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

Transcription

1 STRAIN EFFECTS ON THE VALENCE BAND OF SILICON: PIEZORESISTANCE IN P-TYPE SILICON AND MOBILITY ENHANCEMENT IN STRAINED SILICON PMOSFET By KEHUEY WU A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 5

2 Copyright 5 by Kehuey Wu

3 TABLE OF CONTENTS Page LIST OF TABLES...v LIST OF FIGURES... vi ABSTRACT...x CHAPTER INTRODUCTION.... Relation of Strained Silicon CMOS to Piezoresistive Sensor and Piezoresistivity.... Motivation Valence Band Structure A Preview Focus and Organization of Dissertation... STRAIN EFFECTS ON THE VALENCE BAND AND PIEZORESISTANCE MODEL.... Introduction.... Review of Valence Band Theory and Explanations of Strain Effects on Valence Band Modeling of Piezoresistance in p-type Silicon Calculations of Hole Transfer and Effective Mass Calculation of Relaxation Time and Quantization-Induced Band Splitting Results and Discussion Summary HOLE MOBILITY ENHANCEMENT IN BIAXIAL AND UNIAXIAL STRAINED-SILICON PMOSFET Introduction Mobility Enhancement in Strained-Silicon PMOSFET Discussion Summary...6 iii

4 4 WAFER BENDING EXPERIMENT AND MOBILITY ENHANCEMENT EXTRACTION ON STRAINED-SILICON PMOSFETS Introduction Wafer Bending Experiments on pmosfets Four-Point Bending for Applying Uniaxial Stress Concentric-Ring Bending for Applying Biaxial Stress Uncertainty Analysis Extracting Threshold Voltage, Mobility, and Vertical Effective Field Summary RESULTS AND DISCUSSIONS Mobility Enhancement and π Coefficient versus Stress Discussion Identifying the Main Factor Contributing to the Stress-Induced Drain Current Change Internal Stress in the Channel Stress-Induced Mobility Enhancement at High Temperature Stress-Induced Gate Leakage Current Change Summary SUMMARY, CONTRIBUTIONS, AND RECOMMENDATIONS FOR FUTURE WORK...9 APPENDIX 6. Summary Contributions Recommendations for Future Work... A STRESS-STRAIN RELATION...4 B PIEZORESISTANCE COEFFICIENT AND COORDINATE TRANSFORM.9 C UNCERTAINTY ANALYSIS...3 LIST OF REFERENCES...38 BIOGRAPHICAL SKETCH...46 iv

5 LIST OF TABLES Table Page Values of the inverse mass band parameters and deformation potentials used in the calculations...9 Calculated zeroth-order longitudinal ( ) and transverse ( ) stressed effective masses of the heavy and light holes (in units of m ) for [], [], and [] directions In- and out-of-plane effective masses of the heavy and light holes for uniaxial compression and biaxial tension Stiffness c ij, in units of Pa, and compliance s ij, in units of - Pa -, coefficients of silicon Longitudinal and transverse π coefficients for [], [], and [] directions Experimental data used in Fig Mobility enhancement experimental data and uncertainty for uniaxial longitudinal stresses Mobility enhancement experimental data and uncertainty for uniaxial transverse stresses Mobility enhancement experimental data and uncertainty for biaxial stresses...37 v

6 LIST OF FIGURES Figure Page Definitions of longitudinal and transverse directions for defining π coefficients...3 Schematic diagram of the biaxial strained-si MOSFET on relaxed Si -x Ge x layer Schematic diagram of the uniaxial strained-si pmosfet with the source and drain refilled with SiGe and physical gate length 45nm Strain effect on the valence band of silicon E-k diagram and constant energy surfaces of the heavy- and light-hole and split-off bands near the band edge, k=, for unstressed silicon E-k diagram and constant energy surfaces of the heavy- and light-hole and split-off bands near the band edge, k=, for stressed silicon with a uniaxial compressive stress applied along [] direction E-k diagram and constant energy surfaces of the heavy- and light-hole and split-off bands near the band edge, k=, for stressed silicon with a uniaxial compressive stress applied along [] direction E-k diagram and constant energy surfaces of the heavy- and light-hole and split-off bands near the band edge, k=, for stressed silicon with a uniaxial compressive stress applied along [] direction Top view (observed from [] direction) of the constant energy surfaces with a uniaxial compressive stress applied along [] direction... Stress-induced band splitting vs. stress for [], [], and [] directions...3 The scattering times due to acoustic and optical phonons and surface roughness scatterings...36 Calculated effective masses of heavy and light holes vs. stress using 6 6 strain Hamiltonian Model-predicted longitudinal π coefficient vs. stress for [] direction...4 vi

7 4 Model-predicted longitudinal π coefficient vs. stress for [] direction Model-predicted longitudinal π coefficient vs. stress for [] direction The energies of the heavy- and light-hole and split-off bands vs. stress for [] direction calculated using 4 4 and 6 6 strain Hamiltonians Effective masses vs. stress for the heavy and light holes for [] direction calculated using 4 4 and 6 6 strain Hamiltonians Illustration of uniaxial strained-si pmosfet Illustration of biaxial strained-si pmosfet....5 In-plane effective masses of the heavy and light holes vs. stress for uniaxial compression and biaxial tension Band splitting vs. stress for uniaxial compression and biaxial tension...56 Model-predicted mobility enhancement vs. stress for uniaxial compression and biaxial tension The underlying mechanism of mobility enhancement in uniaxial- and biaxialstrained pmosfets Apparatus used to apply uniaxial stress and schematic of four-point bending Stress at the center of the upper surface of the substrate vs. the deflection of the top pins Stress vs. position and schematic of bending substrate Apparatus used to apply biaxial stress and schematic of concentric-ring bending Stress vs. displacement and schematic of bending plate Finite element analysis simulation of the bending plate (substrate) Uncertainty analysis of the starting point for the four-point and concentric-ring bending experiments Uncertainty analysis of the misalignment of the substrate with respect to the pins for the four-point bending experiment The experimental setup for calibrating the uniaxial stress in the four-point bending experiment vii

8 33 Extracted displacement vs. position curves on the upper surface of the substrate in the four-point bending experiment Extracted radius of curvature vs. position on the upper surface of the substrate in the four-point bending experiment Extracted uniaxial stress vs. position on the upper surface of the substrate in the four-point bending experiment Calibration of the four-point bending experiment Uncertainty analysis for the concentric-ring bending experiment Illustration of extracting threshold voltage Effective hole mobility vs. effective field before and after bending with uniaxial longitudinal tensile and compressive stresses at 6MPa Effective hole mobility vs. effective field before and after bending with uniaxial transverse tensile and compressive stresses at 3MPa Effective hole mobility vs. effective field before and after bending with biaxial tensile stress at 33MPa and compressive stress at 34MPa Mobility enhancement vs. stress π coefficient vs. stress Average effective channel length ratio <r> and variance <σ > vs. gate voltage shift δ for longitudinal stress Average effective channel length ratio <r> and variance <σ > vs. gate voltage shift δ for transverse stress Schematic diagram of doping concentration gradient and current flow pattern near the metallurgical junction between the source/drain and body Simulation result of the internal stress distribution in a pmosfet Mobility enhancement vs. stress at room temperature and C Before and after bending drain current I D and gate current I G vs. gate voltage V G. The after bending gate current coincides with the before bending one Before and after bending drain current I D and gate current I G vs. gate voltage V G. The after bending gate current is much higher than the before bending one viii

9 5 Before and after bending drain current I D and gate current I G vs. gate voltage V G. The after bending gate current is extremely higher than the before bending one Stress-induced gate leakage current change vs. stress Definitions of uniaxial stresses, χ x, χ y, and χ z, and shear stresses, κ xy, κ yx, κ xz, κ zx, κ yz, and κ zy Definitions of strain Measurement errors in X Random and bias errors in gun shots ix

10 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy STRAIN EFFECT ON THE VALENCE BAND OF SILICON: PIEZORESISTANCE IN P-TYPE SILICON AND MOBILITY ENHANCEMENT IN STRAINED-SILICON PMOSFET By Kehuey Wu December 5 Chair: Toshikazu Nishida Cochair: Scott E. Thompson Major Department: Electrical and Computer Engineering This dissertation explores strain effects on the valence band of silicon to explain and model piezoresistance effects in p-type silicon and mobility enhancements in strained-si pmosfets. The strain effects are manifested as changes in the valence band when applying a stress, including band structure alteration, heavy and light hole effective mass changes, band splitting, and hole repopulation. Using the 4 4 k p strain Hamiltonian, the stressed effective masses of the heavy and light holes, band splitting, and hole repopulation are used to analytically model the conductivity and effective mobility changes and the piezoresistance π coefficients. The model predictions agree well with the experiments and other published works. Mobility enhancements and π coefficients are extracted from four-point and concentric-ring wafer bending experiments used to apply external stresses to pmosfet devices. The theoretical results show that the piezoresistance π coefficient is stress-dependent in agreement with the measured π x

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

12 CHAPTER INTRODUCTION Aggressive scaling of complementary-metal-oxide-semiconductor (CMOS) technology has driven the performance improvement of very large scale integrated (VLSI) circuits for years. However, as CMOS technology advances into the nanometer regime, scaling down the channel length of CMOS devices is becoming less effective for performance improvement mainly due to mobility degradation resulting from the high channel doping density, and hence high vertical effective field, in the channel. Strained- Si CMOS provides a very promising approach for mobility enhancement and has been extensively investigated recently [-4]. In addition, as CMOS technology advances into the deep submicron regime, process-induced stresses, for example, shallow trench isolation [5], contact etch stop nitride layer [6], and source and drain silicide [7], etc., may affect device performance and reliability.. 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 [8, 9]. The strain effects responsible for the transduction physics of micromachined piezoresistive sensors is closely related to mobility enhancement in strained silicon CMOS. The strain effect on the valence band of silicon can be used to explain and quantify the piezoresistance effect in p-type strained silicon as well as the hole mobility

13 enhancement in strained-si pmosfets. Details of the strain-stress relation in a material with cubic symmetry such as silicon are discussed in Appendix A. The piezoresistance effect in strained silicon, first discovered by Smith [] fifty years ago, is the stress-induced resistance change. A coefficient, π, used to characterize the piezoresistance is defined as [8] Rχ R π = χ R R =, ( ) χ R where R and R χ are the unstressed and stressed resistances and χ is the uniaxial stress. Since R = ρa / L where ρ is the resistivity, A is the cross-sectional area, and L is the length, the resistance change may be due to a combination of resistivity change and geometry change. However, in semiconductors, the contribution from the geometrical change may be neglected because it is 5 times smaller than the resistivity change [9]. Hence, the π coefficient may be expressed in terms of resistivity change or conversely in terms of conductivity change as follows: R π = χ R ρ σ = =, ( ) χ ρ χ σ where ρ and σ are the resistivity and conductivity respectively. This conductivity change is directly related to mobility change since σ=qµp where q is the electronic charge, µ is the hole mobility, and p is the valence hole concentration. Two types of uniaxial stresses are defined in Fig. in order to distinguish two kinds of π coefficients, longitudinal and transverse [8]. Further details on the piezoresistance coefficient and coordinate transformations for an arbitrary direction are given in Appendix B.

14 3 Figure. Definitions of longitudinal and transverse directions for defining π coefficients (adapted from Ref. [8]). The strain-stress relation is discussed in more detail in Appendix A.

15 4 Longitudinal means that the uniaxial stress, electric field, and electric current are all in the same direction. Transverse means that the electric field is parallel to the electric current but normal to the uniaxial stress. Recently, biaxial strained-si pmosfets using a thick relaxed Si -x Ge x layer to stretch the Si channel has been studied extensively because high biaxial tensile stress can increase the hole mobility [-8]. Figure [9] is a schematic diagram of the biaxial strained-si MOSFET using relaxed Si -x Ge x. Alternatively, applying uniaxial compressive stress along the channel can also enhance the hole mobility [-]. Two approaches have been used to apply uniaxial compressive stress; one method employs source and drain refilled with SiGe [-], another uses highly compressive stress SiN layer [3]. Figure 3 [4] 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 5% hole mobility enhancement can be achieved with about 5MPa [, ]; however, the biaxial tensile stress needs more than GPa 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].. Motivation The piezoresistance effect in n-type strained silicon has been well explained by the many-valley model [], while, in p-type strained silicon, the piezoresistance effect has not yet been fully understood and characterized; most of the previous theoretical

16 Figure. Schematic diagram of the biaxial strained-si MOSFET on relaxed Si -x Ge x layer (adapted from Ref. [9]). 5

17 Figure 3. Schematic diagram of the uniaxial strained-si pmosfet with the source and drain refilled with SiGe and physical gate length 45nm (adapted from Ref. [4]). 6

18 7 works [9,, 3] only show the longitudinal and transverse π coefficients along [] 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 6 N/m ( MPa) to 7 N/m ( 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 []. However, much higher fixed stresses on the order of to MPa in the surface is required to significantly improve the performance of transistor devices (n-channel metal-oxidesemiconductor field-effect transistors (nmosfet) and p-channel metal-oxidesemiconductor field-effect transistors (pmosfet)) [5, 4, 5]. There is a need for a better fundamental understanding of the effect of these high stresses and corresponding strains on the carrier transport properties in advanced MOSFETs and the influence of quantum confinement in nanostructures as well as temperature. Since the piezoresistance effect is precisely the conductivity enhancement obtained in strained-si CMOS, understanding strain effects in semiconductors is vital for continuing performance enhancement in advanced CMOS technologies. An accurate model is needed to estimate the impact of process-induced stresses on the device performance. Such a model can also be used in process and device simulations to estimate the overall impact from various process-induced stresses. Recently, due to the advance of silicon IC technology, the mass production of high precision sensors and the integration of mechanical sensors and electronic circuits (system on chip SOC) are now possible [8]. To design high precision sensors or SOC, a

19 8 more accurate model and better understanding of the piezoresistance effect in silicon are needed..3 Valence Band Structure A Preview The underlying physics of piezoresistance in p-type silicon and mobility enhancement in strained-si pmosfets can be explained by the strain effect on the valence band of silicon. The valence band of silicon consists of three bands, heavy- and light-hole and split-off bands. The heavy- and light-hole bands are degenerate at the band edge and the split-off band is 44meV below the band edge [6, 7]. Applying stress to silicon will lift the degeneracy and alter the valence band structure. As a result, the effective masses of the heavy and light holes will change and holes will repopulate between the heavy- and light-hole bands. To the first order approximation, the contribution from the split-off band can be neglected [7-3] since it is 44meV below the band edge. These stress-induced changes in the valence band are collectively called the strain effect. Figure 4 is the illustration of the strain effects in silicon. Both the unstressed and stressed silicon (with a uniaxial stress applied along the [] direction) are shown in the figure. For unstressed silicon on the left hand side of the figure, the E-k diagram shows the degeneracy of the heavy- and light-hole bands at the band edge and the splitoff band is 44meV below the band edge. The energy surfaces of three hole bands are shown next to the E-k diagram. On the right hand side of the figure are the E-k diagram and the constant energy surfaces of the three hole bands for stressed silicon, assuming a uniaxial compressive stress is applied along the [] direction. As can be seen from the E-k diagram, the degeneracy at the band edge is lifted and the light-hole band rises above the heavy-hole band with a band splitting E, resulting in hole repopulation from the

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

21 heavy- to light-hole bands. In the meantime, the band structures or the shapes of the constant energy surfaces of the heavy- and light-hole bands are altered, causing the changes in the effective masses of the heavy and light holes. Detailed discussion of the strain effects will be presented in Chapter..4 Focus and Organization of Dissertation This dissertation will mainly focus on creating a simple, analytical model that can be easily understood and provide quick, accurate predictions for the piezoresistance in p- type silicon and mobility enhancement in strained-si pmosfets. Most of the theoretical works on strain effects on metal-oxide-semiconductor field-effect transistors (MOSFETs) have employed pseudo-potential [, 3] or k p full band numerical simulations [3, 7]. A simple, analytical model can provide a quick check for numerical simulations as well as provide physical insight. We develop a simple, analytic model using 4 4 k p strain Hamiltonian [7, 8, 3]. The split-off band will be neglected [7-3] but the influence from it is added into the model as a correction. In Chapter, the strain effects on the valence band of silicon will be explained and quantified using 4 4 strain Hamiltonian. The stressed effective masses of the heavy and light holes, band splitting between the heavy- and light-hole bands, and the amount of hole repopulation will be calculated. Then, the longitudinal and transverse π coefficients along three major crystal axes, [], [], and [] directions will be calculated. The result of the strain effects calculated using 4 4 strain Hamiltonian will be compared with the result using 6 6 Hamiltonian to estimate the valid stress range that the model is applicable. Chapter 3 is the calculation and comparison of mobility enhancements in uniaxial- and biaxial-strained Si pmosfets using the results of the

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

23 CHAPTER STRAIN EFFECTS ON THE VALENCE BAND AND PIEZORESISTANCE MODEL. Introduction In this chapter, the strain effects on the valence band will be explained in detail. The valence band theory will be reviewed first. The equations of the constant energy surfaces, effective masses, band splitting, and hole repopulation, which are derived from Kleiner-Roth 4 4 strain Hamiltonian [7, 3], will be presented and explained. The equations will then be used to model the piezoresistance in p-type silicon. The piezoresistance in n- and p-type silicon was discovered by Smith 5 years ago []. The n-type silicon piezoresistance can be well explained by the many-valley model [, 3]. Recently, piezoresistance in p-type silicon has been modeled in terms of stressinduced conductivity change due to two key mechanisms [9,, 3]: (i) the difference in the stressed effective masses of the heavy and light holes and (ii) hole repopulation between the heavy- and light-hole bands due to the stress-induced band splitting. However, previous works [9,, 3] only focus on the piezoresistance along the [] direction. In this chapter, we will extend the previous works to model the piezoresistance along three major crystal axes, [], [], and [] directions. In section., we review the valence band theory and explain the strain effect on the valence band using Kleiner-Roth 4 4 strain Hamiltonian [7, 3]. The valence band structure, band splitting, and hole repopulation between the heavy- and light-hole bands will be explained. In section.3, a model of piezoresistance in p-type silicon is presented. The longitudinal and transverse conductivity effective masses of the heavy and light

24 3 holes, the magnitude of the band splitting and hole repopulation, and the corrections due to the influence of the split-off band are calculated. The relaxation times due to the acoustic and optical phonon scatterings are calculated. Since the model is employed to estimate mobility enhancement on pmosfets, surface roughness scattering is also taken into account. In addition, the quantization effect in the inversion layer of a pmosfet is also considered. Section.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 4 4 with the 6 6 strain Hamiltonians, which includes the split-off band. Finally, section.5 is the summary.. Review of Valence Band Theory and Explanations of Strain Effects on Valence Band Single-crystal silicon is a cubic crystal. Without strain or spin-orbit interaction, the valence band at the band edge, k=, is a sixfold degenerate p multiplet due to cubic symmetry [7]. The sixfold p multiplet is composed of three bands, and each band is twofold degenerate due to spin. The spin-orbit interaction lifts the degeneracy at the band edge, and the sixfold p multiplet is decomposed into a fourfold p 3 multiplet, J=3/ state, and a twofold p multiplet, J=/ state, with splitting energy Λ=44meV between the two p multiplets [6, 7]. The p 3 state consists of two twofold degenerate bands designated as heavy- and light-hole band. The p state is a twofold degenerate band called spin-orbit split-off band. Near the band edge, k=, the constant energy surface for the p 3 state can be determined by k p perturbation [7, 33], approximated as

25 4 4 E ( k) = Ak ± B k + C ( k k + k k + k k ), ( 3 ) x y y z z x and the p state is given by E ( k) = Ak Λ, ( 4 ) where A, B, and C are the inverse mass band parameters determined by cyclotron resonance experiments [7, 33, 34]. The upper and lower signs in Eq. (3) represent the heavy- and light-hole band respectively. Figure 5 shows the E-k diagram and the constant energy surfaces of the heavy- and light-hole and split-off bands near the band edge for unstressed silicon. As seen from the E-k diagram in Fig. 5, the heavy- and light-hole bands are degenerate at the band edge and the split-off band is below the band edge with a splitting energy Λ=44meV. Also seen from Fig. 5, the constant energy surfaces of the heavy- and light-hole bands are distorted, usually called warped or fluted, due to the coupling between them. As for the split-off band, it is decoupled from the heavy- and light-hole bands and has a spherical constant energy surface. Applying stress to silicon will break the cubic symmetry and lift the degeneracy of the fourfold p 3 multiplet at the band edge [7, 8]. If a uniaxial stress is applied along an axis with higher rotational symmetry, for example, [] direction with four-fold rotational symmetry or [] direction with three-fold rotational symmetry, the p 3 state will be decoupled into two ellipsoids. For the [] direction, the constant energy surfaces of the heavy- and light-hole bands become [7, 8] E 3 ( k) = ( A + B) k + ( A B) k + ε, ( 5 )

26 5 Figure 5. E-k diagram and constant energy surfaces of the heavy- and light-hole and split-off bands near the band edge, k=, for unstressed silicon. The heavy- and light-hole bands are degenerate at the band edge, as shown in the E-k diagram. The constant energy surfaces of the heavy- and light-hole bands are distorted or warped. The split-off band is 44meV below the band edge and has a spherical constant energy surface.

27 6 E ( k) = ( A B) k + ( A + B) k ε, ( 6 ) where E ) and E ) are the heavy- and light-hole bands respectively, 3 ( k ( k k + = k x k y, k = k. The band splitting E between the heavy- and light-hole bands z is expressed as E = ε = Du ( s s)χ, ( 7 ) 3 where ε is the energy shift for the heavy- and light-hole bands for the [] direction, χ is the uniaxial stress, D u is the valence band deformation potentials for [] direction, s and s are compliance coefficients of silicon. The definition of compliance coefficients is given in Appendix A. Along the [] direction, the heavy- and light-hole bands become [7, 8] ' E 3 ( k) = ( A + N) k + ( A N) k + ε, ( 8 ) 6 3 ' E ( k) = ( A N) k + ( A + N) k ε, ( 9 ) 6 3 where k + = k k, k =, k and k are along the [ ] and [ ] directions k3 respectively, k 3 is along the [] direction, and N + = 9B 3C is an inverse mass band parameter. The band splitting E between the heavy- and light-hole bands is expressed as ' ' s44 E = ε = Du χ, ( ) 3

28 7 ' where ε is the energy shift for the heavy- and light-hole bands for the [] direction, s 44 is a compliance coefficient of silicon, ' D u is the valence band deformation potential for the [] direction. Figures 6 and 7 show the E-k diagrams and constant energy surfaces of the heavy- and light-hole and split-off bands for stressed silicon with uniaxial compressive stresses applied along the [] and [] directions respectively. As seen from Figs. 6 and 7, the degeneracy of the heavy- and light-hole bands is lifted with a band splitting energy E and the heavy- and light-hole bands become prolate and oblate ellipsoids respectively with axial symmetry about the stress direction [7, 8]. When a uniaxial compressive stress is applied, the light-hole band will rise above the heavy-hole band and holes will transfer from the heavy- to light-hole bands because the energy of the light-hole band is lower than the heavy-hole band and vice versa. Note that the energy axis E of the E-k diagrams represents the electron energy. The hole energy is the negative of the electron energy and in the opposite direction of the electron energy axis. Therefore, by valence band rising or falling it means that the hole energy in the valence band is decreasing or increasing respectively. When applying a uniaxial stress along the two-fold rotational symmetry axis, [] direction, the situation is more complex. The energy surfaces of the heavy- and light-hole bands still are ellipsoids yet have three unequal principal axes and the constant energy surface of the heavy- and light- hole band become [7, 8] where ± k) = k + k + k3 m m m3 ' ( ε + 3 ) h h h E ( ± ε, ( )

29 8 Figure 6. E-k diagram and constant energy surfaces of the heavy- and light-hole and splitoff bands near the band edge, k=, for stressed silicon with a uniaxial compressive stress applied along [] direction. The degeneracy of the heavy- and light-hole bands is lifted with a band splitting energy E and the light-hole band rise above the heavy-hole band. The heavy- and light-hole bands become prolate and oblate ellipsoids respectively with axial symmetry about the stress direction.

30 9 Figure 7. E-k diagram and constant energy surfaces of the heavy- and light-hole and splitoff bands near the band edge, k=, for stressed silicon with a uniaxial compressive stress applied along [] direction. The degeneracy of the heavy- and light-hole bands is lifted with a band splitting energy E and the light-hole band rise above the heavy-hole band. The heavy- and light-hole bands become prolate and oblate ellipsoids respectively with axial symmetry about the stress direction.

31 h m B N = A m η ± η, ( ) h m = A ± Bη, ( 3 ) h m 3 B N = A m η m η, ( 4 ) and η = and + 3β β 3 η =, ( 5 ) + β ' with β = ε ε. The upper signs in Eqs. () (4) belong to the heavy-hole band E 3/ (k) and the lower signs belong to the light-hole band E / (k). In Eq. (), k 3 is the longitudinal direction along the [] direction; k and k are two transverse directions along the [ ] and [] directions respectively. The band splitting E between the heavy- and light-hole bands is expressed as ' In Eq. (), ( 3 ) ε ' ( ε + ) E. ( 6 ) = 3ε ε + is the energy shift for the heavy- and light-hole bands for the [] direction [7]. Figure 8 shows the E-k diagram and constant energy surfaces of the heavy- and light-hole and split-off bands for stressed silicon with a uniaxial compressive stress applied along the [] direction. Figure 9(a) and (b) are the top views of the constant energy surfaces of the light- and heavy-hole bands respectively as observed from the [] direction. As can be seen from Figs. 8 and 9, the constant energy surfaces of the

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

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

34 3 heavy- and light-hole bands are ellipsoids with three unequal principal axes and the lighthole band rises above the heavy-hole band..3 Modeling of Piezoresistance in p-type Silicon.3. Calculations of Hole Transfer and Effective Mass From Eq. () and the illustrations in Fig., the longitudinal and transverse π coefficients can be defined as [3] ρ σ σ χ σ π l = = =, ( 7 ) χ ρ χ σ χ σ l l l ρ σ σ χ σ π t = = =, ( 8 ) χ ρ χ σ χ σ t t t where χ l and χ t are the longitudinal and transverse uniaxial stresses respectively, and σ are the stressed and unstressed conductivity respectively, and σ χ p p σ = ) hh lh q τ ( + qpµ eff mhh mlh, ( 9 ) where µ eff is the effective carrier mobility, q is the electron charge, τ is the hole relaxation time, m hh and m lh are the heavy and light hole conductivity effective mass respectively. For silicon, the resistance change due to the geometrical change is 5 times smaller than the resistivity change [9], therefore, in Eqs. (7) and (8), the contribution from the geometrical change is neglected. Using Eq. (9), Eqs. (7) and (8) can then be expressed as σ σ χ σ π l = = χ σ χ σ l l µ µ = = χ µ χ l eff l χeff µ µ eff eff, ( )

35 4 σ σ χ σ π t = = χ σ χ σ t t µ µ = = χ µ χ t eff t χeff µ µ eff eff, ( ) where µ χeff and µ eff are the stressed and unstressed effective mobility, respectively. In Eq. (9), p p hh + p = lh is the total hole concentration, hh p and p lh are the heavy and light hole concentrations respectively. In order to simplify the model, the contribution from the split-off band is neglected [7-3] because the split-off band is 44meV below the valence band edge [35]. For a non-degenerate p-type silicon, p hh and p lh are given by [36], p hh * πm hhk = ( h B T ) 3 EF E exp( k T B v ) = N vh EF E exp( k T B v ), ( ) p lh * πmlhk = ( h B T ) 3 EF E exp( k T B v ) = N vl EF E exp( k T B v ), ( 3 ) * * where m hh =.49m and m lh =.6m [37, 38] are the density-of-state effective mass of the heavy and light hole respectively, m is the free electron mass, k B is the Boltzmann constant, T is the absolute temperature, E F is the Fermi level, and E v is the energy at the valence band edge. For the unstressed case, the valence band is degenerate at the band edge, k=, the heavy-hole band energy E vh and the light-hole band energy E vl are equal and E vh = E vl = E v. Using Eqs. () and (3), the heavy and light hole concentrations can then be calculated from the doping density p [3]: p *3 *3 = mhh p *3 *3 m + m and mlh p m m p lh =. ( 4 ) *3 *3 + hh hh lh hh lh

36 5 When stress is applied, energy splitting of the heavy- and light-hole bands occur and holes repopulate between the heavy- and light-hole bands. The concentration changes in the heavy-hole band, p hh, and the light-hole band, p lh, can be obtained by differentiating Eqs. () and (3) [39] p hh = N vh EF E exp( k T B vh )( )( E k T B F E vh phh ) = ( E k T B F E vh ), ( 5 ) p lh = N vl EF E exp( k T B vl )( )( E k T B F E vl plh ) = ( E k T B F E vl ), ( 6 ) with p + p =, ( 7 ) hh lh E E = E, ( 8 ) vh vl where Evh and Evl are stress-induced energy shifts of the heavy- and light-hole bands. Using Eqs. (5) (8), we can get [3] p E p E = m. ( 9 ) hh phh = ± * * 3 k BT + ( mhh mlh ) and lh plh * * 3 k ( ) BT + mlh mhh In Eq. (9), the upper and lower signs are for uniaxial tensile and compressive stress respectively. The conductivity effective masses of the heavy and light holes, m hh and m lh, can be derived from the E-k dispersion relations described in section., Eqs. (5), (6), (8), (9), and (),

37 6 h m =. ( 3 ) d E dk For a uniaxial stress applied along [] direction, if the inverse mass band parameters, A and B, are given in units of h m, where m is the free electron mass, using Eqs. (5), (6), and (3), the longitudinal ( ) and transverse ( ) effective masses of the heavy and light holes can be obtained as [7, 8] [] = and A B m hh m hh [] = and A + B m lh [] =, ( 3 ) A + B [] =, ( 3 ) A B m lh where the units of effective masses are normalized by m. Along the [] direction, if N is also given in units of h m are obtained as [7, 8], then using Eqs. (8), (9), and (3), the effective masses m hh [] = and A N 3 m lh [] =, ( 33 ) A + N 3 m hh [] = and A + N 6 [] =. ( 34 ) A N 6 m lh For [] direction, the situation is more complex due to the ellipsoids with three unequal axes. We define m, m, and m 3 as the effective masses corresponding to k, k, and k 3 as defined in Eq. (). The longitudinal effective masses of the heavy and light holes can be obtained using Eqs. (), (4), and (3) [7, 8]

38 7 m3 hh [] = and B N A η η =, ( 35 ) N η m 3 [] lh B A + η + and the two transverse effective masses of the heavy and light holes can be obtained from Eqs. () (3) and (3) = and N η m [] hh B A η + m lh [] =, ( 36 ) B N A + η η m hh [] A + = and Bη m lh [] =, ( 37 ) A Bη where η and η are defined in Eq. (5). At high stress, the influence from the split-off band is no longer negligible and the 4 4 strain Hamiltonian is subject to a correction [7, 8]. The correction can be expressed in terms of effective mass shift added to the stressed effective masses obtained in Eqs. (3)-(37), which are zeroth-order, stress-independent stressed effective masses * m. The experimentally measured effective mass formula [8] * m can be expressed by an empirical m * + = m m = * * m +αχ, ( 38 ) where α is a parameter [7, 8]. For a special case that a uniaxial stress is applied along a higher rotational symmetry axis, the [] direction (four fold) or the [] 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 [] and [] directions are given by [7, 8],

39 8 Λ = ± = [] 4 ε χ α B m and Λ = = [] ε χ α B m m, ( 39 ) Λ = ± = ' [] 3 4 ε χ α N m and Λ = = ' ] [ 3 ε χ α N m m. ( 4 ) In Eqs. (39) and (4), the upper and lower signs are for uniaxial tensile and compressive stress respectively. For [] direction, due to the lower rotational symmetry (two fold), both the heavy and light holes will experience effective mass shifts [7, 8]. For the heavy hole, the effective mass shifts, (/m hh ), (/m hh ), and (/m 3hh ), are expressed as [8] ( ) ( ) + Λ + Λ = = ' ' ' ' ' [] ε ε ε ε ε ε ε ε ε ε χ α N B m hh, ( 4 ) ( ) + Λ = = ' ' [] 3 3 ε ε ε ε ε χ α B m hh, ( 4 ) ( ) ( ) + Λ + + Λ = = ' ' ' ' ' 3 [] ε ε ε ε ε ε ε ε ε ε χ α N B m hh. ( 43 ) For light holes, the three effective mass shifts are given by ( ) ( ) + + Λ + + Λ = = ' ' ' ' ' [] ε ε ε ε ε ε ε ε ε ε χ α N B m lh, ( 44 ) ( ) + + Λ = = ' ' [] 3 3 ε ε ε ε ε χ α B m lh, ( 45 )

40 9 ' ' ( ) ( ) B 3ε ε N ε ε ' = α = χ ε ε. ( 46) m Λ ' + Λ ' + 3 lh [] ε 3ε ε 3ε 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. The values of A, B, and N are given in units of ħ /m [7]. Table. Values of the inverse mass band parameters and deformation potentials used in the calculations. Parameters Symbol Units Values Inverse Mass Band Parameters A h m -4.8 Ref.[7] Inverse Mass Band Parameters B h m -.75 Ref.[7] Inverse Mass Band Parameters N h m Ref.[7] Deformation Potentials Deformation Potentials D u ev 3.4 Ref.[4] ' D u ev 4.4 Ref.[4] The calculated effective mass and band splitting will then be substituted into the piezoresistance model to predict π coefficients. Table lists the zeroth-order stressed effective mass * m calculated using Eqs. (3) - (37) and Fig. shows the stress-induced band splitting vs. stress for [], [], and [] directions calculated using Eqs. (7), (), and (6). In Table, for the [] and [] directions, m is the longitudinal effective mass, i.e., the effective mass calculated using Eq. (3) and the direction of k is parallel to the direction of the stress; m is the transverse effective mass, i.e., the effective mass calculated using Eq. (3) and the direction of k is perpendicular to the direction of the stress. For the [] direction, m, m, and m 3 are the effective masses with the

41 3 Table. Calculated zeroth-order longitudinal ( ) and transverse ( ) stressed effective masses of the heavy and light holes (in units of m ) for [], [], and [] directions. [] m m heavy.8 light. heavy. light.6 [] m m heavy.86 light.4 heavy.7 light.37 [] m [ ] m [ ] m 3 [ ] heavy.6 light.44 heavy. light.6 heavy.54 light.5 direction of k along the directions of k, k, and k 3 respectively and the direction of k 3 is along the stress direction []..3. Calculation of Relaxation Time and Quantization-Induced Band Splitting In a lowly doped surface inversion channel, the relaxation time τ in Eq. (9) is due mainly to three scattering mechanisms: acoustic and optical phonon and surface

42 3.5 []. Band Splitting / ev.5. [] [] Stress / MPa Figure. Stress-induced band splitting vs. stress for [], [], and [] directions.

43 3 roughness scattering. Including surface roughness scattering is necessary because the model will be verified by experiments on hole inversion channel in pmosfets. The relaxation time can be calculated using approximate analytical equations based on certain assumptions. To calculate the acoustic and optical phonon scattering times, the following assumptions have been made: (i) the heavy- and light-hole bands are assumed to be parabolic as shown in Eqs. (5), (6), (8), (9), and () [4], (ii) the silicon is nondegenerate [4], (iii) the acoustic phonon scattering is elastic and the optical phonon scattering is inelastic and the corresponding scattering times depend only on hole energy [4, 4], and (iv) all holes are scattered isotropicly [4, 4]. For the acoustic phonon scattering, the scattering time τ ac is expressed as [4-43] τ ac ( ε ) ( m m ) k T Eac t B = ε, ( 47 ) 4 π h ρ l ul where ε is the energy, E ac = 5. 3eV [4] is the acoustic deformation potential constant of the valence band, m l and m t are longitudinal and transverse effective mass, ρ is the density of silicon, and u l is the longitudinal sound velocity. The total acoustic phonon scattering time is given by [4] where τ ac, hh ( ε ) and τ lh ( ε ) τ = +, ( 48 ) ( ε ) τ ( ε ) τ ( ε ) ac, total ac, hh ac, lh ac, are the acoustic phonon scattering time in the heavy- and light-hole bands respectively. For non-degenerate silicon, the average scattering time for the acoustic phonon can be obtained as [4]

44 ε ε ε τ ac total τ ac total ( ε ) d π kbt kbt =,, exp. ( 49 ) 3 kbt Since acoustic phonon energy is very small compared to the carrier thermal energy [4], the acoustic phonon scattering mainly occurs in the intraband scattering and the stressinduced band splitting will not affect its scattering rate [44, 45]. [4-43] For optical phonon scattering, without stress, the scattering time τ opu is given by τ opu ( ε ) = π D ( m m ) t l kb h Θρ N q Θ T [ ] ( ε + k Θ) + exp Re ( ε k Θ) B B, ( 5 ) where 8 D = 6.6 ev / cm [4] is the optical deformation potential constant of the valence band, Θ = 735K is the Debye temperature and k B Θ = h ω 63meV is the = optical phonon energy [4], N q is the Bose-Einstein phonon distribution [4] and N q [ ( ) ] exp h ω k T = [ exp( Θ ) ] [4]. The Re in Eq. (5) means Re ( A ) = A = T B if A is real; Re ( A ) = if A is a complex number [4]. Applying stress, the stress-induced band splitting will change the scattering rate [44, 45], τ ops ( ε ) = π D ( m m ) t l B h k Θρ N q Θ T [ ] ( ε + k Θ E) + exp Re ( ε k Θ E) B B, ( 5) where E is the band splitting energy. Using Eqs. (48) and (49) with τ ac replaced by τ opu and τ ops in Eqs. (5) and (5) respectively, the average optical phonon scattering time can be obtained.

45 34 For the surface roughness scattering, the scattering time τ sr 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 4 ( 4) ( q) L exp ( Lq) = π, ( 5 ) where =.7Å [46] is the r.m.s value of the roughness asperities and L =.3Å [46] is the roughness correlation length, q = k sin( θ ), and k is the crystal momentum. Using the simple parabolic band approximation, k can be expressed as k = mε h. Then, the surface roughness scattering time τ sr is obtained as [46-48] τ sr ( ε ) e Eeff m = 3 πh π ( cos( θ )) S( q) dθ, ( 53 ) where E eff 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 5MPa stress, the corresponding strain is only about.3%, thus the L and in Eq. (5) is essentially unchanged. In addition, surface roughness scattering is independent of the stress-induced band splitting. Therefore, the surface roughness scattering will not be affected by stress. Using Eqs. (48), (49), (5), and (5) with τ ac replaced by τ sr, the average scattering time due to surface roughness scattering can be obtained. Then the total scattering time τ in Eq. (9) can be calculated: = + +. ( 54 ) τ τ τ τ ac, total op, total sr, total

46 35 Figure shows the calculation results of the scattering times. It is assumed that a longitudinal tensile stress is applied along the [] direction on a () wafer. In calculation of the surface roughness scattering, the vertical effective field is chosen at.7mv/cm because the mobility enhancement in the pmosfet will be extracted at.7mv/cm. As seen from Fig., the surface roughness scattering is the most significant compared to the acoustic and optical phonon scatterings. The optical phonon scattering time increases as the stress increases, while the surface roughness and the acoustic phonon scatterings are independent of stress. Since the mobility enhancement and piezoresistance will be extracted using pmosfets, in addition to the surface roughness scattering, we also must consider the quantization effect in the channel due to surface effective field. Like stress, the surface electric field can lift the degeneracy at the valence band edge and cause band splitting due to the difference in the effective mass of the heavy and light hole. This band splitting must be considered in addition to the stress-induced band splitting. We use the triangular potential approximation [49, 5] to estimate the field-induced band splitting. The quantized energy subbands for the heavy- and light-hole bands can be approximated by [49, 5] E jhh 3 3 hqξs = j mhh / 3 and E jlh 3 3 hqξs = j mlh / 3, j=,,,., ( 55 ) respectively, where ξ s is the surface electric field, m hh and m lh are the heavy and light hole effective mass normal to the surface, and h is the Planck s constant. For our application, only non-degenerate silicon is considered, therefore, we only take into

47 Scattering Time / ps Acoustic Phonon Scattering Optical Phonon Scattering.5 Total Surface Roughness Scattering Stress / MPa Figure The scattering times due to acoustic and optical phonons and surface roughness scatterings. It is assumed that a longitudinal tensile stress is applied along [] direction on a () wafer.

48 37 account the first subband, i.e., j=. Then the field-induced band splitting can be approximated as E ξ = E hh E lh. ( 56 ) The field-induced band splitting will then be added to the stress-induced band splitting to calculate the hole repopulation and the total band splitting is E = E +. ( 57 ) total strain E ξ.4 Results and Discussion In this section, π coefficients will be calculated and compared with the published data [8, ]. Later in Chapter 5, the calculated π coefficients along [] and [] directions will be compared with the experimental results. Using the definitions of π coefficient and conductivity in Eqs. (7) (9), hole concentrations in Eq. (4), hole repopulation in Eq. (9), the zeroth-order stressed effective masses in Eqs. (3) (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, σ χ and σ, can be calculated from p σ χ = q τ hh + p m hhχ hh + plh + p m lhχ lh and phh plh σ = q τ +, ( 58 ) mhhχ mlhχ and the longitudinal and transverse π coefficients can be obtained from Eqs. () and (). To calculate the unstressed conductivity σ 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

49 38 but not insignificant lattice stress, estimated about 6kPa [5-53]. This small lattice stress can lift the degeneracy at the valence band edge and change the shapes of constant energy surfaces of the heavy- and light-hole bands and the effective masses of the heavy and light holes. Figure is a plot of calculated effective masses of holes in top and bottom bands vs. stress using 6 6 strain Hamiltonian [54], assuming a uniaxial compressive stress is applied along the [] direction. On the left hand side of the figure, for silicon, the top band represents the heavy-hole band and the bottom band represents the light-hole band and they are degenerate at the band edge. As uniaxial stress increases to about 6kPa, on the right hand side of the figure, the degeneracy at the band edge is lifted and the top band now represents the light-hole band and the bottom band represents the heavy-hole band. The stressed effective masses of heavy and light holes saturate. As a result, the stressed effective masses should be used in calculation of unstressed conductivity due to the presence of small dopant-induced residual stress. However, this small lattice stress, 6kPa, only causes very small band splitting, as can be seen from Fig., thus the hole population is essentially unchanged. Figures 3, 4, and 5 show the model-predicted longitudinal π coefficient vs. stress for [], [], and [] direction. The published data from Smith [8, ] are also included for comparison. Note that Smith s data were extracted with to MPa uniaxial tensile stress. One important observation from Figs. 3, 4, and 5 is that the π coefficients for uniaxial tensile and compressive stresses are different and stressdependent. The main reasons are two folds: (i) the stress-induced hole repopulation between the heavy- and light-hole band and (ii) the correction to the hole effective mass is stress-dependent as shown in Eq. (38). The discontinuities at zero stress are due to

50 39 Figure. Calculated effective masses of heavy and light holes vs. stress using 6 6 strain Hamiltonian [54], assuming a uniaxial compressive stress is applied along the [] direction. The solid line represents the effective masses of holes in the top band and the dashed line represents the bottom band. On the left hand side of the figure, for the unstressed silicon, the top band represents the heavy-hole band and the bottom band represents the light-hole band. On the right hand side of the figure, for the stressed silicon, the top band represents the lighthole band and the bottom band represents the heavy-hole band. After about 6kPa stress, the degeneracy is lifted and the light-hole band rises above the heavy-hole band and the stressed effective masses of heavy and light holes saturate.

51 4 5 π [] / - Pa - Compression 5 Smith's data Tension Stress / MPa Figure 3. Model-predicted longitudinal π coefficient vs. stress for [] direction. Smith s data [] are included for comparison.

52 4 6 4 π [] / - Pa - Compression Tension Smith's data Stress / MPa Figure 4. Model-predicted longitudinal π coefficient vs. stress for [] direction. Smith s data [] are included for comparison.

53 4 9 π [] / - Pa - Compression 8 7 Tension 6 Smith's data Stress / MPa Figure 5. Model-predicted longitudinal π coefficient vs. stress for [] direction. Smith s data [] are included for comparison.

54 43 relaxation time and quantization effect in the inversion layer of pmosfet as described in subsection.3.. The predictions associated with [] and [] direction will be verified by the experiments presented later in Chapters 4 and 5. In comparison to the previous works [9,, 3], first, all of them only consider π coefficient for [] direction and do not explicitly discuss the stress dependence of π 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. [] consider stress-induced hole transfer between the heavy- and light-hole bands and the effective mass shift for light hole due to the stress-induced coupling between the light-hole and split-off bands. However, they make an assumption that the conductivity change due to the hole transfer is given by = µ µ ' Du s44χ = σ 3k T hh lh σ σ µ hh + µ lh B µ µ hh hh µ lh + µ lh E k T B, ( 59 ) where µ hh and µ lh are the mobility of heavy and light hole, E is the band splitting for [] direction defined in Eq. (), k B is the Boltzmann constant and T is the absolute temperature. The authors do not provide justification or explanation to the assumption. Using Eq. (9), our model predicts the conductivity change due to the hole transfer as µ hh µ lh phh plh E σ = σ, ( 6 ) p µ + p µ p + p k T hh hh lh which can be derived from semiconductors equations. The authors obtain longitudinal and transverse π coefficients as 3 - Pa - and Pa - respectively, without specifying at what stress. Kanda [9] uses the model and result from Suzuki et al. []. Kleimann et al. [3] consider hole transfer and effective mass shift for the light hole due lh hh lh B

55 44 to (i) coupling between the light-hole and split-off bands and (ii) incomplete decoupling between the heavy- and light-hole bands. However, they postulate that the effective mass shift due to the incomplete decoupling does not affect the longitudinal heavy and light hole effective masses. For transverse effective masses of the heavy and light holes, they introduce correction terms proportional to stress to represent the effective mass shift due to incomplete decoupling, m hh = β χ and m lh = β χ, ( 6 ) where β and β are two parameters. The correction terms shown in Eq. (6) are contradictory to the results from Hasegawa [8] and Hensel et al. [7], to which Kleimann et al. refer in their paper. The reason is because the incomplete decoupling effect should decrease as the stress increases and, at very high stress, the heavy- and light-hole bands will decouple completely and the correction terms will disappear, in contradiction to Eq. (6). The authors obtained 89 - Pa - for longitudinal π coefficient for [] direction. As for transverse π coefficient, the authors fit the experimental value, Pa -, and get values for β and β,.4-9 Pa - and Pa - 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. was derived from Kleiner-Roth 4 4 strain Hamiltonian [7, 3], which neglects the spilt-off band. To estimate the error introduced by using the Kleiner-Roth 4 4 strain Hamiltonian [7, 3], we also use the Bir-Pikus 6 6 strain Hamiltonian [55, 56], which takes the split-off band into account, to calculate the

56 45 band splitting and effective masses for [] direction and compare to the results presented in section.3. Figure 6 illustrates the band energy vs. stress calculated from 4 4 and 6 6 strain Hamiltonians for [] direction. The 4 4 strain Hamiltonian overestimates the band splitting between the heavy- and light-hole bands by 4% at 5MPa uniaxial compression but underestimates 6% at 5MPa uniaxial tension. Note that the energy separation between the upper hole band and split-off band is larger at high stress than zero stress, which implies that the hole concentration in the split-off band is even smaller at high stress and neglecting the split-off band in the model is justified. Comparisons of longitudinal and transverse, heavy and light hole effective masses are shown in Figs. 7(a) and (b) respectively. For the longitudinal light hole effective mass, the 4 4 strain Hamiltonian underestimates about % and 5% at 5MPa uniaxial compression and tension respectively, and the deviations are about % 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 4 4 and 6 6 strain Hamiltonians. Based on these comparisons, we conclude that using 4 4 strain Hamiltonian is a good approximation and the model is suitable for the stress less than 5MPa..5 Summary The strain effects on valence band are explained in detail in this chapter. The constant energy surfaces of the heavy- and light-hole bands, heavy and light hole effective masses, stress-induced band splitting, hole repopulation are explained and derived using Kleiner-Roth 4 4 strain Hamiltonian. At high stress, the influence from the split-off band is taken into account by adding an effective mass shift to the light hole

57 46. Light Hole 6x6 Energy / ev -. Light Hole 4x4 Heavy Hole 6x6 & 4x4 -.4 Split-off 4x4 -.6 Split-off 6x Stress / MPa Figure 6. The energies of the heavy- and light-hole and split-off bands vs. stress for [] direction calculated using 4 4 and 6 6 strain Hamiltonians.

58 47 Longitudinal Effective Mass / m Light Hole 6x Light Hole 4x4.7 Heavy Hole 6x6 & 4x4 (a) Stress / MPa Transverse Effective Mass / m (b) Light Hole 4x4 Light Hole 6x6 Heavy Hole 6x6 & 4x Stress / MPa Figure 7. Effective masses vs. stress for the heavy and light holes for [] direction calculated using 4 4 and 6 6 strain Hamiltonians. (a) Longitudinal. (b) Transverse.

59 48 effective mass (for [] and [] directions) and to both heavy and light hole effective masses (for [] direction). The strain effects are used to model piezoresistance in p-type silicon. The longitudinal and transverse π coefficients for three major crystal axes, [], [], and [] directions, are calculated and compared to the published data. The model-predicted π coefficients are stress-dependent and the tensile and compressive π coefficients are different. The reasons are two folds: (i) the stress-induced band splitting causes the stress dependence of hole population and (ii) light hole effective mass (for [] and [] directions) or both heavy and light hole effective masses (for [] direction) are stressdependent due to the influence of the split-off band at high stress. Finally, the valid stress range that the model is applicable is estimated by comparing the band splitting and the heavy and light hole effective masses calculated using 4 4 and 6 6 strain Hamiltonians. The comparisons show that 4 4 strain Hamiltonian is a good approximation to 6 6 strain Hamiltonian at stress under 5MPa, and hence, the piezoresistance model is good with stress less than 5MPa. Later in Chapters 4 and 5, the piezoresistance model will be verified by the experiments.

60 CHAPTER 3 HOLE MOBILITY ENHANCEMENT IN BIAXIAL AND UNIAXIAL STRAINED- SILICON PMOSFET 3. Introduction The underlying mechanisms of mobility enhancement in uniaxial and biaxial strained-si pmosfets are the same as the piezoresistance effect in p-type silicon as seen from Eqs. () and (). As shown in Fig. 8, by uniaxial strained-si pmosfets, we mean that an in-plane uniaxial compressive stress (uniaxial compression) is applied along the channel of pmosfet, i.e., along [] direction in the () plane that contains the channel. By biaxial strained-si pmosfets, we mean that an in-plane biaxial tensile stress (biaxial tension) is applied to the channel in the () plane, which is illustrated in Fig. 9. Biaxial stress, like uniaxial stress, can lift the degeneracy at the valence band edge and cause hole repopulation and mobility change. Uniaxial and biaxial strained-si pmosfets are two important technologies used to enhance the hole mobility [-4]. Thompson et al. [,, 44] compared uniaxial vs. biaxial in terms of device performance and process complexity and concluded that uniaxial strained-si pmosfets is preferable since comparable mobility enhancement is attained at smaller stress (5MPa compared to >GPa) which is retained at high effective field. In this chapter, we will use the strain effects on the valence band described in Chapter to explain quantitatively how uniaxial compressive and biaxial tensile stresses change the hole mobility in pmosfets in the low stress regime (<5MPa). In section 3., the mobility enhancement will be calculated. An equivalent out-of-plane uniaxial compressive stress will be derived for biaxial-strained pmosfet. The valence band 49

61 5 Figure 8. Illustration of uniaxial strained-si pmosfet. The arrows represent the uniaxial compressive stress in the channel. The uniaxial compressive stress can be generated by process, for example, source/drain refilled with SiGe [- ] or highly compressive stressed SiN capping layer [3], or by four-point wafer bending, which will be described in Chapter 4.

62 5 Figure 9. Illustration of biaxial strained-si pmosfet. The arrows represent the biaxial tensile stress in the channel. The biaxial tensile stress can be generated by process, for example, using thick relaxed Si -x Ge x layer to stretch the Si channel [-8], or by concentric-ring wafer bending, which will be described in Chapter 4.

63 5 structure, in- and out-of-plane heavy and light hole effective masses, band splitting, and hole transfer are explained and calculated using the model described in Chapter. Section 3.3 is the discussion. Finally, section 3.4 is the summary. 3. Mobility Enhancement in Strained-Silicon PMOSFET In this section, we will calculate and compare the mobility improvement in uniaxial- and biaxial-strained pmosfets. The equations of constant energy surface and band splitting described in Chapter were derived by Hensel and Feher [7] in terms of uniaxial stress, for example, Eqs. (7) and (). In order to use their equations to model biaxial-strained pmosfets, we will show that in-plane, (), biaxial-tensile stress can be represented by an equivalent out-of-plane uniaxial compressive stress χ bi χuni of the same magnitude along the [] 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 [] direction is given by [7], E = ε = Du S, ( 6 ) 3 where give by D u is the valence band deformation potential for [] direction and the strain is S ( s s ) χ = ε zz ε xx =, ( 63 ) where χ is the uniaxial stress along [] direction, and ε zz and ε xx are the uniaxial strain along the [] and [] directions respectively. The strain-stress relation for material with cubic symmetry is reviewed in Appendix A. In Eq. (63), s and s are compliances of silicon [57] and

64 53 c + c = and ( c c )( c + c ) s s =, ( 64 ) ( c c )( c + c ) c where c =.657 Pa and c =.639 Pa are the normal and off-diagonal stiffnesses of silicon. For in-plane biaxial tensile strain, ε ε = ε, and ε zz can be xx = yy derived from the stress-strain equation with zero shear strains [58], χ xx c χ yy c χ zz c = τ yz τ zx τ xy c c c c c c c 44 c 44 ε xx ε yy ε zz, ( 65 ) c44 where χ xx, χ yy, and χ zz are the uniaxial stresses, and τ xy, τ yz, and τ zx are the shear stresses. On the left hand side of Eq. (65) is the stress tensor, and the right hand side is the elastic stiffness tensor and strain tensor. With in-plane biaxial tensile strain, there is no out-ofplane stress, i.e., χ zz = c ( ε xx + ε yy ) + cε zz = ( c c )ε and S in Eq. (63) becomes ε zz =. Then, ε zz can be obtained as c S = c bi ε ε =, ( 66 ) ( c c ) χ where χ bi is the in-plane biaxial tensile stress in () plane and E χbi = ε, ( 67 ) ( ν )

65 54 where E is the Young s modulus, ν is the Poisson ratio, and E ( ) = ( s + s ) =.85 Pa ν is the biaxial modulus and invariant in the () plane [57]. For a uniaxial compressive stress χuni, S = s ( ) s ( χ ) = χ uni χ uni uni. ( 68 ) ( c c ) Comparing Eqs. (66) and (68), if bi uni χ = χ, then the in-plane biaxial tensile and out-ofplane uniaxial compressive stresses will create the same strain S and thus the same band splitting along [] direction. Therefore, the in-plane biaxial tensile stress represented by an equivalent out-of-plane uniaxial compressive stress magnitude. χ bi can be χuni of the same The equations and band parameters presented in section.3 are used to calculate the effective mass, band splitting, and hole transfer between the heavy- and light-hole bands. Table 3 gives the zeroth-order, in- and out-of-plane heavy and light hole effective Table 3. In- and out-of-plane effective masses of the heavy and light holes for uniaxial compression and biaxial tension. Uniaxial Compression Biaxial Tension Heavy Hole Light Hole Heavy Hole Light Hole In-Plane Out-of-Plane masses for uniaxial compression and biaxial tension. Figure shows the in-plane heavy and light hole effective mass with mass correction vs. stress. The band splitting induced by both stress and quantum confinement vs. stress is shown in Fig.. The mobility enhancement in uniaxial- and biaxial-strained pmosfets are calculated using Eqs. (), (), and (58) and shown in Fig. Two published experimental data points for biaxial-

66 55.6 In-Plane Effective Mass / m Uniaxial Heavy Hole Biaxial Light Hole Biaxial Heavy Hole Uniaxial Light Hole Stress / MPa Figure. In-plane effective masses of the heavy and light holes vs. stress for uniaxial compression and biaxial tension.

67 Biaxial Tension Band Splitting / ev Uniaxial Compression Stress / MPa Figure. Band splitting vs. stress for uniaxial compression and biaxial tension. Both contributions from stress and quantum confinement are included. Doping density is assumed 7 cm -3.

68 Thompson et al. 4 (Uniaxial Compression).4.3 Uniaxial Compression This Work. µ /µ eff. Oberhuber 98 (Biaxial Tension) Biaxial Tension This Work -. Rim et al. (Biaxial Tension) Rim et al. 3 (Biaxial Tension) -..E+7.E+8.E+9 Stress / Pa Figure. Model-predicted mobility enhancement vs. stress for uniaxial compression and biaxial tension. Published theoretical [3] and experimental [5, 8, ] works are also shown for comparison.

69 58 strained pmosfet [5, 8] and one for uniaxial-strained pmosfet [] are included for comparison. Also included is the model prediction for biaxial-strained pmosfet from Oberhuber et al. [3]. In comparison with their theoretical work [3], our model gives better prediction that shows mobility degradation instead of mobility enhancement at low biaxial tensile stress. At 5MPa, the model predicts 48% mobility improvement for uniaxial compression and 4% mobility degradation for biaxial tension, in good agreement with published data [5, 8, ]. 3.3 Discussion The underlying mechanism of the mobility enhancement in uniaxial- and biaxialstrained pmosfets is the hole repopulation from the heavy- to light-hole bands. For uniaxial compression, as shown in Figs. 8 and, since the heavy- and light-hole bands are prolate and oblate ellipsoids, the heavy hole effective mass along the channel in () plane is larger than the light hole and hole repopulation from the heavy- to light-hole bands improves the mobility. For biaxial tension, as seen from Figs. 6 and, the effective mass along the channel in () plane for the heavy hole is actually smaller than the light hole and hole repopulation from the heavy- to light-hole bands results in mobility degradation. In addition, as seen from Fig., the difference in the in-plane effective masses of the heavy and light holes is larger for uniaxial compression than biaxial tension. This causes larger mobility improvement for uniaxial compression. The summary of the underlying mechanism of mobility enhancement in uniaxial- and biaxialstrained pmosfets is illustrated in Fig. 3. 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

70 Figure 3. The underlying mechanism of mobility enhancement in uniaxial- and biaxial-strained pmosfets. The effective mass shown next to the constant energy surfaces are the zeroth-order in-plane effective masses. 59

71 6 interband optical phonon scattering, as shown in Eq. (5). For biaxial tension, it implies that, at certain high stress, the mobility will stop decreasing and start increasing with even higher stress. The band splitting between the heavy- and light-hole bands is ~5meV for 5MPa biaxial tensile stress and ~mev for.gpa, corresponding to x=.8 in Si - xge x. When the band splitting becomes larger than the optical phonon energy, 63meV [4], the interband scattering between the heavy- and light-hole band is suppressed and hole mobility increases. This explains the published biaxial tension data at high Ge concentration [5, 8]. However, due to the limitation of 4 4 strain Hamiltonian, the analytical model cannot provide accurate prediction for hole mobility enhancement at such high stress. 3.4 Summary At low stress (<5MPa), the uniaxial-strained pmosfet is shown to have large mobility improvement due to hole repopulation from the heavy-hole band with larger inplane effective mass to the light-hole band with smaller one. For biaxial-strained pmosfet, because the in-plane effective mass of the heavy hole is smaller than the light hole, hole repopulation from the heavy- to light-hole bands degrades the mobility. Both predictions are in good agreement with the published data, 48% improvement for uniaxial-strained pmosfet and 4% degradation for biaxial-strained pmosfet at 5MPa. 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.

72 CHAPTER 4 WAFER BENDING EXPERIMENT AND MOBILITY ENHANCEMENT EXTRACTION ON STRAINED-SILICON PMOSFETS 4. 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 π coefficients of p-type silicon vs. stress will then be calculated. Concentric-ring and four-point bending apparatus are used to apply six kinds of mechanical stress to the channels of pmosfets, biaxial tensile and compressive and uniaxial longitudinal and transverse, tensile and compressive stress. The stress range used in this experiment is 5MPa to 3MPa. PMOSFETs from 9nm technology [,, 59] with the channels oriented along [] direction on () wafers are used in the experiments. In section 4., wafer bending experiments are presented. First, the four-point bending apparatus used to apply uniaxial stress will be explained in detail and equations for calculating the uniaxial stress will be derived. For the concentric-ring bending jig used to apply biaxial stress, finite element analysis simulation is used due to nonlinear bending. Uncertainty analysis in the applied stress will be performed. Section 4.3 explains the methods to extract threshold voltage, hole mobility, and vertical effective field. Uncertainty analysis for effective mobility will be performed. Section 4.4 is the summary. 6

73 6 4. Wafer Bending Experiments on pmosfets 4.. Four-Point Bending for Applying Uniaxial Stress Uniaxial stress is applied to the channel of a pmosfet using four-point bending. Figure 4(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. 4, the upper and lower surfaces of the substrate will experience uniaxial compressive and tensile stress along [] direction, respectively. The stress on both surfaces can be calculated using the following analysis [6] 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. 4(b), let the deflection at any point on the upper surface be designated by y(x), where y()= and y(l)=. The stress on the upper and lower surfaces at the center of the substrate are given by respectively, where EH EH σ xupper = and σ xlower = ( 69 ) r r E =.689 Pa [57] is the Young s modulus of crystalline silicon along the [] direction on () substrate, H is the substrate thickness, r is the radius of curvature given by [6] = r M EI = Pa z EI z, ( 7 ) where 3 M = Pa is the moment for a x L, and I z = bh is the moment of inertia for a substrate with rectangular cross section and width of b. Eq. (69) can then be expressed as [6]

74 63 (a) (b) Figure 4. Apparatus used to apply uniaxial stress and schematic of four-point bending. (a) The picture of jig. In this picture, uniaxial compressive and tensile stresses are generated on the upper and lower surfaces of the substrate respectively. (b) Schematic of four-point bending. The substrate is simply supported. Four loads applied by cylinders are approximated by four point forces, P. The deflection at any point on the upper surface is designated by y(x).

75 64 MH PaH σ xupper = = and I I z z MH PaH σ xlower = =. ( 7 ) I I z z For x a, the moment M = Px and EI z d y dx = M = Px. ( 7 ) Solving Eq. (7) we get P 3 y = x + Cx + C EI 6 z, ( 73 ) where C and C are integration constants. For a x L, the moment ( x a) Pa M = Px P = and [6] EI z d y dx = M = Pa. ( 74 ) Solving Eq. (74) we obtain Pa y = x + C3x + C4 EI z, ( 75 ) where C 3 and C 4 are integration constants. The four integration constants can be determined from the boundary conditions [6]: (i) the slope dy dx determined from Eqs. (73) and (75) should be equal at x = a, (ii) the slope dy dx = at x = L, 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 =, y =. With these four boundary conditions, Eqs. (73) and (75) become [6] ( L a) P 3 Pa y = x + x EI 6 z x a, ( 76 )

76 65 y = EI z Pa PaL Pa x + x 6 3 a x L. ( 77 ) Using Eqs. (76) and (77), the deflection at x = a and x = L can be calculated [6] Pa L a yx = a =, ( 78 ) EI 3 z y L x= Pa = 4EI z ( 3L 4a ). ( 79 ) Measuring the deflection at x = a, P I z can be obtained as P I z = a Eyx= a L 3 a, ( 8 ) and then the stress on both surfaces of the substrate in Eq. (7) can then be calculated EHy = σ = x a xupper and L a a 3 EHy = σ = x a xlower. ( 8 ) L a a 3 The radius of curvature in Eq. (7) can also be obtained as = r Pa EI z yx= a =. ( 8 ) L a a 3 Finite element analysis (FEA) using ABAQUS [6] is also performed to verify the assumptions used to obtain Eq. (8) and the results are shown in Fig. 5. In Fig. 5, 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

77 66 Stress / MPa Uniaxial Tension (Equation) Maximum Deflection Used Uniaxial Compression (Equation) Deflection / mm Uniaxial Tension (Simulation) Uniaxial Compression (Simulation) Figure 5. 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. (8). Simulated stress values are obtained using finite element analysis with ABAQUS [6].

78 67 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. (8). The calculated stresses using Eq. (8) agree well with the results of finite element analysis for the range of deflection used. Figure 6(a) is a plot of stress vs. relative position y to the neutral axis along the cross section, AA in Fig. 6(b), extracted from simulation. The deflection is assumed as.9mm. Figure 6(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.77mm. As shown in Fig 6(a), the stress vs. position y curve is linear and symmetric about the origin. This result verifies the validity of Eq. (8) within the range of deflection.9mm. The finite element analysis simulations shown in Figs. 5 and 6 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.%. Therefore, the effect of sample location on stress can be neglected for the range of deflection used. 4.. Concentric-Ring Bending for Applying Biaxial Stress Biaxial stress is applied to the channel of a pmosfet using concentric-ring bending. Figures 7(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

79 Stress at top plane 443MPa Stress / MPa Stress at bottom plane -443MPa Position y / µm Stress at neutral axis (a) (b) Figure 6. Stress vs. position and schematic of bending substrate. (a) Plot of stress vs. relative position y to the neutral axis along the cross section, AA in (b), extracted from simulation. The deflection is assumed as.9mm. (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.77mm.

80 69 (a) (b) Figure 7. Apparatus used to apply biaxial stress and schematic of concentric-ring bending. (a) The picture of jig. In this picture, biaxial compressive and tensile stresses are generated on the upper and lower surfaces of the substrate respectively. (b) Schematic of concentric-ring bending. The plate (substrate) is simply supported. The deflection at any point on the upper surface is designated by w(r).

81 7 analysis) to calculate deflections and stresses in a plate. The finite element analysis (FEA) using ABAQUS [6, 6] 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 E v ε xx E ε yy v ε zz = E γ yz γ zx γ xy xx xy xx xz xx v E E v E yx yy yy yz yy v E v E E zz zx zz zy zz G yz G zx G xy χ xx χ yy χ zz, ( 83 ) τ yz τ zx τ xy where E xx = E = E =.3 Pa are the three Young s moduli [57], yy zz G xy = G = G =.796 Pa are the three shear moduli, v v = v =. 79 are yz zx xy = yz zx the three Poisson ratios, χ xx, χ yy, and χ zz are the normal stresses, ε xx, ε yy, and ε zz are the normal strains, τ xy, τ yz, and τ zx are the shear stresses, and γ xy, γ yz, and γ zx 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. 7. The results are shown in Figs. 8 and 9 [6]. In Fig. 8, the stresses along x and y axes, σ xx and σ yy respectively, at the center of the bottom, middle, and top planes vs. the displacement of the smaller ring are presented. As can be seen from Fig. 8, σ xx = σ yy, this result confirms that the stress at the center of the planes is biaxial. In addition, Figure 9 (a) shows the shear stress τ xy at

82 σ xx sigma(xx) sigma(yy) σ yy Bottom Stress at Center (MPa) Middle Maximum Displacement Used Top Displacement of Small Ring (mm) (a) (b) Figure 8. Stress vs. displacement and schematic of bending plate. (a) Finite element analysis simulation of the bending plate (substrate). The radial stresses, σ xx and σ yy, at the center of the top, middle, and bottom planes of the bending plate vs. the displacement of the smaller ring are shown. σ xx =σ yy. (b) Illustration of top, middle, and bottom planes of the plate.

83 7.5 Shear Stress at Center (MPa).5 Maximum Displacement Used Bottom Plane Middle Plane Top Plane Displacement of Small Ring (mm) (a) Load (N) Displacement of Small Ring (mm) (b) Figure 9. 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.

84 73 the center of three planes, and Fig. 9(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.77 mm), the stress on the top and bottom planes are of nearly the same magnitude but opposite sign, as shown in Fig. 8. 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.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. 8. According to Fig. 9(b), the corresponding load is about 45N or lb when displacement reaches.89mm. The maximum deflection achieved in this work is about.46mm, corresponding to about N or 5lb. According to Figs. 8(a) and 9(a), τ xy is about three orders of magnitude smaller than σ xx and σ yy at the center. Detailed analysis of the shear stress shows that τ xy has no effect on mobility enhancement at the center of the concentric ring due to symmetry of τ xy and τ yx. Thus, the mobility enhancement at the center is due to biaxial stress alone. The finite element analysis simulations shown in Figs. 8 and 9 need further investigations due to no systematic studies of grid convergence and no uncertainty

85 74 analysis in sample location variation from exact center. More detailed analysis can be done in future work Uncertainty Analysis In this subsection, the uncertainty in the applied stress will be estimated. There are four major sources of uncertainty in applied uniaxial stress using the four-point bending jig shown in Fig. 4. 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. 3. The average value is used as the distance between the two plates. Finally, repeat the procedure times and calculate the uncertainty with 95% confidence with these values. This value is the uncertainty of the starting point in terms of deflection. The uncertainty of starting point is estimated as.7mm. The second source of uncertainty is the micrometer for setting the displacement [63]. The resolution of the micrometer is / inch (.3mm), the uncertainty is one half of the resolution or about.mm. The total uncertainty in deflection from the starting point and micrometer is mm.

86 75 Figure 3. Uncertainty analysis of the starting point for the four-point and concentric-ring bending experiments. The distance between two plates are measured at the locations designated,, 3, and 4.

87 76 The third source is the variation of the substrate thickness. The typical thickness of a inches (3mm) wafer is 775±µm [64]. The uncertainty in wafer thickness is.mm. The uncertainty in stress from the first three sources can be calculated by differentiating Eq. (8) [65] EH Ey σ = y H L a L a, ( 84 ) a a 3 3 σ σ y = y H +. ( 85 ) H In Eq. (84), the variations in L and a are negligible because they are fixed by the grooves as shown in Fig. 4. In Eq. (85), ( H ) = ( 775). 7 with ( y y) = (.73.57). H is negligible compared, where.57mm is the maximum deflection achieved in the experiment. The total uncertainty in stress is estimated about 4MPa. The fourth uncertainty is from the substrate angle misalignment as shown in Fig. 3 [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 ±, corresponding to about 3MPa uncertainty in stress [66]. The total uncertainty from all four sources is estimated as = 5MPa. 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]

88 77 Figure 3. Uncertainty analysis of the misalignment of the substrate with respect to the pins for the four-point bending experiment.

89 78 = r d y dx dy + dx 3, ( 86 ) assuming the beam deflections occur only due to bending [67]. Substituting into Eq. (69), we get d y EH σ = dx. ( 87 ) 3 dy + dx 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. 3. The optical sensor sweeps across the substrate between the two top pins, and the distance between the sensor and the substrate is measured and recorded. To reference the original unstressed wafer surface, the before bending curve is measured first, and the after bending curves are measured subsequently. Subtracting the before bending curve from the after bending one, the elastic curve is measured and the result is shown in Fig. 33. A similar method was used by Uchida et al. [69] previously. The elastic curve and radius of curvature are extracted from the polished surface of a bare wafer instead of the device wafer due to poor reflectivity on the patterned and passivated device wafer. There is a passivation on top of the device wafer, which typically consists of a phosphorus-doped silicon dioxide layer and then silicon nitride

90 Figure 3. The experimental setup for calibrating the uniaxial stress in the four-point bending experiment. 79

91 8 Displacement y / µm fitting curve y = -E-6x +.69x R =.9975 fitting curve y = -6E-7x +.33x R = Position x / µm Figure 33. Extracted displacement vs. position curves on the upper surface of the substrate in the four-point bending experiment. The experimental displacement curves are LSF to nd order polynomials with R =.9975 and.999 respectively.

92 8 layer. The reflectivity of passivation is too low for optical sensor to operate accurately, therefore, the polished surface of bare wafer is used to extract the elastic curve and radius of curvature. The extracted elastic curves are least-square fit (LSF) to a second order polynomial with R =.9975 and.999. Substituting the resulting LSF second order polynomial, y = - -6 x +.69x and y = - 6 x +.33x 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.% and also about.% 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.4mm, smaller than.7mm 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.mm uncertainty as in uniaxial case. The total uncertainty is mm. 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. 8 to extract the uncertainty in stress. At each pre-set

93 E E E E+5 Radius of Curvature / µm E E E E E E E E+5 Position x / µm Figure 34. Extracted radius of curvature vs. position on the upper surface of the substrate in the four-point bending experiment, corresponding to the upper curve in Fig. 33. The difference between the maximum and minimum radius of curvatures is only about.%.

94 E E E E+8 Stress / Pa -.46E E E E E E+8 Position x / µm Figure 35. Extracted uniaxial stress vs. position on the upper surface of the substrate in the four-point bending experiment, corresponding to the Fig. 34. The difference between the maximum and minimum stresses is only about.%.

95 84 Stress/ MPa Experimental stress data extracted from curvature Calculated value from displacement Displacement / mm Figure 36. Calibration of the four-point bending experiment. The extracted stress values are close to the calculated ones using Eq. (8) and within the uncertainty range at 95% confidence level.

96 85 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.4mm, the uncertainty range of the displacement is.4.45 <.4 < mm and the corresponding uncertainty range of the tensile stress on the bottom plane can be found by projection as 48.8 < 8.8 < 3.6MPa or = 64.8MPa and the uncertainty range for the compressive stress on the top plane is 4. > 66.8 > 87.4MPa or 4. ( 87.4) = 45.MPa. 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 π coefficients will be described. The effective mobility will be extracted from the drain current in the linear region (low drain bias) for a long-channel MOSFET. At low drain bias V DS, the linear drain current of an ideal MOSFET can be approximated as I DS W = µ eff C ox ( V GS V T ) V DS, ( 88 ) L and the effective mobility µ eff can then be expressed as = C DS µ eff, ( 89 ) ox W L I ( V GS V T ) V DS

97 Bottom Plane Stress at Center (MPa) Top Plane Displacement of Smaller Ring (mm) Figure 37. Uncertainty analysis for the concentric-ring bending experiment. At each preset displacement point, the displacement with the uncertainty range at 95% confidence level is projected on the stress vs. displacement curve to get the stress value with the uncertainty range in stress.

98 87 where C ox is the gate oxide capacitance, W and L are the channel width and length respectively, and V GS and V T 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 I DS -V GS 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 I DS -V GS curve and threshold voltage extraction using the Agilent 455C Semiconductor Parameter Analyzer. In Fig. 38(b), an additional curve proportional to the gradient of the I DS -V GS curve, I DS V GS, is also shown to help to determine the point with the largest slope on the I DS -V GS curve. The effective vertical electric field E eff is expressed as [7] E eff Q +ηq ε b inv =, ( 9 ) s where η is a fitting parameter and equal to /3 for holes, Q b is the bulk depletion charge, Q inv ox ( V V ) = C is the inversion charge, and ε s is the dielectric constant of silicon. At GS T the interface of gate oxide and silicon channel, the electric displacement continuity gives E ox ε = E ε, ( 9 ) ox S S where E ox is the electric field in the oxide, ε ox is the dielectric constant of oxide, E S is the silicon surface field at the interface, and E S Q + Q ε inv b =. ( 9 ) S With Eqs. (9) and (9), the bulk depletion charge Q b can be expressed as

99 88 (a) (b) Figure 38. Illustration of extracting threshold voltage. (a)the tangent line with the largest slope intercepts x axis at -.4V, therefore, the threshold voltage V T = -.4V. (b)snapshot of the I DS -V GS characteristic and threshold voltage extraction from Agilent 455C Semiconductor Parameter Analyzer.

100 89 inv ox ox b Q E Q = ε, ( 93 ) and the effective vertical field E eff can then be obtained as ( ) ( )( ) [ ] S t GS GS ox S inv ox ox eff V V V C Q E E ε η ε η ε = =, ( 94 ) where E ox is approximated by V GS / t ox, which is valid for V DS <<V GS [7], and C ox is the gate capacitance. The uncertainty in extracted effective mobility can be estimated using Eq. (89) by evaluating the sensitivity of µ eff for variations in I DS, C ox, L, W, V GS, V T, and V DS as shown in Eqs. (95) and (96) [65], ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) DS DS eff T T eff GS GS eff eff eff ox ox eff DS DS eff eff V V V V V V W W L L C C I I = µ µ µ µ µ µ µ µ, ( 95 ) = = DS DS T T T GS T GS GS T GS GS ox ox DS DS DS DS DS eff eff DS T T T eff eff T GS GS GS eff eff GS eff eff eff eff ox ox ox eff eff ox DS DS DS eff eff DS eff eff V V V V V V V V V V V V W W L L C C I I V V V V V V V V V V V V W W W W L L L L C C C C I I I I µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ. ( 96 )

101 9 For the Agilent 455C Semiconductor Parameter Analyzer, the measurement accuracy are ±.% and ±.% for current and voltage measurement [7] respectively, therefore, 6 ( I I ) = 8, ( V V ) = 4, and ( ) 8 V V = 4 DS DS GS GS DS DS. According to ITRS [7], the uncertainty in gate dimension for 9nm technology is 3.3nm. The gate length and width of pmosfet used in experiment are µm and 5µm respectively, 5 9 therefore, ( L L) = ( 3.3 ) and ( W W ) = ( 3.3 5) 4. The variation in threshold voltage of all measured pmosfets is less than 5mV. The mobility enhancement is extracted at effective field.7mv/cm, corresponding to gate voltage V GS 4 about.8v. With the threshold voltage V T =.3V, ( ) ( V ( V V )) =. 56, and ( ( V V )) =. 36 GS GS T V. T GS T V V =.8, For uncertainty in C ox and doping density, it can be estimated from analyzing the uncertainty in threshold voltage. The threshold voltage V T is expressed as T T V T 4ε qn ψ Si a B = V fb + ψ B +, ( 97 ) Cox where V fb is the flat-band voltage, ψ B is the difference between the Fermi potential and the intrinsic potential, ε Si is the permittivity of silicon, q is the electron charge, and N a is the doping density. There are two main factors that affect the uncertainty in V T, N a and 4 C ox. As shown previously, the uncertainty in threshold voltage is ( ) 4 4 therefore, ( C C ).8 and ( ) ox ox N A N A.8 V V =.8, T T. The total uncertainty in extracted mobility can be estimated as ( µ eff / µ eff ) = = Therefore, µ =.. eff µ eff

102 9 4.4 Summary The four-point and concentric-ring bending apparatus are used to apply uniaxial and biaxial stresses to the pmosfets. An analytic equation is derived for calculating the uniaxial stress from the deflection, and finite element analysis is used to determine the biaxial stress from the deflection due to the nonlinearity of the bending plate. Uncertainty analysis is also performed, and the calibration of the uniaxial stress is within the uncertainty range of the calculated value. The approaches used to extract the mobility, threshold voltage, and vertical effective field are described. These approaches will be used to extract mobility enhancement and calculate the π coefficients in Chapter 5. The uncertainty analysis in extracted effective mobility is also performed.

103 CHAPTER 5 RESULTS AND DISCUSSIONS 5. Mobility Enhancement and π Coefficient versus Stress The devices used in the experiment are from 9nm technology [,, 59] with and µm channel length and Å thin gate oxide thickness. An Agilent 455C Semiconductor Parameter Analyzer is used to measure the I DS -V GS characteristics with gate voltage V GS swept from to -.V and drain voltage V DS fixed at -5mV. The extracted effective mobility vs. effective field of the pmosfets before and after bending with uniaxial longitudinal and transverse and biaxial compressive and tensile stresses are shown in Figs. 39, 4, and 4, respectively. One important observation from these experimental results is that before and after bending, the mobility vs. effective field curves are parallel with each other in all three figures. It implies that the mobility enhancement is independent of effective field for all six types of stresses at this stress level and within this range of effective field. Mobility enhancement µ µ, where µ is the before bending mobility, is extracted at an effective field,.7mv/cm and plotted vs. stress in Fig. 4. This figure compares the experimentally extracted mobility enhancement with the model prediction (Chapters and 3) and published experimental data [73, 74] and numerical simulation [74]. Each data point has 3 to 6 devices measured and the average value is plotted with 95% confidence level uncertainty. The uncertainty analysis will be given in Appendix C. Good agreement is obtained between experiments and analytical model predictions 9

104 longitudinal compression 6MPa 85.4% 8 Mobility / (cm /V.sec) % before bending.% 55 5 Universal Mobility longitudinal tension 6MPa -3.3% Effective Field / (MV/cm) Figure 39. Effective hole mobility vs. effective field before and after bending with uniaxial longitudinal tensile and compressive stresses at 6MPa. The magnitude of mobility change is larger for longitudinal compression than tension. The curve of universal mobility is also included for comparison.

105 % transverse tension 3MPa Mobility / (cm /V.cm) % transverse compression 3MPa before bending 4.9% 5 Universal Mobility -4.% Effective Field / (MV/cm) Figure 4. Effective hole mobility vs. effective field before and after bending with uniaxial transverse tensile and compressive stresses at 3MPa. The magnitude of mobility change is larger for transverse tension than compression. The curve of universal mobility is also included for comparison.

106 % biaxial compression 34MPa Mobility / (cm /V.sec) % biaxial tension 33MPa Universal Mobility before bending 3.% -5.3% Effective Field / (MV/cm) Figure 4. Effective hole mobility vs. effective field before and after bending with biaxial tensile stress at 33MPa and compressive stress at 34MPa. The curve of universal mobility is also included for comparison.

107 96 Thompson et al. 4 Uniaxial Longitudinal Compression.5.4 Symbol: experimental data Line: model prediction Mobility Enhancement / ( µ / µ) Wang et al. 4 Biaxial Compression Uniaxial Transverse Compression Gallon et al. 4 Uniaxial Longitudinal Tension Uniaxial Transverse Tension Biaxial Tension Stress / MPa Figure 4. Mobility enhancement vs. stress for six kinds of stresses, biaxial tensile and compressive and uniaxial longitudinal and transverse, tensile and compressive. The mobility enhancements are extracted at.7mv/cm. The solid lines are the model predictions: blue: this work, orange: Wang et al. [74]. The symbols are experimental data: blue circle: this work, green triangle: Thompson et al. [], orange diamond: Wang et al. [74], and purple square: Gallon et al. [73].

108 97 presented in Chapters and 3. The experimental results from both Gallon et al. [73] and Wang et al. [74] are consistent with this work within the estimated uncertainty. Both of them [73, 74] used four-point bending to apply uniaxial stress. Gallon et al. used beam bending like this work except the authors measured the displacement at the center of substrate and then calculated the stress at the center of substrate EHy center σ =, ( 98 ) 3L 4a where y center is the displacement at the center of substrate. Wang et al. used a more accurate and realistic approach by considering the four-point bending as 3-dimensional plate bending instead of -dimensional beam bending. For their theoretical study, Wang et al. [74] employed a quantum anisotropic transport model using 6-band stress dependent k p Hamiltonian and momentumdependent scattering model. The predictions given by their numerical model for uniaxial longitudinal and transverse stresses also support our analytical model prediction. These comparisons provide independent corroboration of the experimental data and analytical model developed in this study. The π coefficient can be calculated from the results of mobility enhancement vs. stress using Eqs. () and (). Figure 43 is a plot of extracted π coefficient vs. stress. Only longitudinal and transverse, tensile and compressive π coefficients for [] direction and transverse tensile and compressive π coefficients for [] direction are provided since they pertain to [] channel direction pmosfet on () surface with uniaxial and biaxial stresses. Our model predictions and the results from Smith [], Wang et al. [74], and Gallon et al. [73] are also included for comparison. Good

109 98 longitudnal π [].5E-9 Symbol: experimental data Line: model prediction.e-9 Smith's longitudnal π [] Wang et al. longitudnal π[] 5.E- Smith's transverse π [] π coefficient Gallon et al. transverse π [] longitudinal π [].E E- transverse π [] Smith's transverse π [] -.E-9 Gallon et al. trannsverse π [] Wang et al. transverse π [] -.5E-9 stress / MPa Figure 43. π coefficient vs. stress, including longitudinal and transverse π coefficients for [] direction and transverse π coefficient for [] direction. The solid lines are the model predictions: blue: this work, orange: Wang et al. [74]. The symbols are experimental data: blue circle: this work, green triangle: Smith [], orange diamond: Wang et al. [74], and purple square: Gallon et al. [73].

110 99 agreements are obtained between the experimental data and model predictions presented in Chapter. However, compared with Smith s piezoresistance data [], good agreement is obtained only for the longitudinal π coefficient for [] direction. For the transverse π coefficients for [] and [] directions, significant deviations exist between our work, model prediction and experimental data, and Smith s data [], however, they are within the estimated uncertainty. The experimental and theoretical results from Wang et al. [74] and experimental data from Gallon et al. [73] agree with this work. The stress dependence of π coefficient has been verified by experimental data from this work as well as Wang et al. [74] and Gallon et al. [73]. At low stress, the extracted π coefficients are smaller than the model predictions, possibly due to the uncertainty in starting point, causing the actual stress smaller than the calculated stress. 5. Discussion 5.. Identifying the Main Factor Contributing to the Stress-Induced Drain Current Change As can be seen from Eq. (88), there are five possible factors that contribute to the stress-induced linear drain current change, V T, C ox, W, L, and µ eff. Using the same approach as shown in Eqs. (95) and (96), the stress-induced drain current change can be expressed as I I DS DS + V GS VT V µ = µ eff T eff VT VT C + C ox ox V + V DS DS W + W L + L VGS + VGS V T V VGS GS. ( 99 ) Take Fig. 39 with a 6MPa longitudinal compressive stress applied along the channel as an example, the measured before and after bending drain currents I DS show

111 ( I I ) =. 43 DS DS. For the extracted threshold voltage V T, the difference in V T before and after bending is mv and the extracted threshold voltage is about.3v. Therefore, 5 ( V V ) = 4 and ( ( V V )) =. 36 T T V as described in sub-section 4.3., thus T GS T the contribution from the stress-induced threshold voltage variation is =.4-5 and can be neglected. This result is consistent with the published works that the threshold voltage variation due to stress is negligible [73, 75, 76]. For a 6MPa longitudinal stress applied along the [] direction, the corresponding strain is about.3. (the Young s modulus for the [] direction is.689 Pa [57].) At the interface of the gate oxide and Si channel, the strain parallel to the interface is continuous across the interface. Therefore, the silicon dioxide (SiO ) will experience the same strain, about.3, along the [] direction. The Poisson ratio for SiO is about.7 [77] so the corresponding out-of-plane ([] direction) strain due to the Poisson effect is only about.3-4. Since Cox = ε ox tox, 8 ( C C ) = ( t t ) = 5. ox ox ox ox. Therefore, the contribution from the stress-induced change in gate oxide capacitance can be neglected. This result is also consistent with the observation from Matsuda and Kanda [78]. Regarding the channel width W (along the [ ] direction if the channel direction is along the [] direction), for a 6MPa longitudinal stress, due to the small Poisson ratio of Si for the [] and [ ] directions,.64 [57], the stress-induced change in W W W = is very small, ( ) ( ) change in W can also be neglected. = 7.3. Therefore, the stress-induced

112 Since the gate V GS and drain V DS voltages are fixed (within the uncertainty of machine 4-8 ) before and after bending, there are no stress-induced changes of V GS and V DS, i.e., V V = and V V =. GS GS DS DS For the effective channel length L, the situation is more complex because the metallurgical channel length is not necessarily equal to the effective channel length [5]. For the metallurgical channel length, the change due to stress is about.3 for 6MPa longitudinal stress. Therefore, the stress-induced change in metallurgical channel length is also negligible. To experimentally study the effective channel length as a function of stress, we first use the shift and ratio method [5, 79] to extract the effective channel length before and after bending. Figures 44 and 45 show the effective channel length ratio <r> of µm pmosfet, defined as after bending effective channel length L eff,bending divided by before bending L eff, r >=< L eff L >, for longitudinal and transverse <, bending eff stresses applied, respectively. In Fig. 44, the percent decrease of the effective channel length after bending with longitudinal compressive stress and the percent increase with longitudinal tensile stress are close to the percent mobility enhancement and degradation respectively. It implies that the change in effective channel length might be a key factor of the stress-induced drain current change. However, as can be seen from Fig. 45, the decrease and increase of the effective channel length after bending with transverse tensile and compressive stresses, respectively, show a similar result as the longitudinal stresses. Considering the small Poisson ratio,.64 [57], the result for transverse stress in Fig. 45 is contradictory to the conclusion for longitudinal stress in Fig. 44. This contradiction can be resolved by examining the key assumption of the shift and ratio method that the ( VGS V T ) dependence of effective mobility µ eff is a common function for all devices

113 Longitudinal Compression, Mobility Enhancement 8.8% <r> <σ > Vg shift (δ) (a) Longitudinal Tension, Mobility Enhancement -8% <r> <σ > Vg shift (δ) (b) Figure 44. Average effective channel length ratio <r> (blue diamond) and variance <σ > (orange square) vs. gate voltage shift δ for longitudinal stress. (a) Compression. (b) Tension. The numbers indicated on the figures are the ratios with the minimum variance.

114 3 Transverse Compression, Mobility Enhancement -4.4% <r>. <σ > Vg shift (δ) (a) Transverse Tension, Mobility Enhancement 3.8% <r> <σ > Vg shift (δ) (b) Figure 45. Average effective channel length ratio <r> (blue diamond) and variance <σ > (orange square) vs. gate voltage shift δ for transverse stress. (a) Compression. (b) Tension. The numbers indicated on the figures are the ratios with the minimum variance.

115 4 [5]. In the experiment, the threshold voltage of pmosfets is essentially constant, so (V GS - V T ) and hence the effective mobility µ eff are assumed to be constant, which is incorrect for stressed pmosfet. Therefore, the shift and ratio method is not applicable for strained-si pmosfet. In fact, Scott et al. [5] studied the influence of trench isolation-induced stress on the drain current of nmosfets. They compared the drain currents in nmosfets with the same physical gate length but different width of source/drain region, which is defined as the distance between the gate and the trench isolation along the channel direction. Smaller width of source/drain regions will induce higher longitudinal compressive stress in the channel of the device and vice versa. They concluded that the longitudinal compressive stress induced by trench isolation degrades the electron mobility and the drain current of nmosfets. The effective channel lengths are essentially the same for all nmosfets with different widths of source/drain regions because the drain induced barrier lowering (DIBL) is very small between devices with different widths of source/drain. If the variation of the effective channel length is really a key factor, the DIBL should be much higher. Therefore, it is the stress-induced change in the mobility and not the effective channel length that affects the drain current of nmosfets under the stress caused by trench isolation. We can also prove the contribution from the change in effective channel length is negligible by examining the definition of the effective channel length. The effective channel length L eff is defined as the portion controlled by the gate [5]. If the source/drain doping is graded, L eff is greater than the metallurgical channel length L met because part of the source/drain overlap is also controlled by the gate [5]. Figure 46 [8] is an illustration of the definition of L eff and

116 5 Figure 46. Schematic diagram of doping concentration gradient and current flow pattern near the metallurgical junction between the source/drain and body. The dashed lines are contours of constant doping concentration. The dark region is the accumulation layer and the arrows indicate the position where the current starts spreading into bulk source/drain region. (after Taur [5].)

117 6 L eff = L + L, ( ) met ov where L ov is the part of the source/drain overlap controlled by the gate, and L met is the metallurgical channel length. In Fig. 46, the metallurgical junction is defined as when the body doping N d is equal to the source/drain doping N a, N = N a d. The dotted lines represent the doping gradient. The junction depth of the source/drain is designated by x j, and x c is the thickness of the accumulation layer. Due to the doping gradient, the doping density near the metallurgical junction of the source/drain is not as high as in the bulk region of the source/drain, and hence the resistivity is also higher. When a MOSFET is turned on, an inversion layer is formed in the channel region near the interface of Si and gate oxide, while an accumulation layer is formed in the source/drain overlap region L ov. The length of L ov is determined by the point where the carrier concentration of the accumulation layer is equal to the background doping concentration of the source/drain or the resistivity of the accumulation layer is equal to the background resistivity [8]. When the channel current flows across the metallurgical junction, the current will flow through the accumulation layer instead of spreading into the bulk region of source/drain because the background doping concentration is much lower and the resistivity is much higher than the accumulation layer near the metallurgical junction. At the point where the background doping concentration or resistivity is equal to the accumulation layer, the current starts spreading [8]. The distance between this point and the metallurgical junction is L ov. Since this accumulation layer is also induced and controlled by the gate similar to the inversion layer, it needs to be included in the effective channel length [5]. When applying stress to a MOSFET, not only the effective mobility or conductivity of the channel is increasing, but the conductivity of the source/drain is also increasing and

118 7 hence the resistivity is decreasing. It implies that L ov is decreasing. The amount of L ov reduction is approximately proportional to the decrease of background resistivity. According to the International Technology Roadmap for Semiconductors [7], for 9nm technology, the extension lateral abruptness is 4.nm/decade. Assume the worst case that L ov is about nm based on source/drain doping density 5 9 cm -3 and channel doping 7 cm -3. Applying a longitudinal compressive stress 6MPa to a MOSFET along the channel direction [], the effective mobility or conductivity will increase about % and thus the resistivity will drop about %. It means L ov will reduce approximately %, to about 8nm which is the worst case since the piezoresistance coefficient decreases at high doping concentration [8]. For a pmosfet with an µm channel length, which are used in the experiment, the change in L ov is about.4µm and the change in the metallurgical channel length is about.3 µm. The total change in the effective L L 5 channel length is about.53 µm and ( ) =.8. This small amount of change in effective channel length cannot explain the much larger change in the linear drain current. Therefore, the contribution from the change in effective channel length can also be neglected. From Fig. 39, the mobility enhancement at effective field.7mv/cm is.4% and ( ) =. 4 µ. Considering the uncertainty in extracted mobility eff µ eff enhancement 4 ( ) = 3.9 µ as described in sub-section 4.3., the stress-induced eff µ eff uncertainty =. mobility enhancement is still close to ( I I ) 43 DS DS. Based on the above discussion, we conclude that the main factor contributing to the stress-induced change in the linear drain current is effective mobility change and affirm the validity of extracting

119 8 mobility enhancement from Eq. (88). It is also shown that the shift and ratio method is not applicable for extracting the effective channel length from strained-si MOSFETs. 5.. Internal Stress in the Channel The 9nm technology [,, 59] used SiGe in source and drain to induce about 6MPa longitudinal compressive stress in the channel of a 45nm channel length pmosfet as shown in Fig. 3 [4]. For longer channel pmosfets, the channel compressive stress from SiGe source/drain is reduced as shown by the stress simulation results using the FLOOPS-ISE process simulator in Fig. 47 [8, 83]. In Fig. 47, the channel length is µm; outside the source/drain is shallow trench isolation (STI); and the source/drain is filled with Si.83 Ge.7 as shown in Fig. 3. The longitudinal compressive stress at the center of the channel is reduced to about 85MPa due to the longer channel length while it is about MPa near the source/drain metallurgical junction. When reducing the channel length to 45nm, the simulated channel compressive stress is increased to about 6MPa, consistent with the simulation result shown in Fig. 3. Therefore, the µm long channel length device was used for the mobility extraction to investigate the effect of applied external stress Stress-Induced Mobility Enhancement at High Temperature So far we only discussed the models and experiments at room temperature. However, the devices normally operate not at room temperature but at a much higher temperature, about C. In this subsection, the model prediction of mobility enhancement at C will be given and compared to the one at room temperature. The experimental data will also be provided.

120 9 Figure 47. Simulation result of the internal stress distribution in a pmosfet. The channel length is µm and the source/drain is built with Si.83 Ge.7. Outside the source/drain is shallow trench isolation (STI). Near the Si channel and gate oxide interface, the stress is about -85MPa (compressive stress) at the center and about -MPa (compressive stress) near the source/drain junction.