experiment .............................................. 84 37 Uncertainty analysis for the concentricring 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 shift 6 for longitudinal stress....................................................... ............... 102 45 Average effective channel length ratio 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 Stressinduced gate leakage current change vs. stress. ............... ...................117 53 Definitions ofuniaxial stresses, x, Zy, and Zz, and shear stresses, ixy, iKy, Kxz, K s, Ky 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 PTYPE SILICON AND MOBILITY ENHANCEMENT IN STRAINEDSILICON 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 ptype silicon and mobility enhancements in strainedSi 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 kp 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 fourpoint and concentricring wafer bending experiments used to apply external stresses to pMOSFET devices. The theoretical results show that the piezoresistance r coefficient is stressdependent in agreement with the measured n coefficients. The analytical model predictions for mobility enhancements in uniaxial and biaxial strainedSi 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 stressinduced 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 stressinduced 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 stressinduced band splitting suppresses the interband optical phonon scattering which reduces the mobility degradation. CHAPTER 1 INTRODUCTION Aggressive scaling of complementarymetaloxidesemiconductor (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 [114]. In addition, as CMOS technology advances into the deep submicron regime, processinduced 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 ptype strained silicon as well as the hole mobility enhancement in strainedSi pMOSFETs. Details of the strainstress 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 stressinduced 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 crosssectional 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 strainstress 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 strainedSi pMOSFETs using a thick relaxed SilxGex layer to stretch the Si channel has been studied extensively because high biaxial tensile stress can increase the hole mobility [18]. Figure 2 [9] is a schematic diagram of the biaxial strainedSi MOSFET using relaxed SilxGex. Alternatively, applying uniaxial compressive stress along the channel can also enhance the hole mobility [1012]. Two approaches have been used to apply uniaxial compressive stress; one method employs source and drain refilled with SiGe [1012], 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 ntype strained silicon has been well explained by the manyvalley model [21], while, in ptype 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 strainedSi MOSFET on relaxed SilxGex 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 strainedSi 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 (nchannel metaloxide semiconductor fieldeffect transistors (nMOSFET) and pchannel metaloxide semiconductor fieldeffect 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 strainedSi 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 processinduced stresses on the device performance. Such a model can also be used in process and device simulations to estimate the overall impact from various processinduced 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 StructureA Preview The underlying physics of piezoresistance in ptype silicon and mobility enhancement in strainedSi 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 lighthole and splitoff bands. The heavy and lighthole bands are degenerate at the band edge and the splitoff 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 lighthole bands. To the first order approximation, the contribution from the splitoff band can be neglected [2730] since it is 44meV below the band edge. These stressinduced 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 Ek diagram shows the degeneracy of the heavy and lighthole 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 Ek diagram. On the right hand side of the figure are the Ek 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 Ek diagram, the degeneracy at the band edge is lifted and the lighthole band rises above the heavyhole band with a band splitting AE, resulting in hole repopulation from the Unstressed 4 [001] 1IKeavy Hole Band Light Hole Band ' Splitoff SBand E(k) Stressed t [111] [111] P k , Light Hole Band Hole tUniaxial Stress 11[1111 I Heavy Hole Band * Splitoff Band Figure 4. Strain effect on the valence band of silicon. On the left hand side are the Ek diagram and the constant energy surfaces of the heavy and lighthole and splitoff bands for unstressed silicon. The degeneracy of the heavy and lighthole bands at the band edge is also shown. The Ek diagram and the constant energy surfaces of the three hole bands are shown on the right hand side. The degeneracy is lifted and the lighthole band rises above the heavyhole band with a band splitting AE, causing the hole repopulation from the heavy to lighthole bands. The shapes of the constant energy surfaces of the heavy and lighthole bands are altered and effective masses of the heavy and light holes are also changed. E(k) heavy to lighthole bands. In the meantime, the band structures or the shapes of the constant energy surfaces of the heavy and lighthole 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 strainedSi pMOSFETs. Most of the theoretical works on strain effects on metaloxidesemiconductor fieldeffect transistors (MOSFETs) have employed pseudopotential [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 4x4 kp strain Hamiltonian [27, 28, 32]. The splitoff band will be neglected [2730] 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 lighthole 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 biaxialstrained 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 fourpoint and concentricring 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 FLOOPSISE. 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 KleinerRoth 4x4 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 n and ptype silicon was discovered by Smith 50 years ago [20]. The ntype silicon piezoresistance can be well explained by the manyvalley model [21, 23]. Recently, piezoresistance in ptype 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 lighthole bands due to the stressinduced 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 KleinerRoth 4x4 strain Hamiltonian [27, 32]. The valence band structure, band splitting, and hole repopulation between the heavy and lighthole bands will be explained. In section 2.3, a model of piezoresistance in ptype 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 splitoff 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 splitoff band. Finally, section 2.5 is the summary. 2.2 Review of Valence Band Theory and Explanations of Strain Effects on Valence Band Singlecrystal silicon is a cubic crystal. Without strain or spinorbit 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 spinorbit 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 lighthole band. The pl/2 state is a twofold degenerate band called spinorbit splitoff band. Near the band edge, k=0, the constant energy surface for the p3/2 state can be determined by kp 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 lighthole band respectively. Figure 5 shows the Ek diagram and the constant energy surfaces of the heavy and lighthole and splitoff bands near the band edge for unstressed silicon. As seen from the Ek diagram in Fig. 5, the heavy and lighthole bands are degenerate at the band edge and the splitoff 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 lighthole bands are distorted, usually called "warped" or fluted," due to the coupling between them. As for the splitoff band, it is decoupled from the heavy and lighthole 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 fourfold rotational symmetry or [111] direction with threefold rotational symmetry, the p3/2 state will be decoupled into two ellipsoids. For the [001] direction, the constant energy surfaces of the heavy and lighthole bands become [27, 28] E (k)=(A+ B)k2 +(AB)k2 + (5) Unstressed S[001] V IS ea am am a A=44meV me. .. . S[01o] Heavy Hole Band Light Hole Band ' "a Splitoff Band Figure 5. Ek diagram and constant energy surfaces of the heavy and lighthole and splitoff bands near the band edge, k=0, for unstressed silicon. The heavy and lighthole bands are degenerate at the band edge, as shown in the Ek diagram. The constant energy surfaces of the heavy and lighthole bands are distorted or warped. The splitoff 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 lighthole bands respectively, k2 = k + k2 k2 k The band splitting AEoo1 between the heavy and lighthole bands is expressed as AEooi = 2E0 = 2. 2D (s, S12), (7) 3 where Eo is the energy shift for the heavy and lighthole 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 lighthole 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 lighthole bands is expressed as AE, = 2. =2D s44 (10) where Es is the energy shift for the heavy and lighthole 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 Ek diagrams and constant energy surfaces of the heavy and lighthole and splitoff 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 lighthole bands is lifted with a band splitting energy AE and the heavy and lighthole bands become prolate and oblate ellipsoids respectively with axial symmetry about the stress direction [27, 28]. When a uniaxial compressive stress is applied, the lighthole band will rise above the heavyhole band and holes will transfer from the heavy to lighthole bands because the energy of the lighthole band is lower than the heavyhole band and vice versa. Note that the energy axis E of the Ek 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 twofold rotational symmetry axis, [110] direction, the situation is more complex. The energy surfaces of the heavy and lighthole 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 Splitoff i emO Band Figure 6. Ek diagram and constant energy surfaces of the heavy and lighthole 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 lighthole bands is lifted with a band splitting energy AE and the lighthole band rise above the heavyhole band. The heavy and lighthole 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 Splitoff SBand Figure 7. Ek diagram and constant energy surfaces of the heavy and lighthole 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 lighthole bands is lifted with a band splitting energy AE and the lighthole band rise above the heavyhole band. The heavy and lighthole 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 = AT1 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 heavyhole band E3/2(k) and the lower signs belong to the lighthole 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 lighthole 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 lighthole bands for the [110] direction [27]. Figure 8 shows the Ek diagram and constant energy surfaces of the heavy and lighthole and splitoff 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 heavyhole 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'"" Splitoff Band Figure 8. Ek diagram and constant energy surfaces of the heavy and lighthole 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 lighthole bands is lifted with a band splitting energy AE and the lighthole band rise above the heavyhole band. The heavy and lighthole bands still are ellipsoids yet have three unequal principal axes. E(k) AEi 0.06 0.02 OD k2 0 [001] 004 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) Lighthole band. (b) Heavyhole band. heavy and lighthole bands are ellipsoids with three unequal principal axes and the light hole band rises above the heavyhole band. 2.3 Modeling of Piezoresistance in pType 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 splitoff band is neglected [2730] because the splitoff band is 44meV below the valence band edge [35]. For a nondegenerate ptype 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 densityofstate 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 heavyhole band energy Evh and the lighthole 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 lighthole bands occur and holes repopulate between the heavy and lighthole bands. The concentration changes in the heavyhole band, Aphh, and the lighthole 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 stressinduced energy shifts of the heavy and lighthole bands. Using Eqs. (25) (28), we can get [23] Aphh = Phh AE and Ap F Plh AE (29) ^phkBT 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 Ek 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 mlIh[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] AN/3 and mih[ll A'1] (33) mlhh[111] = and milh[111] / (34) A+N/6 AN/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) AA 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 AB]71 where q7 and 72 are defined in Eq. (15). At high stress, the influence from the splitoff 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 zerothorder, stressindependent 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 zerothorder stressed effective mass m* calculated using Eqs. (31) (37) and Fig. 10 shows the stressinduced 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; mL 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 zerothorder 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 mL 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 QuantizationInduced 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. Stressinduced 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 lighthole 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 [4143] 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 lighthole bands respectively. For nondegenerate 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 [4143] 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 BoseEinstein 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 stressinduced 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 [4648] 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 stressinduced 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 stressinduced band splitting. We use the triangular potential approximation [49, 50] to estimate the fieldinduced band splitting. The quantized energy subbands for the heavy and lighthole 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 nondegenerate 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 fieldinduced band splitting can be approximated as AE = Elhh E11h. (56) The fieldinduced band splitting will then be added to the stressinduced 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 zerothorder 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 [5153]. 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 lighthole 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 heavyhole band and the bottom band represents the lighthole 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 lighthole band and the bottom band represents the heavyhole 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 dopantinduced 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 modelpredicted 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 stressinduced hole repopulation between the heavy and lighthole band and (ii) the correction to the hole effective mass is stressdependent as shown in Eq. (38). The discontinuities at zero stress are due to E \ E S0.4 II S0.3 * LU V 0.2 \ oMMUNo 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 heavyhole band and the bottom band represents the lighthole 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 heavyhole band. After about 60kPa stress, the degeneracy is lifted and the lighthole band rises above the heavyhole band and the stressed effective masses of heavy and light holes saturate. Compression 300 Tension Smith's data 100 Stress / M Pa Figure 13. Modelpredicted 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. Modelpredicted 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. Modelpredicted 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 stressinduced hole transfer between the heavy and lighthole bands and the effective mass shift for light hole due to the stressinduced coupling between the lighthole and splitoff 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 10Pa1 and 56x 101Pa1 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 lighthole and splitoff bands and (ii) incomplete decoupling between the heavy and lighthole 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 lighthole bands will decouple completely and the correction terms will disappear, in contradiction to Eq. (61). The authors obtained 89x 1011Pa1 for longitudinal coefficientt for [111] direction. As for transverse r coefficient, the authors fit the experimental value, 44.5x1011Pa1, and get values for /3 and /2, 2.4x109Pa1 and 2.4x109Pa1 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 KleinerRoth 4x4 strain Hamiltonian [27, 32], which neglects the spiltoff band. To estimate the error introduced by using the KleinerRoth 4x4 strain Hamiltonian [27, 32], we also use the BirPikus 6x6 strain Hamiltonian [55, 56], which takes the splitoff 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 lighthole 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 splitoff band is larger at high stress than zero stress, which implies that the hole concentration in the splitoff band is even smaller at high stress and neglecting the splitoff 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 lighthole bands, heavy and light hole effective masses, stressinduced band splitting, hole repopulation are explained and derived using KleinerRoth 4x4 strain Hamiltonian. At high stress, the influence from the splitoff 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 Splitoff 4x4 \ 0.04 \ Splitoff 6x6 0.06 I1 .I I 500 250 0 250 500 Stress / MPa Figure 16. The energies of the heavy and lighthole and splitoff 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 ptype 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 modelpredicted z coefficients are stressdependent and the tensile and compressive r coefficients are different. The reasons are two folds: (i) the stressinduced 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 splitoff 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 strainedSi pMOSFETs are the same as the piezoresistance effect in ptype silicon as seen from Eqs. (20) and (21). As shown in Fig. 18, by uniaxial strainedSi pMOSFETs, we mean that an inplane 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 strainedSi pMOSFETs, we mean that an inplane 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 strainedSi pMOSFETs are two important technologies used to enhance the hole mobility [114]. Thompson et al. [11, 12, 44] compared uniaxial vs. biaxial in terms of device performance and process complexity and concluded that uniaxial strainedSi 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 outofplane uniaxial compressive stress will be derived for biaxialstrained pMOSFET. The valence band [001] [110] [[1 10] Figure 18. Illustration of uniaxial strainedSi 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 fourpoint wafer bending, which will be described in Chapter 4. [001] [110] [[11 0] Figure 19. Illustration of biaxial strainedSi 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 SilxGex layer to stretch the Si channel [18], or by concentricring wafer bending, which will be described in Chapter 4. structure, in and outofplane 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 StrainedSilicon PMOSFET In this section, we will calculate and compare the mobility improvement in uniaxial and biaxialstrained 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 biaxialstrained pMOSFETs, we will show that inplane, (001), biaxialtensile stress %b, can be represented by an equivalent outofplane 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 strainstress 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 offdiagonal stiffnesses of silicon. For inplane biaxial tensile strain, E = Syy = E, and E= can be derived from the stressstrain 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 inplane biaxial tensile strain, there is no outof 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 inplane biaxial tensile stress in (001) plane and E where E is the Young's modulus, v is the Poisson ratio, and E/(1v)= 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 inplane biaxial tensile and outof plane uniaxial compressive stresses will create the same strain S and thus the same band splitting along [001] direction. Therefore, the inplane biaxial tensile stress %b, can be represented by an equivalent outofplane 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 lighthole bands. Table 3 gives the zerothorder, in and outofplane heavy and light hole effective Table 3. In and outofplane 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 InPlane 0.54 0.15 0.21 0.26 OutofPlane 0.21 0.26 0.28 0.20 masses for uniaxial compression and biaxial tension. Figure 20 shows the inplane 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 biaxialstrained 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. Inplane 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"cm3. 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. Modelpredicted 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 uniaxialstrained pMOSFET [12] are included for comparison. Also included is the model prediction for biaxialstrained 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 lighthole bands. For uniaxial compression, as shown in Figs. 8 and 20, since the heavy and lighthole 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 lighthole 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 lighthole bands results in mobility degradation. In addition, as seen from Fig. 20, the difference in the inplane 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 biaxialstrained pMOSFETs. The effective mass shown next to the constant energy surfaces are the zerothorder inplane 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 lighthole 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 lighthole 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 uniaxialstrained pMOSFET is shown to have large mobility improvement due to hole repopulation from the heavyhole band with larger in plane effective mass to the lighthole band with smaller one. For biaxialstrained pMOSFET, because the inplane effective mass of the heavy hole is smaller than the light hole, hole repopulation from the heavy to lighthole bands degrades the mobility. Both predictions are in good agreement with the published data, 48% improvement for uniaxialstrained pMOSFET and 4% degradation for biaxialstrained 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 STRAINEDSILICON 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 ptype silicon vs. stress will then be calculated. Concentricring and fourpoint 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 fourpoint bending apparatus used to apply uniaxial stress will be explained in detail and equations for calculating the uniaxial stress will be derived. For the concentricring 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 FourPoint Bending for Applying Uniaxial Stress Uniaxial stress is applied to the channel of a pMOSFET using fourpoint 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 fourpoint 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 fourpoint 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 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 oxIower 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 ConcentricRing Bending for Applying Biaxial Stress Biaxial stress is applied to the channel of a pMOSFET using concentricring 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 concentricring 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 concentricring 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 fourpoint 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 fourpoint and concentricring 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 77520tm [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 fourpoint 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 phosphorusdoped 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 fourpoint 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 fourpoint bending experiment. The experimental displacement curves are LSF to 2nd order polynomials with R2=0.9975 and 0.9929 respectively. __, fitting curve X2 + y = 1E06x + 0.0069x + 8.4543  R2 = 0.9975 , fitting curve y = 6E07x2 + 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 leastsquare fit (LSF) to a second order polynomial with R2=0.9975 and 0.9929. Substituting the resulting LSF second order polynomial, y = 106x2 +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 preset 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 fourpoint 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 fourpoint 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 fourpoint 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 longchannel 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 concentricring 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 IDsVGS 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 IDsVGS 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 IDsVGS curve, JIDS/IVGs, is also shown to help to determine the point with the largest slope on the IDSVGS 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 S2.E04 4.E04 (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 \_^_ _ /^^~ 2sU. 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 IDSVGS 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< 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 ) 