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STRAIN EFFECTS ON HOLE MOBILITY OF SILICON AND GERMANIUM PTYPE METALOXIDESEMICONDUCTOR FIELDEFFECTTRANSISTORS By GUANGYU SUN 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 2007 2007 Guangyu Sun To my dear wife Anita, and my parents ACKNOWLEDGMENTS I am grateful to all the people who made this dissertation possible and because of whom my graduate experience has been one that I will cherish forever. First and foremost I thank my advisor, Dr. Scott E. Thompson, for giving me an invaluable opportunity to work on challenging and extremely interesting projects over the past four years. He has ahli made himself available for help and advice and there has never been an occasion when I have knocked on his door and he has not given me time. He taught me how to solve a problem starting from a simple model, and how to develop it. It has been a pleasure to work with and learn from such an extraordinary individual. I thank Dr. Jerry G. Fossum, Dr. Huikai Xie, Dr. C'!i I1!.ih. r Stanton, and Dr. Jing Guo for agreeing to serve on my dissertation committee and for sparing their invaluable time reviewing the manuscript. I also thank Dr. Toshi Nishida for a lot of helpful discussions and kind help. My colleagues have given me a lot of assistance in the course of my Ph.D. studies. Dr. Yongke Sun helped me greatly to understand the physics model, and we ah,v had fruitful discussions. Dr. Toshi Numata also gave me good advice and some insightful ideas. I also thank Jisong Lim, Sagar Suthram, and all other group members who made my life here more interesting. I acknowledge help and support from some of the staff members, in particular, Shannon Chillingworth, Teresa Stevens and Marcy Lee, who gave me much indispensable assistance. I owe my deepest thanks to my family. I thank my mother and father, and my wife, Anita, who have alv,v stood by me. I thank them for all their love and support. Words cannot express the gratitude I owe them. It is impossible to remember all, and I apologize to those I have inadvertently left out. TABLE OF CONTENTS page ACKNOW LEDGMENTS ................................. 4 LIST OF FIGURES .................................... 7 LIST OF TABLES ....................... ............ 11 ABSTRACT ....................... ........... ...... 12 CHAPTER 1 INTRODUCTION AND OVERVIEW ................... .... 14 1.1 History of Strain in Semiconductors .......... ............ 15 1.2 Apply Strain to A Transistor ................... .... 17 1.3 Main Contributions of My Research .......... ............ 18 1.4 Brief Description of The Dissertation .................. ... 19 2 K P MODEL AND HOLE MOBILITY ............. .... .. 21 2.1 The k p Method ............... . . .... 21 2.1.1 Introduction to k p Method ............. .... . 21 2.1.2 Kane's M odel .................. ........... .. 25 2.1.3 LuttingerKohn's Hamiltonian ................... ... .. 28 2.2 Hole Mobility in Inversion L iv. rs ................ .... .. 32 2.2.1 Selfconsistent Procedure .................. ... .. 32 2.2.2 Hole M obility .................. ........... 33 2.3 Scattering Mechanisms .................. ........ .. .. 34 2.3.1 Phonon Scattering. .................. ........ .. 34 2.3.2 Surface Roughness Scattering ............... . .. 36 2.4 Summ ary .................. ............... .. .. 38 3 STRAIN EFFECTS ON SILICON PMOSFETS ................ .. 39 3.1 Piezoresistance Coefficients and Hole Mobility . . ..... 40 3.1.1 Piezoresistance Coefficients ................ .... .. 40 3.1.2 Hole Mobility vs Surface Orientation ................ .. 41 3.1.3 Hole Mobility and Vertical Electric Field . . ...... 42 3.1.4 Strainenhanced Hole Mobility ................... ... .. 42 3.2 Bulk Silicon Valence Band Structure ............... . .. 48 3.2.1 Dispersion Relation .................. ........ .. 48 3.2.2 Hole Effective Masses .................. .. .... .. .. 49 3.2.3 Valence Band under Super Low Strain . . ...... 56 3.2.4 Energy Contours .................. ......... .. 56 3.3 Strain Effects on Silicon Inversion L rs .................. .. 60 3.3.1 Quantum Confinement and Subband Splitting . . ... 60 3.3.2 Confinement of (110) Si ............... 3.3.3 Straininduced Hole Repopulation .. ....... 3.3.4 Scattering Rate .. ............... 3.3.5 Mass and Scattering Rate Contribution ...... 3.4 Sum m ary . . . . . . . 4 STRAIN EFFECTS ON NONCLASSICAL DEVICES .... 4.1 Single 4.1.1 4.1.2 Gate SOI pMOS . Hole Mobility vs Sili Strainenhanced Hol 4.2 Doublegate pMOSFETs 4.2.1 (001) SDG pMOS 4.2.2 Strain Effect on FinI 4.3 Summary .. ....... con Thickness ......... e Mobility of SOI SGpMOS . ETs ............... 5 STRAIN EFFECTS ON GERMANIUM PMOSFETS .... 5.1 Germanium Hole Mobility ............... 5.1.1 Biaxial Tensile Stress .. ............. 5.1.2 Biaxial Compressive Stress ............ 5.1.3 Uniaxial Compressive Stress ........... 5.2 Strain Altered Bulk Ge Valence Band Structure ..... 5.2.1 Ek Diagram s .. ................ 5.2.2 Effective M ass .. ................ 5.2.3 Energy Contours .. .............. 5.3 Discussion Of Hole Mobility Enhancement ........ 5.3.1 Straininduced Subband Splitting .. ....... 5.3.2 Biaxial Stress on (001) Ge ............ 5.3.3 Uniaxial Compression on (001) Ge ......... 5.3.4 Uniaxial Compression on (110) Ge ......... 5.4 Sum m ary . . . . . . . 6 SUMMARY AND SUGGESTIONS TO FUTURE WORK . 6.1 Sum m ary . . . . . . . 6.2 Recommendations for Future Work .. .......... APPENDIX A STRESS AND STRAIN . ................ B PIEZORESISTANCE .. .................. REFERENCES .......... ............... . . 122 . . 133 BIOGRAPHICAL SKETCH .............................. ....... LIST OF FIGURES Figure page 11 Schematic diagram of biaxial tensile stressed SiMOSFET on relaxed Sil_Ge1 1ivr . . ..... . . . . . .. .. 17 12 Uniaxial stressed SiMOSFET with Sil_1Ge1 Source/Drain or highly stressed capping 1 rv.r . . . . . . . . . .18 31 Hole mobility vs device surface orientation for relaxed silicon . . ... 41 32 Hole mobility vs inversion charge density for relaxed silicon. Both measurements and simulation show larger mobility on (110) devices. . . 43 33 Hole mobility vs stress with inversion charge density 1 x 1013/cm2. ........ 44 34 Calculated strain induced hole mobility enhancement factor vs. experimental data for (001)oriented pMOS. .................. ..... 45 35 Hole mobility enhancement factor vs uniaxial stress for different channel doping. 45 36 Calculated strain induced hole mobility enhancement factor vs. stress for (001) oriented pMOS with different inversion charge density. ............ ..47 37 Ek relation for silicon under (a) no stress; (b) 1GPa biaxial tensile stress; and (c) 1GPa uniaxial compressive stress. .............. .... 50 38 Normalized Ek diagram of the top band under different amount of stress. Larger stress warps more region of the band. The energy at F point for all curves is set to zero only for comparison purpose. ................ ...... 51 39 C'!i i,,, I direction effective masses for bulk silicon under (a) biaxial tensile stress; and (b) uniaxial compressive stress. ............... .... 52 310 Twodimensional densityofstates effective masses for bulk silicon under (a) bi axial tensile stress; and (b) uniaxial compressive stress. ............. .53 311 Outofplane effective masses for bulk silicon under (a) biaxial tensile stress; and (b) uniaxial compressive stress. ................ .... 54 312 Hole effective mass change under very small stress. The change in this stress region explains the "d( I. oi ii ,ly of the hole effective mass between the re laxed and highly stressed Si. ............... .......... .. 57 313 The 25meV energy contours for unstressed Si: (a) Heavyhole; (b) Lighthole. .58 314 The 25meV energy contours for biaxial tensile stressed Si: (a) Top band; (b) Bottom band . ............... ............... .. 59 315 The 25meV energy contours for uniaxially compressive stressed Si: (a) Top band; (b) Bottom band ............... .............. .. 59 316 Quantum well and subbands energy levels under transverse electric field. . 61 317 Schematic plot of strain effect on subband splitting, the field effect is additive to uniaxial compression and subtractive to biaxial tension. . . .. 64 318 Subband splitting between the top two subbands under different stress. . 65 319 Outofplane effective masses for (110) surface oriented bulk silicon under uni axial com pressive stress. .................. .......... ..66 320 The 2D energy contours (25, 50, 75, and 100 meV) for bulk (001)Si. Uniaxial compressive stress changes hole effective mass more significantly than biaxial tensile stress .................... .................. .. 68 321 Confined 2D energy contours (25, 50, 75, and 100 meV) for (001)Si. The con tours are identical to the bulk counterparts. ................ . 69 322 The 2D energy contours (25, 50, 75, and 100 meV) for bulk (110)Si under (a) no stress; (b) uniaxial stress along ( 110); and (c) uniaxial stress along (111). .70 323 Confined 2D energy contours (25, 50, 75, and 100 meV) for (110)Si. The con fined contours are totally different from their bulk counterparts which , '.. r significant confinement effect. .................. ...... 71 324 Ground state subband hole population under different stress. . .... 72 325 Stress effect on the 2 dimensional densityofstates of the ground state subband. 74 326 Two dimensional densityofstates at E=4kT. .................. 75 327 Strain effect on (a) acoustic phonon, and (b) optical phonon scattering rate. .. 76 328 Strain effect on surface roughness scattering rate. .... . ... 77 329 Hole mobility gain contribution from (a) effective mass reduction; and (b) phonon scattering rate suppression for pMOSFETs under biaxial and uniaxial stress. 79 41 Hole mobility vs SOI thickness for single gate SOI pMOS. The mobility decreases with the thickness due to structural confinement. ............... .. 83 42 Hole mobility for single gate SOI pMOS vs uniaxial stress at charge density p 1 x 1013/cm 2 . . . . . . . . . 84 43 Hole mobility enhancement factor of UTB SOI SG devices vs uniaxial compres sive stress at charge density p 1 x 1013/cm2. ................. 85 44 Subband splitting UTB SOI SG devices vs uniaxial compressive stress at charge density p 1 x 1013/cm 2. .................. .. .......... 86 45 Comparison of the subband splitting of double gate and single gate MOSFETs. 87 46 Hole mobility of SDG devices under uniaxial compressive stress at charge den sity p = 1013/cm 2. ................................ 88 47 Hole mobility enhancement factor of SDG MOSFETs vs uniaxial compressive stress at charge density p 1 x 1013/c2. ............ . 89 48 Hole mobility of FinFETs under uniaxial stress compared with bulk (110)oriented devices at charge density p 1 x 1013/cm2. ................ . 90 49 Hole mobility enhancement factor of FinFETs under uniaxial compressive stress at charge density p 1 x 1013/cm2. ............... .... 91 410 Hole mobility gain contribution from effective mass and phonon scattering sup pression under uniaxial compression for (110)/(110) FinFETs compared with SG (110)/(110) pMOSFETs at charge density p 1 x 1013/c2. . ... 92 51 Germanium hole mobility vs effective electric field. .. . . ..... 97 52 Germanium and silicon hole mobility under biaxial tensile stress where the in version hole concentration is 1 x 1013/cm2. ............... .. .. 98 53 Germanium and silicon hole mobility under biaxial compressive stress where the inversion hole concentration is 1 x 1013/cm2. ................ 99 54 Germanium and silicon hole mobility on (001)oriented device under uniaxial compressive stress where the inversion hole concentration is 1 x 1013/cm2. .. 100 55 Germanium and silicon hole mobility on (110)oriented device under uniaxial compressive stress where the inversion hole concentration is 1 x 1013/cm2. .. 101 56 Ek diagrams for Ge under (a) no stress; (b) 1 GPa biaxial tensile stress; (c) 1 GPa biaxial compressive stress; and (d) 1 GPa uniaxial compressive stress. .. 103 57 Conductivity effective mass vs biaxial tensile stress: (a) C('!h im, I direction (<110>) and (b) outofplane direction. ............... ......... 104 58 Conductivity effective mass vs biaxial compressive stress: (a) C(! ,ii,, I direction (<110>) and (b) outofplane direction. ............... ...... 105 59 Conductivity effective mass vs uniaxial compressive stress: (a) C(, ,ii,, I direc tion (<110>) and (b) outofplane direction. ................ . 106 510 25meV energy contours for unstressed Ge: (a) Heavyhole; (b) Lighthole ..... 108 511 25meV energy contours for biaxial compressive stressed Ge: (a) Top band; (b) Bottom band . ............... ............... .. 108 512 25meV energy contours for biaxial tensile stressed Ge: (a) Top band; (b) Bot tom band ................... ............ ...... 109 513 25meV energy contours for uniaxially compressive stressed Ge: (a) Top band; (b) Bottom band .................. ................ .. 109 514 Ge subband splitting under different stress. ................... 110 515 Normalized ground state subband Ek diagram vs biaxial compressive stress. .. 112 516 Two dimensional densityofstates of the ground state subband for Si and Ge at (a)E 5meV; (b)E 2kT 52meV under uniaxial compressive stress. ...... ..113 517 Phonon scattering rate vs uniaxial compressive stress: (a) Acoustic phonon, and (b) optical phonon. .................. .......... 115 518 Surface roughness scattering rate vs uniaxial compressive stress for Ge and Si. 116 519 Mobility enhancement contribution from effective mass change (solid lines) and phonon scattering rate change (dashed lines) for Si and Ge under uniaxial com pressive stress .............. ................. .. 117 520 Confined 2D energy contours for (001)oriented Ge pMOS with uniaxial com pressive stress .................. ................. .. 117 521 Confined 2D energy contours for (H0)oriented Ge pMOS with uniaxial com pressive stress .................. ................. .. 118 A1 Stress distribution on crystals. .................. ....... 123 LIST OF TABLES Table page 21 LuttingerKohn parameters, deformation potentials and splitoff energy for sili con and germanium ................ ............. .. 31 31 Calculated and measured piezoresistance coefficients for Si pMOSFETs with (001) or (110) surface orientation. The first value of each pair is from measure ments and the second is from calculation. .................. ..... 40 A1 Elastic stiffnesses Cij in units of 101N/m2 and compliances Sij in units of 1011m2/N126 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 EFFECTS ON HOLE MOBILITY OF SILICON AND GERMANIUM PTYPE METALOXIDESEMICONDUCTOR FIELDEFFECTTRANSISTORS By Guangyu Sun August 2007 ('C! i: Scott E. Thompson Major: Electrical and Computer Engineering My research explores the strain enhanced hole mobility in silicon (Si) and germanium (Ge) ptype metaloxidesemiconductor fieldeffecttransistors (pMOSFETs). The piezore sistance coefficients are calculated and measured via wafer bending experiments. With good agreement in the measured and calculated small stress piezoresistance coefficients, k p calculations are used to give physical insights into hole mobility enhancement at large stress (3 GPa for Si and 6 GPa for Ge) for stresses of technological importance: inplane biaxial and channeldirection uniaxial stress on (001) and (110)surface oriented pMOSFETs with (110) and (111) channels. The mathematical definition of strain and stress is introduced and the transformation between the strain and stress tensor is demonstrated. Selfconsistent calculation of Schrodinger Equation and Poisson Equation is applied to study the potential and subband energy levels in the inversion 1vi. Subband structures, twodimensional (2D) density ofstates (DOS), hole effective mass, phonon and surface roughness scattering rate are evaluated numerically and the hole mobility is obtained from a linearization of Boltzmann Equation. The results show that hole mobility saturates at large stress. Under biaxial tensile stress, the hole mobility is degraded at small stress due to the subtractive nature of the strain and quantum confinement effects. At large stress, hole mobility is improved via the suppression of the phonon scattering. Biaxial compressive stress improves hole mobility slightly. Uniaxial compressive stress enhances the hole mobility monotonically as the stress increases. In (001) surface oriented pMOSFETs, the maximum enhancement factor is 35(0', for Si and 1,1111', for Ge. The enhancement of (110) pMOSFETs is smaller than (001) pMOSFETs due to the strong quantum confinement and low DOS of the ground state subband. For (001) pMOSFETs, the dominant factor to improve the hole mobility is the hole effective mass reduction at small stress and phonon scattering rate suppression at large stress. For (110) pMOSFETs, the hole effective mass and phonon scattering rate are constant at large stress due to the saturation of the subband splitting and DOS caused by the strong confinement. Strain effects on nonclassical devices (singlegate (SG) silicononinsulator (SOI) and doublegate (DG) pMOSFETs) are also investigated. The calculation shows that the mobility enhancement for SG SOI and DG (001) pMOSFETs is similar to traditional Si pMOSFETs. Hole mobility enhancement in FinFETs is more than traditional (110) pMOSFETs due to the subband modulation. CHAPTER 1 INTRODUCTION AND OVERVIEW Metaloxidesemiconductor fieldeffect transistors (\ OSFETs) have been scaled down .,.ressively to achieve density, speed and power improvement since 1960s [1]. As the channel length is scaled to submicron even nanoscale level, the simple scaling of complementary metaloxidesemiconductor (C\ IOS) devices brings severe shortchannel effects (SCEs) such as threshold voltage rolloff, degraded subthreshold slope, and drain induced barrier lowering (DIBL). Oxide thickness has to be reduced to sub10 nm (about 1 nm in the stateoftheart technology) and channel doping has to be increased up to 1019/cm3 in order to maintain good control of the channel [1]. The thin oxide and the high channel doping result in high vertical electric field in the channel that severely reduces the carrier mobility. Further scaling of the devices does not bring performance improvement due to carrier mobility degradation. With nothing to replace silicon C'\ OS devices in the near future and the need to maintain performance improvements and Moore's law, feature enhanced Si C'\ IOS technology has been recognized as the driver for the microelectronics industry. Strain is one key feature to enhance the performance of Si MOSFETs. Biaxial tensile strain has been investigated both experimentally and theoretically in C'\ IOS technology [2, 3, 4]. It improves the electron mobility [5], but degrades the hole mobility at low stress range (< 500MPa) [3]. Recently, uniaxial stress has been applied to Intel's 90, 65, and 45nm technologies to improve the drive current without significantly increased manufacturing complexity [5, 6]. The goal of this dissertation is to provide physical insights into the strain enhanced hole mobility in Si and Ge pMOSFETs. Before we investigate the hole mobility, the his tory of strain technology and the methods to apply strain to a transistor are is discussed in this chapter. The organization of the dissertation is also introduced. 1.1 History of Strain in Semiconductors The epitaxial growth of semiconductor lw.ri is not new. The basis of 1i, I Iivs experimental guide, piezoresistance [7, 8], and the theoretical approach to the strain effect, i.e., deformation potential theory, can be traced back to the 1950s. But not until in the early 1980's did scientists and engineers start to realize that strain could be a powerful tool to modify the band structure of semiconductors in a beneficial and predictable way [9, 10]. Deformation potential theory, which defines the concept of strain induced energy shift of the semiconductor, was first developed to account for the coupling between the acoustic waves and electrons in solids by Bardeen and Shockley [11], who stated that the local shift of energy bands by the acoustic phonon would be produced by an equivalent extrinsic strain, hence the energy shifts by both intrinsic and extrinsic strain can be described in the same deformation potential framework. The deformation potential theory was applied by Herring and Vogt [12] in 1955 in their transport studies of semiconductor conduction bands. A set of symbols, E, was used to label the deformation potentials. Herring and Vogt [12] also summarized the independent deformation potentials constrained by symmetry at different conduction band valleys. At the F point, another set of symbols are commonly used: ac, a,, b, and d, where a,, b, and d are three independent valence band deformation potentials which have a correspondence to the Luttinger parameters [13] employ, .1 in band calculations. The k p method we use in this work relies on these three deformation potentials to account for the strain effects. Smith measured the piezoresistance coefficients for n and ptype strained bulk silicon and germanium in 1954 [7]. This was the first experimental work that studied strain effects on semiconductor transport. Herring and Vogt used Shockley's band model and ascribed the electron mobility change to two strain effects, "electron transfer effect" and intervalley scattering rate change caused by valley energy shift [12]. This is essentially the same physics that explains the strain enhanced mobility in silicon nchannel MOSFETs. Piezoresistance coefficients are widely used in the industry due to its simplicity in representing the semiconductor transport properties (mobility, resistance, and et al.) under strain. It is defined as the relative resistance change with the stress applied on the semiconductor. Piezoresistance coefficient (7r) can be expressed as 1 Ap) lAp (1 1) a p where a is the applied stress and p is the resistivity of the semiconductor. In 1968, Colman [8] measured the piezoresistance coefficients in ptype inversion l ir. This was the first time that strain effect on hole transport was investigated in the inversion lri~. The similarity and difference of the piezoresistance coefficients compared with the bulk silicon was explained qualitatively in that work. The first silicon nchannel MOSFET which used biaxial stress to improve the electron mobility was demonstrated by Welser et al. [14] in 1992. The work showed that the electron mobility was improved by 2.2 times. A biaxial stressed silicon pchannel MOSFET was first reported by N i I1: et al. [15] in 1993 where the hole mobility was enhanced by 1.5 times. In 1995, Rim [16] showed the hole mobility enhancement in silicon pMOSFETs on top of Sil_Gex substrate with different germanium components. The idea of using longitudinal uniaxial stress to improve the performance of MOSFETs was activated by Ito et al. [17] and Shimizu et al. [18] in the late 1990's through the investigations of introducing high stress capping lI r deposited on MOSFETs to induce channel stress. Gannavaram et al. [19] proposed Sil_Ge1 in the source and drain region for higher boron activation and reduced external resistance which also furnished a technically convenient means to employ uniaxial channel stress. These studies opened the gate to use strain as active factor in VLSI device design and resulted in extensive industrial applications. 1.2 Apply Strain to A Transistor Strain in the channel of Si and Ge MOSFETs is achieved by applying mechanical stress to the wafer. The properties, and relations of strain and stress can be found in the appendix. Here we first introduce how to apply biaxial and uniaxial stress in Si MOSFETs. For (001) wafer, biaxial tensile stress in Si MOSFETs is applied to the channel by using the Sil_Gex substrate. The lattice mismatch stretches silicon atoms in both (100) and (010) directions which is illustrated in Figure 11. The percentage of germanium content in the substrate determines the magnitude of the strain. This inplane tensile strain can also be achieved by applying uniaxial compressive stress from the outofplane direction [20] with capping 1~v. r. The outofplane uniaxial compression is equivalent to the inplane biaxial tension in determining the transport properties of Si. The details are shown in the appendix. For Ge MOSFETs, biaxial tensile stress is not applicable due to its large lattice constant. Biaxial compressive stress is usually introduced by applying Si or SilxGe substrate. I train" "r !'a^ d_ Relaxed SiGe Substrate Figure 11. Schematic diagram of biaxial tensile stressed SiMOSFET on relaxed SilzGez l,v,r Uniaxial stress can be applied from outofplane, inplane longitudinal (parallel to the channel), or inplane transverse (perpendicular to the channel) direction. The inplane longitudinal stress is applied to the channel by either doping germanium to source and drain or depositing compressive or tensile capping 1i, r on top of the device which is shown in Figure 12 [20]. 45nm Gae High stress film ptype MOSFET ntype MOSFET Figure 12. Uniaxial stressed SiMOSFET with Sil_,Ge, Source/Drain or highly stressed capping li ,r Without further clarification, uniaxial stress in this work represents inplane uniaxial longitudinal stress. It is normally along (110) since it is the classical channel direction. Biaxial stress means inplane biaxial stress. For (O110)oriented wafer, biaxial stress is employ, ,1 in both parallel and perpendicular direction to the channel (<110> and <100> directions). The strain in those two directions are not as same as (001)oriented wafer (<100> and <010>directions) due to the different Young's Modulus in <110> and <100> directions. 1.3 Main Contributions of My Research Strain enhanced hole mobility has been reported experimentally at small stress. Little theoretical work has been done to provide physical insights into hole mobility enhancement under large stress, especially uniaxial compressive stress. Strain effects on hybrid ((ll0)surface oriented), nonclassical and Ge pMOSFETs are not understood either. In this work, piezoresistance coefficients are calculated and measured on (110) Si pMOSFETs. Physics of uniaxial stress enhanced hole mobility in (110) pMOSFETs is studied for the first time. The hole mobility dependence on device surface orientation is calculated and the different quantum confinement effect is discussed. Straininduced changes in hole effective mass, subband structures, densityofstates (DOS), phonon and surface roughness scattering rate are analyzed numerically. The results show that under uniaxial stress, 350' and 1 1I 1'. mobility enhancement are achieved in (001) Si and Ge pMOSFETs, respectively. The more enhancement in Ge pMOSFETs is due to smaller hole effective mass of Ge under stress. In (110) Si and Ge pMOSFETs, it is reported for the first time that the maximum enhancement factor is only 10i1' due to the strong quantum confinement undermining the strain effect. Strain induced hole mobility enhancement is studied theoretically for the first time in ultrathinbody (UTB) nonclassical pMOSFETs, including singlegate (SG) silicon oninsulator (SOI), (001) symmetrical doublegate (SDG) pMOSFETs, and (110) ptype FinFETs. For SG SOI pMOSFETs, the strain effects are as same as traditional Si pMOSFETs. For (001) SDG pMOSFETs and (110) FinFETs, subband modulation is found when the channel thickness is smaller than 20 nm. As the stress increases, the mobility enhancement in (001) SDG pMOSFETs is comparable to traditional SG pMOSFETs. For FinFETs, the form factors are much smaller than SG (110) p MOSFETs and the change with stress is larger which , ii. r more reduction of the phonon scattering rate. Therefore, the straininduced hole mobility enhancement (2111'.) is larger than single gate (110) pMOSFETs (1011' .). 1.4 Brief Description of The Dissertation The main purpose of my research is to provide a simple but accurate physical insight into strain effects on hole mobility in Si and Ge inversion l, i~. We begin by introducing the physics model. A sixband k p model with strain effects is derived and finite difference method (FDM) is introduced briefly. Selfconsistent calculation of Schridinger Equation and Poisson Equation is discussed. The isotropic approximation of scattering rate calculation is showed. In the calculation of the hole mobility, the Kubo Greenwood formula, which is from a linearization of Boltzmann Equation, is introduced. Strain enhanced hole mobility in singlegate Si pMOSFETs is then discussed. The unstrained Si hole mobility versus device surface orientation and vertical electric field is calculated. Hole mobility under biaxial and uniaxial stress in (001) and (110) pMOSFETs is showed. The band structure of bulk silicon under strain is discussed. In the Si inversion 1i. i~, the confined energy contours, subband splitting, hole population in ground state subband, twodimensional (2D) densityofstates (DOS), phonon and surface roughness scattering rate are evaluated. The difference of strain induced hole mobility enhancement in (001) and (110) pMOSFETs under biaxial and uniaxial stress is explained. Uniaxial straininduced hole mobility enhancement is calculated for UTB non classical pMOSFETs, including singlegate SOI, (001) SDG pMOSFETs, and (110) ptype FinFETs. The similarity and difference from the traditional Si pMOSFETs are discussed and physical insights are given. Strain induced hole mobility enhancement in Ge pMOSFETs is discussed. Un strained hole mobility in (001) and (110) Ge pMOSFETs is calculated. Strain effect on hole mobility in Sil_Gex with arbitrary Ge components is evaluated. To understand the physics, the bulk valence band structure and hole effective mass with strain effects are calculated. In the inversion l. ,i, the subband structure, 2D DOS and scattering rate are calculated and their relation to hole mobility is analyzed. We conclude with the results that we obtain in this dissertation and sl::. 1 possible future research on strained Si and Ge. CHAPTER 2 K P MODEL AND HOLE MOBILITY Global descriptions of the dispersion relations of bulk materials can be obtained via pseudopotential or tightbinding methods [21]. However, such global solution over the whole Brillouin zone is unnecessary for many aspects of semiconductor electronic properties. What is needed is the knowledge of the dispersion relations over a small k around the band extrema [21]. k p method is widely used in i i. I,1 vIs quantum well and quantum dots calculation due to its simplicity and accuracy regarding the properties in the vicinity of conduction band and valence band edges which govern most optical and electronic phenomena. To study the uniaxial or biaxial strain effect on hole mobility in the inversion 1~,l. s, a 6band k p model, LuttingerKohn's Hamiltonian [13], is utilized in this work. In this chapter, k p method and the derivation of the luttingerKohn's Hamiltonian is introduced first. Then the procedure calculating the hole mobility is explained. Finally, the evaluation of scattering mechanisms, mainly the phonon and surface roughness II. 11. i i. is discussed. In the calculation of the hole mobility with strain effect in the inversion 1~,l s, Schr6dinger Equation and Poisson Equation are solved selfconsistently to simulate the potential energy in the channel. The subband structure and the twodimensional density ofstates (2D DOS) of each subband are calculated and the scattering relaxation time is evaluated in k space. Finally, hole mobility is obtained from a linearization of the Boltzmann equation. 2.1 The k p Method 2.1.1 Introduction to k p Method The k p method [21, 22, 23] is essentially based on the perturbation theory and was first introduced by Bardeen [24] and Seitz [25]. It is also referred to as effective mass theory in literatures. The k p method is most useful for analyzing the band structure near the extrema (ko) of the band. In the case of the band structure near the F point, i.e. valence band edge of silicon and germanium, ko = 0. For an electron in a periodic potential V(r) V(r + R), (21) where R = nal + n2a2 + n3a3, and al, a2, a3 are the lattice vectors, and ni, 72, and n3 are integers, the electron wave function can be described by the Schr6dinger equation H b(r) + V(r) (r) [2 V2 + V(r) b (r) E (k)b (r) (22) 2mo I 2mo I where p h= V/i is the momentum operator, mo is the free electron mass, and V(r) represents the potential including the effective lattice periodic potential caused by the nuclei, ions and core electrons or the potential due to the exchange correlation, impurities, etc. The solution of the Schr6dinger equation H k(r) E k(r) (23) satisfies the condition bk(r + R) e= ikrk(r) (24) k (r) = eikr k(r) (25) where Uk(r +R) Uk(r), (26) and k is the wave vector. Equations 24, 25 and 26 is the Bloch theorem, which gives the properties of the wave function of an electron in a periodic potential V(r). The eigenvalues for Equation 23 can be categorized into a series of bands E, n = 1, 2,... [26] due to the perturbation of the periodic potential at the Brillouin zone edge. Consider the Schr6dinger equation in the nth band with a wave vector k, P + V(r) blk(r) E (k) .k(r). (27) 2mo I Inserting the Bloch function Equation 25 into Equation 27, we have [p2 h2k2 h 1 + + k p + V(r) Uk(r) E(k)Unk(r). (28) 2mo 2mo mo Including the spinorbit interaction term S(7 x VV) p (29) in the Hamiltonian and simplifying the equation, Equation 28 becomes p2 h 2k2 hk h h + + P + (x VV) + 2a x VV) p + V(r) Unk 2mo 2mo mo \ i 1,2 j 1 ',,* EE,(k)Unk(r). (210) where c is the speed of light and a is the Pauli spin matrix. a has the components [22] 0 1 0 i 1 0 (rx = y= z= (211) 1 0 i 0 0 1 Rewriting the Hamiltonian in Equation 2.1.1, we have [Ho + W(k)]unk = EnkUnk, (212) where p2 h Ho =2 + (a x VV) p + V(r) (213) 2mo *1 i, and hk ( h N h2k2 W(k) = p+ ( x VV) + (214) mo 1 ,,:2 2mo Since only W(k) depends on wave vector k, Equation 213 can be used to evaluate the band property at ko. If the Hamiltonian Ho has a complete set of orthonormal eigenfunctions at k = 0, uo, i.e., Houno Enouno, (215) theoretically any function with lattice periodicity can be expanded using eigenfunctions uno. Substituting the expression (216) Unk = ZCn(k)unmo in into Equation 2.1.1, multiplying from the left by u*0, integrating and using the orthonor mality of the basis functions, we have (Eno Ek + )6nm Un0 + 1, 2 (7 x VV) Uno) c(k) 0. m 2mo mo .2 (217) Solving this matrix equation gives us both the exact eigenstates and eigenenergies. As we mentioned earlier, only the dispersion relations over a small k range around the band extrema are important describing the electronic properties of the semiconductor. Only energetically .,ili i,:ent bands are normally considered when studying the k expansion of one specific band for simpleness. To pursue acceptable solutions when k increases, one has to increase the number of the basis states, or consider higher order perturbations, or even both. Neglecting the nondiagonal terms in Equation 217 for small k, the eigenfunction is Unk Uno, and the corresponding eigenvalue is given by Enk = Eno + > The solution can be improved using the second order perturbation theory, i.e. S2k2 UnO I H' umo) (umo H' uno) Enk = Eno + + (218) 2mo mn Eno ~ Emo where H m' 1 i ,,, 2 ( x ) (219) (uo (P + 4moc2(xvv) uno) = was applied in the calculation, which holds for a cubic lattice periodic Hamiltonian due to the crystal symmetry. If we write S= p + h ( x VV) (220) 1 t, ,,,,. 2 the second order eigenenergies can be written as Sh2k2 h2 lrnm, k2 Enk = Eno + m+ (221) 2mo0 'n mn n Emo Equation 221 can also be expressed as Enk = Eno+ ( + kk3, (222) 2 a,/3 m a/3 where 1 1 2 _ _ m 6oa, + 2 E (223) m* mo ',O mT, En0o Emo m* in Equation 223 is the effective mass tensor, and a, = x, y, z. The effective mass generally is anisotropic and k dependent. In the vicinity of the F point, sometimes m* can be treated as kindependent, since at this level of approximation, the eigenenergies close to the F point only depend quadratically on k [22, 23]. 2.1.2 Kane's Model Expanding in a complete set of orthonormal basis states in Equation 217 gives exact solutions of both eigenenergies and eigenfunctions. In reality,it is almost impossible to include a complete set of basis states, therefore only strongly coupled bands are included in usual k p formalism, and the influence of the energetically distant bands is treated as perturbation. In Kane's model for Si, Ge, or IIIV semiconductors, four bands are considered as strongly couples bandsthe conduction, heavyhole (HH), lighthole (LH), and te spin orbit splitoff (SO) bands are considered, which have double degeneracy with their spin counterparts. The rest bands are treated as perturbation and can be analyzed with the second order perturbation theory. Our goal is to find the eigenvalue E of Equation 2.1.1 with eigenfunction Unk(r) = aUno(r) (224) The band edge functions uo(r) are Conduction band: IS T), S 1) for eigenenergy E, (stype), Valence band: IX T), IY T), I T), IX 1), IY 1), Z 1) for eigenenergy E, (ptype). Normally the following eight basis functions are chosen )iS ), xiY T Z), IX+lY T) and TiS ), eigt bsisY ), Ists frY )K The eight basis states for Kane's model are 1 1 = 2' 3 U2= , 2 3 2' 1 4 =  , 24 1 5 ' 3 6 =  , 2 3 U7 =  , 7 ' 1 Us = , 2 IS T) = S ), IHH T) '(X + iY) ), v2 1 2 LHT) (X+iY) + IZT), ISO =) \(X + iY)1) + Z ), = IS 1) = ), = HH 1) = (X iY) ), = LH 1) = SO ) v/z l(X iY) T) + Z ), (X Y) ) Z ). (225) This set of basis states is a unitary transformation of the basis functions and the eigen functions of the Hamiltonian 213. The eigenenergies for IS), IHH), ILH) and ISO) at k = 0 are E,, 0, 0, A, respectively, where E, is the band gap, and the energy of the top of the valence band (HH and LH) is chosen to be 0. A is the splitoff band energy, which is 44meV for Si and 296meV for Ge. At this level of approximation, the bands are still flat because the Hamiltonian 213 is kindependent. Including W(k) in Equation 214 into the Hamiltonian, and defining Kane's parameter as we obtain a matrix expression for Eg + h 2 2mo Pk_  Pk+  Pk+ 0 0  3Pk,  Pk v1P3 Pk+ h2k2 2mo 0 0 0 0 0 0 0 'Pk_ 0 h2 k2 2TnO 0 o P Pk 0 0 0 ih P (SIZ), mo the Hamiltonian H = 3Pk_ 0 0 A + 2mo SPk, 0 0 0 0 0  3Pk, Pk, Eg + h2k2 Pk+ SPk_ 3Pk_ (226) Ho + W(k), i.e., 0 Pk, 0 0 0 0 0 0 Pk_ Pk+ hk2 0 2m0O 0 h2k2 o mo o o Pkz 0 0 0 Pk+ 0 0 A + (227) where k+ = k + iky, k_ = kx iky, and k,, ky, k, are the cartesian components of k. The Hamiltonian 227 is easy to diagonalize to find the eigenenergies and eigenstates as functions of k. We have eight eigenenergies, but due to spin degeneracy, there are only four different eigenenergies listed below. For the conduction band, 1 1 4P2 2P2 S + + + A) m mo+ 32E, 3+2(E, + A) For the light hole and splitoff bands, k2k2/ Elh = 2mlh A 2k2 Eso = A  2mso ' 1 mlh 1 mso 1 4P2 p2 mo 3k2E,' 1 2P2 Smo 2+) mo 3h (E9 + A) h2k2 Ec = Eg +  2m,c (228) (229) (230) For the heavy hole band we have h2k2 1 1 Ehh = (231) 2mhh mhh mo These results are not complete since the effects of higher bands have not been included. They will be taken into account next when discussing the LuttingerKohn model. 2.1.3 LuttingerKohn's Hamiltonian For Si and Ge hole transport, we are only interested in the six valence bands (doubly degenerate HH, LH, and SO). The coupling to the two conduction bands in Kane's model is ignored due to the large band gap. It is convenient to use Lbwdin's perturbation method [27] where the six valence bands are treated in class A and the rest bands are put in class B. We label class A with subscript n and class B with subscript 7. Wave function uk(r) can be expanded as A B Uk(r) (k) (r) + a.(k) uo(r). (232) n 7 C'! .... the eigenstates for class A, we have 3 )  3 1 1 U3 I )= LH 1) = I(X+ iY) +z 2'2 2, 311 2 U3  ) ILH1)= (X iY) + Z{), 2'2 6 3 33 1 = I ) = HH 1) = (X Y) ), 11 1 1 U5 I ) ISO ) =t(X + iY) +) Z ), 2' 23 1 1 1 1 U6 , ) ISO 1) = (X Y) T) Z 1). (233) 2With Lwdin's method we only need to solve the eigenequation With Liwdin's method we only need to solve the eigenequation A n(U n B H HH. UA H,, + E  i7nn 0 E Hjn = (ujo IHIuo) ) 2k2 [Ej(0) + ] 6 2mo H' = (ujo k Inuo) Tmo Let UjA = Dj, DjT can be expressed as Dj = Ej(0)6j where D"W is defined as h2 n"D3 6j,5 in 2mo (239) To express DjT explicitly, we difine hkO 38 So af + D a8 B +E ,7 mo(Eo  Pj/P7n  E7) h2 Ao = 2mo h2 Bo = 2moo h2 B x x 1,1' Eo E 1, 7 Eo E7 k2P TPT + ,, y G h2 B pxp y py Co  i ,, Eo E7 Then define the Luttinger parameters 71, 72, and 73 as where Ej,)a,(k) 0 (234) B Hj,+ E Y3j,n (235) H7 H' E E7 (j, ne A) (236) (j A,7 A) (237) (238) (240) h2 1 h71 (ao + 2Bo) 2mo 3 h2 1 2 72 ( ao B) 2mo 6 2 73 C (241) 2mo 6 Finally we obtain the LuttingerKohn Hamiltonian P +Q S R 0 IS v2R 12 2 3 1 S+ PQ 0 R v2Q S R+ 0 PQ S S+ v2Q 22 H _) 3 1 (242) T 3 3) ' S+ Q 2s VFR P+ 0o 2 1 1 where, P_= 7(i (k2 1(k 2 + k 2), Zmo 2 2mo 3[(kx + 2I1 , S = 2 373(k iky)kz. (243) When strain is present in the semiconductor, P, Q, R, and S in Equation 243 can be resolved to two parts: k p terms (Pk, Qk, Rk, and Sk) and strain terms (Pe, Qe, Re, and Se). They can be expressed as [13] P = Pk + P Q2 = Qk + ,, R = Rk + R S SSk+ S,, Pk ( h2 71 (k 2+ k 2+ k ), Qk 72(k + k 2k ), Rk [72 (k2 k ) + 2i ., I ], (244) 2mo2 Pe = a(6xzz + 6yy + 6zz), Q = b(xx + ySy 2eZZ), 2 C v3 b(xx yy) id(xy Se = d( where ij is the symmetric strain tensor as shown in C'! ipter 1; av, b, and d are the BirPikus deformation potentials for valence band; A is the spinorbit splitoff energy, and the basis function Ij, m) denotes the Bloch wave function at the zone center. Energy zero is taken to be the top of the unstrained valence band. Table 1 shows the parameters for silicon and germanium [28]. Table 21. LuttingerKohn parameters, deformation potentials and splitoff energy for silicon and germanium. 71 72 73 av(eV) b(eV) d(eV) A(eV) Si 4.22 0.39 1.44 2.46 2.35 5.3 0.044 Ge 13.35 4.25 5.69 2.09 2.55 5.3 0.296 2.2 Hole Mobility in Inversion Layers 2.2.1 Selfconsistent Procedure For Si or Ge pMOSFETs, holes are confined in the zdirection quantum well formed by the Si/SiO2 interface and the valence band edge. Since the hole energy is not continu ous along zdirection kz should be replaced by i in Equation 245. Coordinate system transformation is needed to calculate cases with other surface orientation. Subband energy can be evaluated by solving Schr6dinger Equation, [H(k, z) + V(z)]k(z) = E(k)k(z) (245) where V(z) defines the potential energy in the quantum well. Triangular potential approximation is widely used in simulations for simplicity. Stern [29] stated that it should not be used when mobile charges are present. In order to accurately simulate the potential in the quantum well, Schr6dinger Equation 245 is solved selfconsistently with Poisson Equation d2 q2 dz2VH(z) = [p(z) n(z) + N(z) N (z)] (246) where p(z) and n(z) are mobile hole and electron density, ND+(z) and NA(z) are space charge density. To numerically evaluate Schr6dinger Equation and Poisson Equation, Finite Difference Method is utilized. The equations are evaluated on a z mesh of N. points in the interval (0, zax) [3, 30, 31], where zax here is the sum of the thickness of silicon 1.ir and oxide ly .r. This yields a 6Nz x 6Nz eigenvalue problem of the tridiagonal block form [3]. Schr6dinger Equation becomes H_ H,_1 H+ 0 0 _ S0 H_ H H+ 0 =E(k) (247) 0 0 H_ Hii H ' where each ,' = (zi) is a sixcomponent columnvector Qj(zi), the index j running over the k p basis, and H_, Hi, H+ = H_ are 6 x 6 blockdiagonal difference operators, functions of the inplane wavevector k. In principle, the potential V(z) results from three terms: an imageterm, ', .(z); an exchange and correlation potential, V,,(z); and the Hartree term, VH(z) [3, 30]. Fischetti [3] ,.. 1 that the image potential cancels the manybody corrections given by the exchange and correlation term and the Hartree term is focused as the solution of the selfconsistent calculation of Schr6dinger Equation 245 and Poisson Equation 246. 2.2.2 Hole Mobility The hole mobility in inversion l1..is can be calculated from a linearization of the Boltzmann equation. The xx component of the mobility tensor can be expressed as [3] e 2T E) K (E,) 4h2742kBTp, JO J f E,. Dk K,(E,w) x (a KE,2 V) [K,(E, ), ] fo(E)[1 fo(E)] (248) \0x K, (E,6) where p8 = Ep, is the total hole concentration in the inversion ' r, p, is the hole density of subband v, Tf') (K, Q) is the xcomponent of the momentum relaxation time in subband v, and fo(E) = + (249) 1 exp )T is the FermiDirac distribution function. The evaluation of densityofstates (DOS) and a term need further consideration. In energy space a maximum kinetic energy Emax for each subband is selected in order to account correctly for the thermal occupation of the topmost subband. In our calculation, we assumed Emx = 120meV and divided the energy space to 1200 uniform parts, then evaluated DOS and E in each part. 2.3 Scattering Mechanisms Phonon i l 1. i ii impurity scattering and surface roughness scattering are involved in C'\ OS transistors. In the linear region of pMOSFETs, neither chargedimpurity nor neutralimpurity scattering is important [3], hence they are neglected in our calculation. Only phonon scattering and surface roughness scattering are investigated. 2.3.1 Phonon Scattering Carriers migrate through the crystal with properties determined by the periodic potential associated with the array of ions at the lattice points [32]. Vibration of the ions about their equilibrium positions introduces interaction between electrons and the ions. This interaction induces transitions between different states. And this process is called phonon scattering. Phonon scattering can be categorized to acoustic phonon scattering and optical phonon scattering based on the phase of the vibration of the 2 different atoms in one primitive cell. Both contribute to the momentum relaxation time. Acoustic phonon energy is negligible compared with carrier energy, while optical phonon energy is about 61.3meV for silicon and 37meV for germanium at long wavelength limit. When strain is applied to the i I I1 the HH and LH degeneracy is lifted at Fpoint, as we mentioned previously. Therefore, the interband optical phonon scattering will be limited due to band splitting and mobility is enhanced. In fact, this is only significant when strain is high and the band splitting is beyond the optical phonon energy. The reasoning will be shown in the following section. One should also notice that the anisotropic nature of silicon valence bands makes the modeling of scattering rate a complicated task. Since we only need considering scattering in F valley for holes with the diamond crystal structure, equipartition approximation [32] is used where we replace the anisotropic holephonon matrix element with appropriate angleaveraged quantities. First for acoustic phonon, relaxation time r can be expressed as [3] 1 2rkBT2~yy 2 e 7Fp,[E,(K)] (2 50) Tc hpu1 where Eff = 7.18eV is the effective acoustic deformation potential of the valence band, p, is the 2dimensional densityofstates of subband v which is defined as K(E, ) p0~ O[E E)1E d E, (251) oK K,(E,yL) The twodimensional carrier scattering rate for the phononassisted transitions of a carrier from an initial state in the pth subband and a final state in the vth subband is proportional to the form factor 1 1 r+foc F, W = + II, (q,)2 dqz, (252) 27W,,, 27 oo where I, (q) (z ) iqz (v) (z) dz. (253) The form factor F,, illustrates the interaction between initial state and final state due to the wave function overlapping, where "() (z) or < ")(z) is the envelope function at k for subband p or v, respectively. z is the coordinate perpendicular to the Si/SiO2 interface, and q, is the change in the component perpenticular to the interfaces of the carrier momentum in a transition from the pth subband to the vth subband. Following Price's pioneering work, W,, can be expressed as 1 fZmax 2 2I I S270 dz k ''(z) ',k (z) (254) If the final state is also pth subband, W,, represents the effective quantum well width for the pth subband. Since the acoustic phonon energy is small compared with subband splitting or even the thermal energy kT, acoustic phonon scattering is an equalenergy scattering process [32]. The scattering rate solely depends on the densityofstates of the final states. Strain effect on acoustic phonon scattering is smaller than that on optical phonon scattering which is shown in our simulation. Second, the optical phonon scattering relaxation time is expressed as [3] 1 D2 fo [E,(K) hu 1op 1 1 DZ p, V[E,(K) Fhw, p] x t nj p + (255) To Pw~ op 1 fo[E,(K)1 2 2 For absorption and emission, respectively, where Dp = 13.24 x 108eV/cm is the optical deformation potential constant of the valence band, huw = 61.3eV is the silicon optical phonon energy. Optical phonon scattering is not significantly reduced for stress < 1GPa since the subband splitting is less than the optical phonon energy. 2.3.2 Surface Roughness Scattering In MOSFETs, carriers are confined close to the channeloxide interface in strong inversion region. Thermal movement of carriers also results in collision with the interface and hence affects the carrier mobility. This interaction depends heavily on the roughness of the interface. Therefore, this scattering mechanism is called surface roughness scat tering. Surface roughness scattering can be neglected when the transverse electric field is small, since not many carriers are present and they are not strongly confined to the channeloxide interface. But when the electric field is high (carrier density over 1013/cm2), surface roughness scattering must be taken into account in mobility calculation. Unfortunately, people are still unable to model the roughness scattering accu rately [33, 3]. The early formulation by Prange and Nee, Saitoh, and Ando is still the best model available [3]. Different roughness parameters are used in different references. Here, we'll use Gamiz' model and corresponding parameters [34]. As we know, the surface roughness scattering is caused by the roughness of the surface and hence the abrupt potential change at Si/SiO2 interface. 2 assumptions are needed in the simplification of the problem [34]. The first assumption is to consider the interface between silicon and oxide is an abrupt boundary which randomly varies according to a function A of the parallel coordinate, r, A(r). Another assumption is that the potential V(z) close to the interface can be expressed by 8V(z) V[z + A(r)] V(z) + A(r) () (256) Oz The scattering rate can be expressed as [34], EP[Ep(K)]2 )AV(z) ,(z)dz AL2 TsR (k) h Mv ,' (Z A, ox2 d (2 57) + (I+ L22 In this equation Av() is approximately equal to the effective electric field, which means the scattering rate is proportional to the square of the electric field. Therefore, surface roughness scattering becomes more significant when electric field reaches higher level. Different values for L and A are taken by different researchers to explain the exper imental data. Here, we use L = 20.4nm and A = 4nm [3] for silicon as ir'i I. 1 by Fischetti. n = 0.5 [34] is chosen in this work. 2.4 Summary The physics model used in the dissertation is reviewed in this chapter. The history of k p method is introduced. The derivation of Kane's model and LuttingerKohn Hamiltonian is showed. The calculation procedure of the hole mobility in inversion 1i. r is introduced. Phonon and surface roughness scattering are taken into account as the main scattering mechanisms. Form factors and their impact on scattering rate are discussed. In this work, MATLAB and C codes are written to calculate the hole effective mass, band and subband structures, and hole mobility in Si and Ge pMOSFETs. To calculate hole mobility dependence on device surface orientation, coordinate transformation is performed to calculate hole mobility in (110), (111), and (112) oriented Si and Ge to account for the different quantum confinement conditions. For different surfaces, different surface roughness parameters are utilized to fit the interface roughness condition. DOS and form factors are calculated in the whole k space. CHAPTER 3 STRAIN EFFECTS ON SILICON PMOSFETS Hole transport in the inversion li. r of silicon pMOSFETs under arbitrary stress and device surface orientation is discussed in this chapter. Piezoresistance coefficients are cal culated and measured at stress up to 300 MPa via waferbending experiments for stresses of technological importance: uniaxial compressive and biaxial tensile stress on (001) and (10O)surface oriented devices. With good agreement in the measured vs calculated low stress piezoresistance coefficients, k p calculation are used to give insight at high stress (13 GPa). The results show that biaxial tensile stress degrades the hole mobility at low stress due to the quantum confinement offsetting the strain effect. Uniaxial stress on (001)/<110>, (110)/<110>, and (110)/<111> devices improves the hole mobility mono tonically. Unstressed (110)oriented devices have superior mobility over (001)oriented devices due to the strong quantum confinement causing smaller conductivity effective mass of the holes. When the stress is present, the confinement of (110)oriented devices under mines the stress effect, hence the enhancement factor for (110)oriented devices is less than (001)oriented devices. Hole mobility enhancement saturates as the stress increases. At high stress, the maximum hole mobility for (001)/<110>, (110)/<110>, and (110)/<111> devices is comparable. Physical insights are given to explain the difference between biaxial and uniaxial stress, and the difference of (110) and (001) pMOSFETs. The bulk silicon valence band structure under uniaxial compressive or biaxial tensile strain is shown and the difference in effective mass change is calculated. The difference of the vertical electric field (quantum confienment) effect on (001) and (110)oriented pMOSFETs is explained. Subband splitting, ground state subband hole population, and two dimensional (2D) densityof states (DOS) of subbands are calculated under stress. Scattering rate change with stress is also discussed. 3.1 Piezoresistance Coefficients and Hole Mobility Calculated and measured piezoresistance coefficients, and calculated hole mobility vs stress and surface orientation of Si pMOSFETs are covered in this section. 3.1.1 Piezoresistance Coefficients Piezoresistance coefficients are widely used as an effective approach characterizing the resistance change at low stress [7, 8]. Table 31 compared measured and calculated piezoresistance coefficients. In the measurements, the stress is applied using 4point or concentricring bending of the wafers. The piezoresistance coefficients are obtained through the linear regression of the measured resistance versus stress. The actual strain in the devices is measured through the resistance change of a strain gauge mounted on the sample, and via the laserdetected curvature change of bent wafer. In Table 31, 7rL, TTT, and 7Biaxial represent longitudinal, transverse, and biaxial piezoresistance coefficients, respectively. Table 31. Calculated and measured piezoresistance coefficients for Si pMOSFETs with (001) or (110) surface orientation. The first value of each pair is from measurements and the second is from calculation. Substrate (001) (110) Channel <110> <110> <110> 7L 71.7 [6]/72.2 27.3/34 86 [35/79.1 T2 33.8 [6]/ 45.8 5.1/6.6 50 [35]/ 43 7Biaxial 40/35.7 35.7/28.7 15.1/10.2 Both measured and calculated results in table 31 show that under uniaxial longitu dinal stress, (110)/<111> devices have the largest piezoresistance coefficient, followed by the (001)/<110> devices. The piezoresistance coefficient of (110)/<110> devices is the lowest. Under uniaxial transverse stress, the piezoresistance coefficients are smaller than longitudial stress for all pMOSFETs. The table also shows that the biaxial tensile strain increases the channel resistance and hence degrades the hole mobility at low stress. 3.1.2 Hole Mobility vs Surface Orientation Surface and channel orientation dependence of electron and hole mobility has been investigated experimentally since 1960's. Sato [36] reported that for ptype devices with <110> channel, the mobility is the highest in (110)oriented and lowest in (001)oriented pMOSFETs. The hole mobility on a few surface orientations is simulated and compared with Sato's experimental results [36, 37] in Figure 31. Good agreement is found between the calculation and the experimental data. Two different surface roughness models [3, 34] are used in the calculation. Both models are quite accurate and in the following results, Gamiz' surface roughness model is utilized. 250 With Gamiz' surface S200 roughness model *0 *) * S150 0 S100A. 2 50 With Fischetti's surface > 50 With Fischetti's surface roughness model 0 (001) (112) (111) (110) Surface Orientation Figure 31. Hole mobility vs device surface orientation for relaxed silicon with <110> channel. The hole mobility is highest on (110) and lowest on (001) devices. Different surface roughness scattering models are used in the simulation(solid: Gamiz 1999; dotted: Fischetti 2003). 3.1.3 Hole Mobility and Vertical Electric Field The calculated hole mobility versus the effective electric field of unstressed (001)/<110> and (110)/<110> Si pMOSFETs are compared with experimental mobility curves [38, 39, 40] in Figure 32. The agreement between the calculation and the experimental results Ii. I1 this work use reasonable scattering mechanisms. Normally (110)Si has smoother interface with the gate dielectric materials [41, 42], hence the surface roughness scattering rate is lower than (001)oriented devices. Lee [43] even sl. 1. ,1 that the effective field in (110)oriented devices is smaller than (001)oriented devices, which also indicates smaller surface roughness scattering rate considering that the scattering rate is inversely propor tional to the effective electric field [3, 34]. The smaller surface roughness scattering rate is partly responsible for the higher hole mobility on unstressed (110)oriented deices than that of the (001)oriented devices. To fit the appropriate surface roughness condition, the roughness parameters used are L = 2.6nm, A = 0.4nm for (001)oriented pMOSFETs and L = 1.03nm, A 0.27nm for (110)oriented pMOSFETs in this work. The same sur face roughness scattering model is utilized in the mobility calculation even when the strain is present, assuming that the processinduced strain (uniaxial strain) does not change the Si/SiO2 interface properties [3, 44]. 3.1.4 Strainenhanced Hole Mobility Figure 33 shows the hole mobility versus (up to 3 GPa) stress at inversion charge density pi, = 1 x 1013/cm2 and channel doping density ND = 1 x 1017/cm3) for (001)/<110>, (110)/<110>, and (110)/<111> pMOSFETs. Uniaxial compressive stress improves the hole mobility monotonically as the stress increases. The hole mobility enhancement saturates at large stress (3 GPa). Under uniaxial longitudinal compres sive stress, the maximum hole mobility enhancement factor is 350'. for (001)/<110> pMOSFETs, 150'. for (110)/<111> pMOSFETs, and 10l'. for (110)/<110> p MOSFETs. At 3 GPa uniaxial stress, (001) and (110) pMOSFETs have comparable hole (11 0)/ SYang, 2003 , izuno, 2003 Mizuno, 2003 Takagi, 19 Takagi, 19 Q2 (001)/<110> 0.3 0.6 Effective Electric Field / MV/cm Figure 32. Hole mobility vs inversion charge density for relaxed silicon. Both measurements and simulation show larger mobility on (110) devices. 350 300 250 200 150 100 0 0) 0 E .o 0.9 .. mobility. Under biaxial tensile stress, the maximum hole mobility enhancement factor is about 1(I0 ' 400 (110)/<111> uniaxial 3 300 (110)/<110> uniaxial o 200 . y^ (001)/<110> uniaxial o 100 _ (001)/<110> biaxial 0 0 1 2 3 Stress / GPa Figure 33. Hole mobility vs stress with inversion charge density 1 x 1013/cm2. The enhancement factor is the highest for (001)/<110> devices and lowest for (110)/<110> devices. At high stress (3 GPa), three uniaxial stress cases have similar hole mobility. Calculated straininduced hole mobility enhancement factor of (001)oriented pMOSFETs is shown in Figure 34 comparing with experimental data [45, 46, 47, 48, 49, 5, 50, 51]. Good agreement is found between the calculated and measured data. In Figure 33 and 34, the channel doping density is set to be 1 x 1017/cm3 in the calculation. The inversion charge density is 1 x 1013/cm2. In contemporary technology, the actual channel doping is up to 1 x 1019/cm3. The mobility enhancement factor is calculated with different channel doping density at inversion charge density of 1 x 1013/cm2 in Figure 35. The enhancement factors are similar for all three doping levels. For simplicity, the rest of the work will use channel doping 1 x 1017/cm3. 0.01 Strain Exx=Eyy Figure 34. ci E (D LU >^ uniaxial biaxial A EJV _^D Calculated strain induced hole mobility enhancement factor vs. experimental data for (001)oriented pMOS. 0L 0 U HL 1 2 3 Uniaxial Compressive Stress / GPa Figure 35. Hole mobility enhancement factor vs uniaxial stress for different channel doping. Uniaxial Compression A Lee 2005 o Rim 2003 Washington 2006 Thompson 2005 A Smith 2005 Biaxial Tension Wang 2004 0.02 Figure 36 compares the hole mobility enhancement factor for different inversion charge density. The figure shows that the enhancement factor decreases as the inversion charge density increases. This is because with more inversion charge, holes are populated to the higher energy levels in the valence band, while the stress only affects the vicinities of F point. This causes the average change of the hole effective mass decrease. More inversion charges increases the electric field in the channel which undermines the strain effect. The detail will be addressed later in this chapter. With the straininduced hole mobility change as we showed here, physical insights of the difference of biaxial and uniaxial stress, and the difference between (001) and (110)oriented pMOSFETs is given in the next sections. Straininduced silicon valence band structure change, subband structure caused by the transverse electric field, and the hole effective mass and scattering rate change with the strain are analyzed. E co 0 0 1 2 Biaxial Tensile Stress / GPa (a) c 3 E :5 0 a) 0^2 0 o 0 0 0 1 2 3 Biaxial Tensile Stress / GPa (b) Figure 36. Calculated strain induced hole mobility enhancement factor vs. stress for (001)oriented pMOS with different inversion charge density. 3.2 Bulk Silicon Valence Band Structure Carrier mobility is determined by the scattering rate and effective mass of the carrier based on Drude's model: P = (31) where r is the carrier momentum relaxation time that is inversely proportional to scattering rate and m* is the carrier conductivity effective mass. In silicon inversion l v. rs, carriers are confined in a potential such that their motion in one direction (perpendicular to the siliconoxide interface) is restricted and the electronic behavior of these carriers is typically twodimensional (2D). The mobility of the 2D hole gas is different from the 3D holes in bulk silicon. But the simplicity of the bulk band structure calculation can give us insights to how the effective masses of the holes change with the stress and help understand how the quantum confinement modifies the subband position and splitting which is important to 2D hole mobility. Therefore, bulk valence band structure is discussed in this section before we move to the Si pMOSFETs. 3.2.1 Dispersion Relation The Ek diagrams of unstressed, 1 GPa biaxial tensile stressed and 1 GPa uniaxial compressive stressed silicon valence band are shown in figure 37. For the unstressed silicon, the Heavyhole (HH) and Lighthole (LH) bands are degenerate at Fpoint. This is 4fold degeneracy taking into account the spin. The Spinorbital Splitoff (SO) band is 44 meV below HH and LH bands. When stress is applied, the degeneracy of HH and LH bands is lifted as shown in figure 37 (b) and (c). These two bands are also referred to as the top and the second band indicating the split energy levels. The band splitting results in the band warping which changes the effective mass of the holes. In the meantime, the splitting causes the repopulation of the holes in the system. When the stress is large and the splitting is high, most holes will locate in the top band based on FermiDirac distribution function as long as the densityofstates (DOS) of the topmost band is not significantly less than that of the next bands. The repopulation of the holes alters the average hole effective mass and phonon scattering change. Figure 37 shows that the stress only affect the band property close to F point. The figures show that away from the zone center, the band structure is almost identical to the unstressed silicon. Figure 38 illustrates that more region around the zone center and more carriers are affected by the stress when the stress increases. Therefore, the strain effect cannot be explained only by the properties at the F point. Instead, the statistics of the whole system should be considered. Figure 38 i. I; that as the stress increases from 500 MPa to 1.5 GPa, the band warping and effective mass at F point change very little. The next subsection will also show this. In the process, more holes are affected by the stress, therefore the average hole behaviors will still change. We showed in Figure 36 that the mobility enhancement factor decreases as the amount of inversion charges increases. This can be understood as follows. For devices with more inversion charges, more holes occupy the higher energy states when the inversion charge density increases. At the same stress, the average change induced by stress is smaller than the cases with fewer inversion charges. 3.2.2 Hole Effective Masses To better understand the stress effect on hole transport, the hole effective masses at Fpoint of top and bottom bands under different stress are shown in figure 39, 310 and 311. Figure 39 shows the < 110 >direction effective masses, figure 310 shows the 2dimensional densityofstate effective masses, and the outofplane < 001 >direction effective masses are illustrated in figure 311. Figure 39, 310 and 311 also ii.;. 1 that with strain, the HH and LH bands are no longer "pure" HH or LH anymore due to strong coupling of the wave functions. The property of each band depends heavily on the <( i I 1 orientation. A single band can be HHlike along one direction, but LHlike along another. In general, if the < i i1 I1 shows compressive strain along one direction, the top band is LHlike along this specific 0.2 0 Wave vector k / 2/a (b) 0.2 0 Wave vector k / 2/a Figure 37. Ek relation for silicon under (a) no stress; (c) 1GPa uniaxial compressive stress. (b) 1GPa biaxial tensile stress; and E 0 S0.05 1.0 GPa 0.1 1.5 GPa Uniaxial Compressive Stress 0.15 0.1 0 0.1 Wave Vector k / A (b) Figure 38. Normalized Ek diagram of the top band under different amount of stress. Larger stress warps more region of the band. The energy at F point for all curves is set to zero only for comparison purpose. 0.3 0 E E () 0.2 0 1 1.5 2 Biaxial Tensile Stress / GPa 0 0.5 1 1.5 2 2.5 3 Uniaxial Compressive Stress / GPa (b) Figure 39. C'I io,,, I direction effective masses for bulk silicon under (a) biaxial tensile stress; and (b) uniaxial compressive stress. STop Band Bottom Band  0.3 0 E E () uJ 0.2 0 0 0.5 1 1.5 2 2.5 3 Uniaxial Compressive Stress / GPa (b) Figure 310. Twodimensional densityofstates effective masses for bulk silicon under (a) biaxial tensile stress; and (b) uniaxial compressive stress. 0.5 1 1.5 2 2.5 Biaxial Tensile Stress / GPa (a) STop Band Bottom Band I Bottom Band 1 1.5 2 Biaxial Tensile Stress / GPa Top Band Bottom Band 0 0.5 1 1.5 2 2.5 Uniaxial Compressive Stress / GPa Figure 311. Outofplane effective masses for bulk silicon under (a) biaxial tensile stress; and (b) uniaxial compressive stress. 0 E ; 0.25 E D 0.2 uJ F Top Band L~L n 0 n 0.3 0 E E () () 4 LuJ 0.2 F direction; if the ( iI I1 experiences tensile strain along a direction, the top band is HHlike along this direction. For example, when inplane biaxial tensile stress is applied to the xy plane of a silicon sample, in xy plane, the sample experiences tensile strain, the top band is HHlike inplane, as shown in Figure 39. Along zdirection (outofplane), the sample shows tensile strain as we shoed in ('!i lpter 1. The top band is LHlike along this direction as shown in Figure 311. This is a very important issue for biaxial tensile stress. As we will show in the following section, the transverse electric field effect offsets the biaxial stress effect and causes the hole mobility degradation at low stress. Similar analysis can be applied to uniaxial compressive stress. Under uniaxial compression, the <110> channel direction experiences compressive strain, therefore the top band is LHlike along the channel. At the same time, the outofplane direction experiences tensile strain, the top band is HHlike outofplane. The spinorbital splitoff (SO) band is also coupled with HH and LH band when strain is present. This band is not as important due to the large energy separation from HH and LH bands and hence very few holes locate in this band. As stated previously that normal MOSFETs have < 110 > direction as the channel direction, conductivity effective mass along this direction affects the hole mobility directly according to Drude's model, a.k.a equation 31. Figure 39 tells us that compared with the biaxial tensile stress, the uniaxial compressive stress induces much smaller top band effective mass which it. 1; greater hole mobility improvement is expected for uniaxial compressive stress. Twodimensional densityofstates effective masses as shown in Figure 310 gives a qualitative estimation of the 2D densityofstates of the holes in each band. The 2D DOS is not directly related to the bulk electronic properties of semiconductors. In the inversion lI.is, large 2D DOS of the ground state subband ,t.i most holes locating in this subband. This reduces intersubband phonon scattering possibility. In the meantime, if the ground state subband has very low conductivity effective mass, the large DOS actually lowers the average hole conductivity effective mass in the system. 2D DOS will be explained in a lot detail in the following section. <001> outofplane effective mass is a important parameter defining the subband energy levels in the inversion 1lv r as will explained in the following section. 3.2.3 Valence Band under Super Low Strain If we compare the hole effective mass of unstrained bulk Si with Figure 39, 310 and 311, a significant discontinuity can be found at low strain (stress < 1 MPa). As we mentioned before, HH band becomes LHlike along <110> direction under uniaxial compression and along outofplane direction under biaxial tension. In the hole mobility calculation, the discontinuity of the hole effective mass is also a confusing question, although it is not important in industries since any single transistor would have much larger strain in the channel in the process. To understand the "d ... il oiily hole effective mass at F point is calculated for super low stress [52] as shown in 312. The figures show that under uniaxial compressive stress, the HH band is alvb, HHlike and the LH band is albvi LHlike outofplane. Along the <110> direction, the effective mass curves cross over at about 3 kPa where HH band becomes LHlike and LH band becomes HHlike. Biaxial tensile stress acts differently. The inplane HH and LH bands are still HHlike and LHlike, respectively. In the outofplane direction, the HH band becomes LHlike and LH band becomes HHlike as the stress is greater than 1 kPa. As the stress increases beyond 100 kPa, the conductivity effective mass does not change at F point. The average effective mass change of the system comes from the fact that more region of the bands is affected by the stress. 3.2.4 Energy Contours Strain altered energy contours are straightforward describing the strain effect on semiconductor band structures. The 25meV energy contours for I i, ihole and lighthole bands are shown in figure 313 for unstressed bulk silicon. The anisotropic nature of the Si valence band is clearly shown. Using the simple parabolic approximation E = *, where 0.1' 0.01 0.1 1 10 Biaxial Tensile Stress / kPa (a) 0.1 ...... ...... .. .. ...... 0.01 0.1 1 10 100 Uniaxial Compressive Stress / kPa (b) Figure 312. Hole effective mass change under very small stress. The change in this stress region explains the "dh i...i fliii y of the hole effective mass between the relaxed and highly stressed Si. E stands for energy and m* is the effective mass, the thinner the contour is along one direction, the smaller the effective mass is along that direction. The contours show that the HH band has very large effective mass along <110> direction. When stress is applied, the band structure is distorted as shown in figure 314 for 1GPa biaxial tensile stress and figure 315 for 1GPa uniaxial compressive stress. The contours, as well as Ek relation curves, show strain induces lower conductivity effective mass along <110> direction for the top band. The effective masses for unstressed bulk silicon are 0.59mo for HH and 0.151,,, for LH band where mo is the free electron mass. Those two numbers become 0.28mo/0.22mo for 1GPa biaxial tensile stress and 0.11mo/0.2mo for 1GPa uniaxial compressive stress. The bottom band effective masses do not show enhancement compared with the LH band mass of the unstressed silicon. Again, for bulk electronic transport, uniaxial compressive stress should enhance the hole mobility as stress increases, since the top band is LHlike along <110> direction. Biaxial tensile stress does not have the mass advantage since the top band is HHlike inplane. The possible mobility enhancement comes only from band splitting causing phonon scattering rate reduction. For holes in the inversion lizr's, the statement is still true as we will show next. 0.1 0.1 O 0  0.1 01 01 0 0.1 .0 0.10 01 01 01 0 0.1 0.1 (a) (b) Figure 313. The 25meV energy contours for unstressed Si: (a) Heavyhole; (b) Lighthole. 0.1 0 0.1 0.1 01 0 0.1 0. 1 0 0.1 0.1 Figure 314. The 25meV energy contours for biaxial tensile stressed Si: (a) Top band; (b) Bottom band. 0.1 0 0.1 0 0.1 0.1 01 0 0.1 0.1 Figure 315. The 25meV energy contours for uniaxially compressive stressed Si: (a) Top band; (b) Bottom band. 3.3 Strain Effects on Silicon Inversion Layers As we mentioned before, in the silicon inversion 1. ri, the carriers are confined in the potential well formed by the Si/Si02 interface and the valence band edge of the silicon. The motion of the holes is continuous in the horizontal xy plane, but quantized in zdirection [29]. The quantum confinement leaves a set of two dimensional subbands in kspace (kx, ky). The subband structures are affected by both the stress and the transverse electric field. In pMOSFETs, the topmost two subbands (4 counting the spin), the ground state and the first excited state subbands, contain most of the holes and analyzing those two subbands gives us qualitative understanding of the hole transport properties. Therefore, those two subbands will be focused in the following discussions to explain the strain effects, although up to 12 subbands are actually taken into account in the hole mobility calculation. In this section, we shall explain why the biaxial tensile stress and uniaxial com pressive stress affect the subband structure and the hole mobility differently under the transverse electric field. The difference of (001) and (110)oriented devices under uniaxial stress will also be studied. 3.3.1 Quantum Confinement and Subband Splitting Carriers are confined in a potential well very close to the silicon surface in the inversion l1vr of a MOSFET. The well is formed by the oxide barrier and the silicon conduction band or valence band depending on electrons or holes as the carriers [1]. Taking holes (pMOS) as an example, the conduction and valence bands bend up (bend down for nMOS) towards the surface due to the applied negative gate bias at strong inversion region. This means hole motion in zdirection that is perpendicular to the silicon surface is restricted and thus is quantized, leaving only a 2dimensional momentum or k vector which characterizes motion in a plane normal to the confining potential. Therefore, the inversion l1vr holes (or electrons) must be treated quantum mechanically as 2 dimensional (2D). Figure 316 illustrates the quantum well and quantized subbands [51], qualitatively. The band bending at the surface can be characterized as potential V(z). Accurate modeling of V(z) requires numerically solving coupled Schrodinger's and Poisson's Equations selfconsistently. This is one of the main efforts of this work. The details of the method can be found in Chapter 2. SiO2/Si STop of the well Hol distribution of the ground state /D 2/3 E. E(j=0) E [2hqE, + 3 ) 4 a40 4 1 X LU S\ /E(j=1) 60 Valence band edge Hole energy level shift due to quantization Figure 316. Quantum well and subbands energy levels under transverse electric field. The complex calculation procedure somehow prevents people understanding the physics behind stress and electric field effect. To give the physical insights into the relation between those two effects, triangular potential approximation is utilized to estimate the subband energy levels. The triangular potential approximation states that the band bending solely depends on depletion charges under subthreshold condition when the mobile charge density is negligible. The potential V(z) is replaced by eEeffz, where Eeff is the effective electric field in the depletion l1 .r. Triangular potential approximation is not a good approximation calculating accurate subband energies for strong inversion region, but the physics can still be explained qualitatively. Solving Schrodinger's equation, [H(k, z) + V(z)lPk(z) = E(k)fk(z) (32) one will get the subband energies. The energy of subband i can be expressed as [1], Ei= 0i + i = 0, t, 2,... (33) 4 '(u'2' 4 where h is plank constant, e is the electron charge, and m* is the outofplane hole effective mass, also known as confinement effective mass. This effective field is defined as the average electric field perpendicular to the Si SiO2 interface experienced by the carriers in the channel. It can be expressed in terms of the depletion and inversion charge densities: Es = (Q 11l + T Qi.) (34) MOSFETs where the effective field is over 0.5MV/cm throughout this work. This equation for the effective electric field is an empirical equation. It may not be accurate to model the carrier transport for devices with surface orientation other than (001) or other device structures such as silicononinsulator (SOI) devices or doublegated (DG) devices. Equation 33 shows that the subband energy of holes is inversely proportional to the outofplane effective mass of the holes. With the transverse electric field, the subband that is HHlike outofplane is shifted up (lower energy for holes) and the subband that is LHlike outofplane is shifted down (higher energy). Figure 311 and 39 show that in (001)oriented devices, biaxial tensile strain shifts the outofplane LHlike band up which is the inplane HHlike band. The electric field effect offsets the biaxial tensile strain effect. At low strain, this can be understood as follows. When the biaxial strain is very small, i.e. 10 MPa, and the subband energy levels is dominated by the electric field effect, the ground state subband is HHlike outofplane and LHlike along the channel. As we increase the strain and keep the electric field constant, the energy splitting between the ground state and the first excited state will decrease and at some stress level, the two subbands will cross each other. The process is showed in Figure 317 schematically. If the strain continues in, I iii. the strain becomes dominant determining the subband energies and structures. During the process, the average hole effective mass increases since holes transfer from the inplane LHlike subband to the HHlike subband. This increasing effective mass is responsible to the initial mobility degradation under biaxial tensile strain which is observed both in experiments and our calculation. The mobility enhancement shown in Figure 33 comes from the suppressed intersubband phonon scattering rate due to the high subband splitting as will be shown later. Under uniaxial compressive strain, the top band is HHlike outofplane and LHlike along the channel, which i' . I the strain and the electric field effects are additive. Based on the similar an !1, i both the uniaxial compressive strain and the quantum confinement effects shift up the outofplane HHlike band which is LHlike along the channel. Therefore the ground state subband is alvi  LHlike along the channel and the average effective mass decreases monotonically as the stress increases. The calculated subband splitting between the ground state and the first excited state is showed in Figure 318 for different stress and surface orientation. For biaxial stress, the splitting is zero at 500 MPa which sir.. I the crossingover of the HHlike and LHlike subbands. For all uniaxial stress cases, the subband splitting increases with the stress. Like (001)/<110> devices, the ground state subband of both (110)/<110> and (110)/<111> devices is HHlike outofplane and LHlike along the channel under uniaxial compressive stress. The difference is tat the outofplane effective mass of the ground state subband in (110)oriented devices is much larger 319 than that of the Low Vertical Field High Vertical Field Uniaxial Biaxial Si02 ...... E0 Second \ EtO E.. Second Esecond EV Ev Band splitting due to strain Splitting (increases) / (decreases) under confinement Figure 317. Schematic plot of strain effect on subband splitting, the field effect is additive to uniaxial compression and subtractive to biaxial tension. (001)oriented devices, which results in much larger subband splitting at low stress. The splitting for (110)oriented devices does not change as much as (001)oriented devices, and the splitting saturates much faster with the stress compared with (001)devices. This is due to the strong quantum confinement underminging the strain effect, which is not observed in (001)oriented devices. In general, inplane compressive stress is desirable for pMOS, since it causes the silicon top valence band to be HHlike outofplane and LHlike inplane, which is additive to the electric field effect. <110> uniaxial compressive stress is the best choice because it gives very small conductivity effective mass. 3.3.2 Confinement of (110) Si Figure 318 shows the difference of the subband splitting between (001) and (110) oriented devices. Figure 33 shows that the maximum enhancement factor at 3 GPa stress for (001)oriented devices under uniaxial stress is much larger than (110)oriented devices. 120 a) (001)/<110> uniaxial  E 100 60 0 80 E 80  ~/ 'A " cl) 60 0 "(110)/<110> uniaxial 40 0 ( (110)/<111> uniaxial U0 20 0 S (001)/<110> biaxial 0 I  0 1 2 3 Stress / GPa Figure 318. Subband splitting between the top two subbands under different stress. To explain the physics, the bulk and confined 2D energy contours of the ground state subband for (001) and (110)oriented Si are shown in Figure 320, 321, 322, and 323. The figures show that for (001)/<110> devices, the ground state hole effective mass decreases with uniaxial compressive stress (LHlike) along the channel), but the reduction is not as notable under biaxial stress. Compared with the bulk Si energy contours, the electric field does not modify the subband structure in kx ky plane for (001)oriented devices (it does affect the subband splitting though). The conductivity effective masses along the channel direction are almost identical to those of bulk counterparts. The confinement effect is much more significant on (110)oriented devices. The confined effective mass of the ground state subband is very low along <110> and <111> direction even for unstressed Si, which explains why unstressed (110)oriented devices have superior hole mobility over (001)oriented devices (the confinement effect is also significant in (111) and (112) pMOSFETs (31), though the hole effective mass is larger than that in (110) 2.51 0 I lup DdlIU E S1.5 0) tU 0.5 Bottom Band 0 0.5 1 1.5 2 2.5 3 Uniaxial Compressive Stress / GPa Figure 319. Outofplane effective masses for (110) surface oriented bulk silicon under uniaxial compressive stress. 66 pMOSFETs). Furthermore, for (110)/<110> devices, stress shows very little effects on the confined contours and the effective masses hardly change. For (110)/<111> devices, the 2D contours are warped much more significantly and the effective mass decreases more than (110)/<110> devices with uniaxial stress. This difference explains why the hole mobility of (110)/<110> devices and (110)/<111> devices respond differently under uniaxial stress. 3.3.3 Straininduced Hole Repopulation Strain induced hole population in the ground state subband is shown in Figure 324. For (001) devices under uniaxial stress, the initial decrease of the hole population is due to the decreased DOS near F point. As stress increases, the increasing subband splitting causes the hole population increasing and the average conductivity effective mass keeps decreasing since the ground state subband is LHlike along the channel under uniaxial compressvie stress. For biaxial stress, the decrease of the hole population at low stress again reflects the initial confinement effect lifting the inplane LHlike subband and reducing the subband splitting( 318). This inplane LHlike subband is shifted down as the stress increases and the inplane HHlike subband is shifted up. After the crossingover of the two subbands, the ground state subband population starts increasing with the stress. For (110)oriented devices under uniaxial compressive stress, the ground state hole population increases with the stress, but it saturates at much lower stress compared with (001)oriented devices which is consistent with the subband splitting change. The hole population of (110)oriented devices is ah,i lower than (001)oriented devices under uniaxial compressive stress, although the subband splitting is much larger. The subband splitting and hole population difference of (001) and (110)oriented devices can be explained by the ground state subband 2D DOS as shown in Figure 325. DOS difference also ii.. ; the different straininduced mobility change. Both figures show that (001) oriented devices have larger DOS than (110)oriented devices. For (001)/<110> devices under uniaxial compressive stress, although the first excited state subband is HHlike 0.15 <110> 1GPa Uniaxial Compression 0.15 0 0.15 k x (b) 0.15 <110> 1 GPa Biaxial Tension 0.15 0 0.15 k x (c) Figure 320. The 2D energy contours (25, 50, 75, and 100 meV) for bulk (001)Si. Uniaxial compressive stress changes hole effective mass more significantly than biaxial tensile stress. 0.15 0.15<110> 1 GPa Uniaxial Compression 0.15 k0 0.15 k (b) 0.15 Figure 321. Confined 2D energy contours (25, 50, 75, and 100 meV) for (001)Si. The contours are identical to the bulk counterparts. 0.15 1GPa Uniaxial Compression 0 0.15 0 0.15 k x (b) 0.15 1 GPa Uniaxial Compression 111> 0 0.15 0 0.15 k x (c) Figure 322. The 2D energy contours (25, 50, 75, and 100 meV) for bulk (110)Si under (a) no stress; (b) uniaxial stress along ( 110); and (c) uniaxial stress along (111). 70 Unstressed Si <111> 0W 0.15 0 0.15 k (a) 0.15 1 GPa Uniaxial Compression <110> 0 0.15 0 0.15 k (b) 0.15 1 GPa Uniaxial Compression <111> 0 0.15 k0 0.15 x (c) Figure 323. Confined 2D energy contours (25, 50, 75, and 100 meV) for (110)Si. The confined contours are totally different from their bulk counterparts which s i: 1 significant confinement effect. along <110> channel, the subband splitting and the high 2D DOS of the ground state subband (compared with (110)oriented devices) assures most holes populating to the ground state subband as shown in Figure 324. The decreasing DOS in Figure 325 (b) for both biaxial and uniaxial stress of (001)oriented devices also ii. i that the phonon scattering rate decreases with te stress. The DOS of (110)oriented devices does not change with the stress especially at high stress region (13 GPa) which ii.' i the phonon scattering rate should not change much. 1.0 (001)/<110> uniaxial ~0.8 o O 0.6 0 o / (110)/<110> uniaxial 0.4 C (110)/<111> uniaxial c) (001)/<110> biaxial 0.2 0 1 2 3 Stress / GPa Figure 324. Ground state subband hole population under different stress. As we mentioned in the previous section, the stress does not warp the band structure evenly in the whole kspace. This can also be seen from the DOS change in Figure 325 (b) where the DOS at Energy E = 52meV (2kT where T = 300k) is shown. Taking uniaxial stress on (001) devices as an example, when the stress is low, only a small region close to F point is affected and becomes LHlike along <110> direction (still HHlike along transverse and outofplane direction), while the rest of the band with higher energy (including the energy level showed here) does not respond to the stress yet. As the stress increases, more region is affected and becomes LHlike along the channel. The initial constant DOS at low stress in Figure 325 (b) ,ii. 1 when the stress is lower than about 500 MPa, the stress is too small to warp the band at this energy level. When the stress increases, DOS starts decreasing because the stress starts warping the band at this energy and the <110> direction becomes LHlike. The DOS curve becomes flat again when the stress effect saturates for this energy level. For (001) pMOSFETs under biaxial stress, Figure 325 (b) does not show a DOS peak like Figure 325 (a) which means the position crossingover of the top two subbands only happens close to F point, and the HHlike band is alv,v on top out of that region. For (110) pMOSFETs, the DOS is constant with the stress, which is due to the strong quantum confinement effect. To discover the strain effect, 2D DOS at 4kT (102meV at T 300K) is shown in Figure 326. For (001) pMOSFETs, the curves have the similar trend compared with the DOS curves at 2kT. The only difference is that the DOS starts to decrease at higher stress. For (110) pMOSFETs, DOS decreases at low stress and the change is not as significantly as (001) pMOSFETs. Figure 325 and 326 i.. 1 that the strain in (110) pMOSFETs only warps the subband at high energy region due to the strong quantum confinement. The strain induced mobility change should be less than (001) pMOSFETs, since smaller portion of holes locate at high energy compared with F point. 3.3.4 Scattering Rate Besides effective mass change, hole mobility is inversely proportional to the scattering rate. Phonon scattering and surface roughness scattering are focused in this work, since they are the predominant scattering mechanisms when the effective electric field in the channel is over 0.5MV/cm [3, 1]. Figure 327 shows that for (001)oriented devices, the phonon scattering rate does not change much when the stress is lower than 500 MPa. This indicates that at low stress, the hole mobility enhancement (or degradation) is almost purely caused by the effective mass 3x1014 2x1014 1x1014 0.0 S6x1014 0 O a, o u 4xl 014 (r) a, Cz > 2x1014 o 70 C\ 0.0 1 2 Stress / GPa Stress / GPa (b) Figure 325. Stress effect on the 2 dimensional densityofstates of the ground state subband at (a) the top of the subband (E=0); (b) E=2kT. (110)devices have much smaller 2D DOS which limits the ground state hole population (larger intersubband phonon scattering). Another observation is that DOS of (110)devices does not change with stress. (001)/<110> uniaxial (1 0)/<1 uniaxial (110)/<111> uniaxial  ftr m m (110)/<110> uniaxial I 6x1014 E o (001)/<110> uniaxial a( 3 4x1014 \ (001)/<110> biaxial 2x1014  a) S((11 0)/<111> uniaxial > ^ 2x1014  o4 0.0 0 1 2 3 Stress / GPa Figure 326. Two dimensional densityofstates at E=4kT. change. When the stress increases from 500 MPa to 3 GPa, the phonon scattering rate decreases by 5('. for both acoustic phonon and optical phonon scattering, the phonon scattering rate reduction overweighs the effective mass change to become the main driving force to improve the hole mobility in this stress range, especially for biaxial stress. Unlike (001)oriented devices, phonon scattering rate changes more at low stress region rather than high stress region for (O110)oriented devices under uniaxial compressive stress. This is consistent with Figure 318 and 324 that the subband splitting and the ground state subband hole population only increase at low stress. The constant phonon scattering rate at high stress explains why the hole mobility of (110)/<111> devices at 3 GPa is not significantly larger than (001)/<110> or (110)/<110> devices, regardless of the largest piezoresistance coefficient at low stress. Figure 328 shows that the surface roughness scattering rate increases with stress for (001)oriented devices. This is due to the increasing hole population in the ground 4x1012 (001)/<110> uniaxial c 3x1012 (001)/<110> biaxial 0 ) 0 S 110 (110)/<110>uniaxial 0 C/) (110)/<111> uniaxial 0  0 1 2 3 Stress / GPa (a) 1x1013 c/) 000< OD (001)/<110> uniaxial ( 6x1012 %mo 0 24x1012 "_ (110)/<110> uniaxial O (110)/<111> uniaxial 0 0 1 2 3 Stress / GPa (b) Figure 327. Strain effect on (a) acoustic phonon, and (b) optical phonon scattering rate. Optical phonon scattering is the dominant scattering mechanism improving the mobility. Phonon scattering rate changes mainly in high stress region for (001)devices and low stress region for (HO)devices. 4x1012. U)a (D U) 3x1 a) ~cc CzC Cc 2x1 a)r CZ a) C/) 0 C/) 012 012[ 012 Stress / GPa Figure 328. Strain effect on surface roughness scattering rate of holes in the inversion lwvr. As stress increases, the scattering rate increases for (001)devices due to the increasing occupation in the ground state subband which brings the centroids of the holes closer to the Si/SiO2 interface. state subband which brings the centroids of the holes closer to the Si/SiO2 interface. The magnitude of the surface roughness scattering rate is much smaller than the phonon scattering rate and therefore the increasing surface roughness scattering does not affect the hole mobility as much. The surface roughness scattering rate for (110)oriented devices does not change much with the stress, which is consistent with the fact that the ground state subband hole population is relatively constant with the stress. 3.3.5 Mass and Scattering Rate Contribution Figure 329 illustrates the stressinduced hole mobility enhancement contribution from hole effective mass and phonon scattering rate reduction, respectively. Under uniaxial compression, (001)/(110) pMOSFETs have the largest mobility improvement from both aspects. Compared with (110)/(111) pMOSFETs, (110)/(110) pMOSFETs (001)/<110> uniaxial (001)/<110> biaxial (110)/<111> uniaxial (110)/<110> uniaxial have smaller effective mass gain but larger phonon scattering rate gain. For (001) p MOSFETs under biaxial tension, the mobility enhancement is purely from the suppression of the phonon scattering rate. 3.4 Summary From the results of the selfconsistent calculation of Schrodinger's Equation and Poisson's Equation, we notice that the subband splitting between the ground state and the first excited state decreases as the biaxial stress increases when the stress is smaller than 600MPa, but the splitting increases with uniaxial compressive stress. The difference is due to the subtractive or additive nature between the quantum confinement effect and the stress effect which causes the increase or decrease of the average effective mass of the holes in the inversion l~ivr. As the stress keeps in' i i i: the stress effect outweighs the confinement effect for both stresses and the subband splitting increases so much that the intersubband phonon scattering rate reduces and hence the hole mobility increases. Uniaxial stress on (110) devices improves the hole mobility too. But the improvement is not as much as (001)oriented devices. This is due to the strong confinement effect on (110)oriented devices undermining the stress effect. When no stress is present, the confinement effect swaps the subband structure and reduces the hole effective mass around the Fpoint. This effective mass advantage over the (001)oriented unstressed pMOS causes that the hole mobility is much larger. When the stress is applied, the effective mass change is not as significant, neither does the subband splitting. Therefore, the mobility enhancement with the stress is not supposed to be as much as the (001)oriented pMOS. It is also noticed that the subband splitting saturates when the stress reaches 2 or 3 GPa, so does the effective mass. This leads to the saturation of the stress enhanced hole mobility. 1.5 a) E a) o LU > 0 Ill 3o 1.0 0.5 0.0 0.5 1.5 ci a) E i ai O 1.0 0.5 0.0 0.5 1 2 Stress / GPa (a) 0 1 2 Stress / GPa (b) Figure 329. Hole mobility gain contribution from (a) effective mass reduction; and (b) phonon scattering rate suppression for pMOSFETs under biaxial and uniaxial stress. 79 CHAPTER 4 STRAIN EFFECTS ON NONCLASSICAL DEVICES As the silicon C\ IOS technology is scaled to sub100 nm, even sub50 nm scale, further simple scaling of the classical bulk devices is limited by the short channel effects (SCEs) and does not bring performance improvement. The ultrathin body (UTB) silicon oninsulator (SOI) transistor architecture [54, 55, 56, 57, 58] has been considered possible replacement for the bulk MOSFETs. The basic idea of SOI Ci\OS fabrication [54, 56] is to build traditional transistor structure on a very thin lv, r of crystalline Si which is separated from the substrate by a thick buried oxide li r (BOX). Compared with the bulk C'\ OS, UTB SOI technology brings benefits such as reduced junction capacitance which increases switching speed, no body effect since the body potential is not tied to the ground or Vdd but can rise to the same potential as the source, low subsurface leakage current, and et al.. SOI MOSFETs are often distinguished as partially depleted (PD) transistors that the Si thickness is larger than the maximum depletion width and fullydepleted (FD) SOI transistors that the Si is thinner than the maximum depletion width. FD SOI technology [1] add additional performance enhancements over PD SOI including low vertical electric field in the channel (higher mobility) due to the fact that most FDSOI transistors have undoped channel, further reduction of the junction capacitance, and better scalability. Although FD SOI technology has better scalability than classical device structures, it is still difficult to scale the device to sub20 nm scale. In shortchannel FD SOI MOSFETs, the thick BOX acts like a wide gate depletion region and is vulnerable to sourcedrain field penetration and results in severe shortchannel effects [1, 59, 60]. To better control the channel, doublegate (DG) transistors, especially FinFETs, have been investigated theoretically and experimentally [61, 62, 63, 64]. DGMOSFETs have better scalability than singlegate (SG) SOI transistors and are considered promising candidates for sub20nm technologies [62]. Overall, SOI SG devices and DG devices have been shown to increase circuit performance and reduce active power consumption. These nonclassical device structures are the future of the C'\ OS technology. With the research of strain effects on bulk silicon devices, strained silicon UTB FETs draw the attention of researchers as such devices may combine the strain induced transport property enhancements with their scaling advantages. Stress enhanced hole mobility in SOIdevices has been investigated experimentally in recent years [65, 66, 67, 68, 69, 70]. In 2003, Rim [45] reported the biaxial tensile stressed SOIpMOS hole mobility with dependence of strain and inversion charge density. Zhang [71] showed the hole mobility enhancement under low uniaxial longitudinal and transverse stress. (10O)surface SOI devices with strain effects are also investigated [72]. Those results are consistent with the measured and calculated results for bulk Si devices that are showed in the last chapter. Strain research on double gate devices lags that on bulk devices and even single gate SOI devices partly due to the difficulty employing stress to the channel without damaging the properties of the channel and Si/SiO2 interfaces. Due to the better scalability and higher hole mobility, more attention has been drawn to (110)oriented FinFETs over planar DG FETs. Collaert [73] investigated strain effect on electron and hole mobility enhancement on FinFETs. Shin [74] and his colleagues investigated multiple stress effects on ptype FinFETs using wafer bending method. Verheyen [75] reported "'. drive current improvement of ptype multiple gate FET devices with germanium doped source and drain. Although hole mobility enhancement is observed in those experiments, the actual stress in the fin is unknown. Theoretically, strain effects on FinFETs are much less understood. With the knowledge of stress enhancing hole mobility in bulk devices, it's important to understand how that stress alters the hole mobility in FinFETs. Uniaxial compressive stress will be focused in this work since it provides the greatest hole mobility improvement than other stress on bulk devices. Another reason is that for (110)oriented FinFETs, the stress in the channel is normally uniaxial longitudinal stress even if SiGe S100 0)~m ^ L    UK p=6x1012/cm2 E 80 0 60 4 / p=1.2x1013/cm2 40 I I I 0 5 10 15 20 SOI Thickness / nm Figure 41. Hole mobility vs SOI thickness for single gate SOI pMOS. The mobility decreases with the thickness due to structural confinement. SOI thickness decreases. This does not bring smaller intersubband scattering rate. The rapidly increasing form factor actually keeps the scattering rate increasing. Another issue related to the silicon thickness is subband modulation. Both measure ments and MonteCarlo simulation show that the phononlimited mobility increases at very thin SOI thickness [67, 69, 77]. This issue only happens to nMOS. Uchida's mea surements show there is no such mobility peak in ptype UTB SOI FETs [67], which is consistent with our calculation. 4.1.2 Strainenhanced Hole Mobility of SOI SGpMOS Rim [45] reported that biaxial tensile strain improves (or degrades) the hole mobility as same as it does to the bulk devices, which is supported by our calculation. Uniaxial compressive strain is focused in this chapter due to its much larger mobility enhancement factor than biaxial tensile strain. Figure 42 shows the singlegate SOI pMOS hole mobility vs uniaxial compressive stress comparing with bulk Si devices. Calculated curves for SOI thickness of 3 nm and 5 nm are shown in the figure. Simulation results for thicker SOI are not included because they almost overlap with the bulk device curve. 400 o Conventional Si (001)/<110> 300 \ / = E 1 tsoi = 5 nm O 200 2 100 .0 Pinv= 1xi 013/cm2 0 0 0 1 2 3 Uniaxial Compressive Stress / GPa Figure 42. Hole mobility for single gate SOI pMOS vs uniaxial stress at charge density p 1 x 1013/cm2. The hole mobility enhancement factor for SOI pMOS with SOI thickness of 3 nm is shown in Figure 43. The enhancement factor for SOI devices is similar to the case of bulk devices at low stress, but larger than bulk FETs at high stress. As we mentioned in C'! lpter 3 that for (001)oriented Si pMOS, the mobility is enhanced mainly due to the decreased hole effective mass at low stress. At high stress, phonon scattering rate reduction due to the increasing subband splitting is the main driving force to improve the mobility. The overlapping curves at low stress sI: 1 the effective mass gain should be similar for both cases. Calculation shows that the structure of each subband in SOI pMOS is as same as the bulk counterpart which also ,.':. i the effective mass change for both cases should be the same. Figure 44 shows the subband splitting of the ground state and the first excited state subbands for SOI and bulk FETs. The larger splitting for SOI devices I, :.  ; more intersubband phonon scattering rate change, which is responsible for the larger mobility enhancement. 5 tso = 3 nm E t0so = 5 nm : S 3 y00 o / .' Traditional Si (001)/<110 0 0 Pinv= 1xx1013/cm2 0 1 2 3 Uniaxial Compressive Stress / GPa Figure 43. Hole mobility enhancement factor of UTB SOI SG devices vs uniaxial compressive stress at charge density p 1 x 1013/cm2. Uchida reported that as the SOI thickness reduces down to 23 nm, the fluctuation of the Si/SiO2 interface is the main factor to limit the carrier mobility [67, 69]. Therefore, the large hole mobility enhancement as shown in Figure 43 cannot be obtained in real devices. A new surface roughness model is needed to solve this problem. In our discussion of the doublegate devices including FinFETs later in this chapter, the smallest Si thickness we consider would be 5 nm. 4.2 Doublegate pMOSFETs Due to the overwhelming research effort on FinFETs, FinFETs are focused in this section. For (001)oriented DG pMOS, only symmetricaldoublegate MOSFETs are considered here. Unlike single gate devices, double gate MOSFETs have two surface 140 0 120 Stsot = 3 nm 0 100 \ / . 80 /) 60 c 40 / Traditional Si (001)/<110> = 20 20 U0nx Pinv = 1x1013/cm2 0 1 2 3 Uniaxial Compressive Stress / GPa Figure 44. Subband splitting UTB SOI SG devices vs uniaxial compressive stress at charge density p 1 x 1013/cm2. channels. The wave functions of the two channels interact and one energy level splits to two according to Pauli's exclusive principle (subband modulation). Schematic comparison of the subband splitting for bulk and double gate devices is showed in Figure 45. The subband splitting for SDG MOSFETs and FinFETs is very small when the Si thickness is over 5 nm (5 meV when tsi = 5nm, 3 meV when tsi = 15nm). If the Si thickness is below 5 nm, the strong interaction of the two surface channel causes the subband splitting increasing drastically (i.e. 18 meV for tsi = 3nm). EE E V ... top Second E Ethird SG FET SDG FET Figure 45. Comparison of the subband splitting of double gate and single gate MOSFETs. 4.2.1 (001) SDG pMOS The hole mobility and the mobility enhancement factor for SDG pMOSFETs are shown in Figure 46 and 47, respectively. Double gate devices have higher mobility than traditional bulk transistors mainly due to the undoped body, much smaller channel effective electric field and bulk inversion [1]. Figure 46 shows that the hole mobility decreases as the silicon thickness decreases. The reason is as same as single gate SOI devices and has been explained in last section. The mobility enhancement factor of SDG pMOS in Figure 47 is very similar to the bulk case, but the mechanisms are a little different. The first excited subband (very close to the ground state) provides smaller average effective mass to help the mobility S500 ) 3tsi = 10 nm > 400 N tsi = 5 nm S3000 S200 Traditional Si (001)/<110> 0 100 I / 1 0 Pin, = x1 013/cm2 0 1 2 3 Uniaxial Compressive Stress / GPa Figure 46. Hole mobility of SDG devices under uniaxial compressive stress at charge density p 1 x 1013/c 2. 88 enhancement, but at the same time it also brings larger intersubband phonon scattering rate. Those two factors balance each other. Therefore the SDG devices show a little larger mobility enhancement at low stress, but a little lower enhancement at high stress. The difference is slim and the average effect is very similar to singlegate devices. 5 P 1Pin xlx 013/cm2 cI 4 S=L ts = 5 nm . 1 0 > CZ SG Si (001)/<110> T5 1 o 0 1 2 3 Uniaxial Compressive Stress / GPa Figure 47. Hole mobility enhancement factor of SDG MOSFETs vs uniaxial compressive stress at charge density p 1 x 1013/cm2. 4.2.2 Strain Effect on FinFETs The total hole mobility of the FinFET with respect to the stress is shown in Figure 4 8, comparing with the singlegate (110) and (001)oriented ptype devices at the inversion charge density of 1 x 1013/cm22. In the calculation of the singlegate devices, the doping density is taken to be 1 x 1017/cm3. This is a low doping density compared with the contemporary C'\ IOS technology. Even so, the FinFET shows significantly greater mobility than the bulk devices. If larger doping density is applied, the mobility advantage of the FinFET would be even larger. When 3 GPa uniaxial compressive stress is applied to a FinFET, about 3:1i i'. enhancement of the mobility is expected, compared to only 2111 I' enhancement for a bulk (110)oriented transistor as shown in Figure 49. Even though the (001)oriented pMOS shows greater relative enhancement (over !1111'.), the absolute mobility is still lower than that of the FinFET due to its low mobility with no stress. 700 Pinv= 1x1013/cm 2 ( 600 ) 600 FinFET 500 E 400 Bulk (001) FET 300 S200 ,k 0 F 0 100 Bulk (110) FET I 0 1 1 0 1 2 3 Uniaxial Compressive Stress / GPa Figure 48. Hole mobility of FinFETs under uniaxial stress compared with bulk (10O)oriented devices at charge density p 1 x 1013/cm2. We mentioned in the last chapter that 2D DOS of the topmost subband in (110) oriented devices is very small near F point no matter if the stress is present and the stress does not warp the subbands much. Therefore the average effective mass does not change as much as standard (001)oriented devices when uniaxial stress is present. Regarding FinFETs, strong subband modulation is observed where the topmost 2 subbands are close to each other (like (001) SDG pMOSFETs) as we illustrated in Figure 45. This extra subband is so close to the ground state subband and it acts like increasing the DOS of the ground state subband. More importantly, the band bending at the Si/SiO2 interface is E 3  S<] / FinFETs (110)/<110> .co S/ w... w SG (110)/<110> 0 0 1 2 3 Uniaxial Compressive Stress / GPa Figure 49. Hole mobility enhancement factor of FinFETs under uniaxial compressive stress at charge density p 1 x 1013/cm2. 91 very small in FinFETs when gate bias is applied and the ground state subband is much closer to the Fermilevel than that of the singlegate FETs (in both cases, the ground state subbands are on top of Fermilevel). With the same total amount of holes in both systems, the lower ground state subband level keeps more holes close to F point that can be affected by the strain and the electric field. Although the topmost two subbands in FinFETs are close to each other, the form factors are extremely small (only about 1/6 of single gate case) between these two subbands. This results in smaller phonon scattering than single gate devices, and the change of the scattering rate with stress is larger than SG pMOSFETs. 2 0 4 0 4 FinFETs (110)/<11 O0> S 0 1 3 (110 \ 11 1  0 1 2 3 Uniaxial Stress / GPa Figure 410. Hole mobility gain contribution from effective mass and phonon scattering suppression under uniaxial compression for (110)/(110) FinFETs compared with SG (110)/(110) pMOSFETs at charge density p 1 x 1013/c2. To understand the hole mobility difference between FinFETs and traditional single gate (110)/(110), the hole mobility gain contribution from effective mass change and phonon scattering rate change is shown in Figure 410. It shows that phonon scattering rate change is the main factor to improve the hole mobility for both FinFETs and bulk pMOSFETs. Both the effective mass and phonon scattering rate for FinFETs change are larger than single gate (110)/(110) devices, which leads to higher mobility enhancement. Smaller surface roughness scattering rate due to small electric field in FinFETs also contributes to the higher mobility enhancement. The calculation also shows the enhancement is not a strong function of the silicon thickness of the fin as the fin thickness is above 5 nm. If the fin is thinner than that, more subband splitting is observed (about 18 meV for 3 nm of the fin thickness). Since the splitting is still not too large, our analysis about the effective mass I ,i, true. Surface roughness scattering rate is much larger and the hole mobility enhancement would not be as large as that for thicker fin. An accurate surface roughness model for such devices would be necessary to evaluate the mobility change numerically. 4.3 Summary Strain effects on SOI MOSFETs, including planar symmetrical DG devices and (110)oriented FinFETs are discussed in this chapter. For single gate SOI pMOS, the mobility decreases as the SOI l, r thickness decreases due to increasing phonon and surface roughness scattering rate. The hole mobility enhancement under stress is similar to that of bulk silicon devices unless when the SOI thickness is so small that the surface roughness scattering outdominates the phonon scattering. For double gate devices, subband splitting is drastically smaller than the bulk devices due to the interaction of the quantum states of the two surface channels. For (001) oriented planar symmetrical DG pMOS, the structure of each subband is still identical to the counterpart in the bulk devices. The extra effective mass gain is canceled by the inter subband phonon scattering and the total hole mobility enhancement is similar to the bulk FETs at low stress. But when the stress is over 2 GPa, the effective mass gain is saturate. The mobility gain is less than that of bulk FETs due to the larger intersubband optical phonon scattering. This effect is not that significant due to the smaller form factors. As of the FinFETs, the extra subband provides much more effective mass gain while the phonon scattering rate is similar to the bulk devices. This causes that the mobility enhancement is higher than bulk FETs. Although the mobility enhancement factor for FinFETs (3 times) is not as large as (001)oriented bulk pMOS (>4 times), FinFETs still have much higher mobility due to its high initial mobility without stress. Together with the better scalability, FinFETs will be strong candidate for C'\ OS technology under 20 nm scale. CHAPTER 5 STRAIN EFFECTS ON GERMANIUM PMOSFETS As shortchanneleffects (SCEs) prevent the simple scaling of traditional Si MOSFETs achieving historical performance improvement, new material, as well as feature enhanced technology (strain technology), attract attention of the researchers. Germanium is one of those new materials due to its large electron and hole mobility. With the strained silicon technology in the industry, it's a interesting topic to discover how the strain affects the electron and hole mobility in germanium MOSFETs. Germanium has been of special interest in high speed C '\OS technology for years [78, 79]. The bulk germanium hole mobility is larger than that of other semiconductor materials, and its electron and hole mobility are much less disparate than other materials. In 1989, germanium hole mobility of 770cm2/V sec in a pMOSFET was exhibited by Martin [80] and his coworkers using SiO2 as the gate insulator. Since then, more and more work [81, 82] has been done on germanium or SiGe channel pMOS [83, 84, 85]. In order to reduce the surface roughness and limit the bandtoband tunneling issue, silicongermanium or SiSiGe dual channel is also used in some applications. Different gate dielectric materials [86, 87, 88] have been utilized to find the best material to limit the surface roughness at the interface between gate insulator and germanium channel. Due to the uncertainty in the surface roughness and the surface states, different hole mobility values have been reported in those publications. In recent years, with the strain technology applied to silicon C'\ OS, strain effect is also investigated on germanium MOSFETs [87, 89, 90, 91, 92]. The strain is normally achieved by applying SiGe substrate underneath the germanium or SiGe channel. But most of the work stays only in experiments, the physical insights of the strain effect on germanium MOSFETs have not been discussed carefully. The only available theoretical works are some Monte Carlo simulations [93, 94, 95]. The goal of this chapter is to give physical insights of strain effects on germanium utilizing k p calculation. In this chapter, straininduced hole mobility change of Ge and SilzGez in pMOS inversion lwiri~ is investigated. The hole mobility vs electric field and surface orientation is showed. Strainenhanced hole mobility is calculated for different Ge concentration in Si1lGex. To understand the difference between Ge and Si, hole effective mass, band and subband splitting, and twodimensional densityofstates are calculated and their effects on hole mobility is analyzed. Phonon and surface roughness scattering is also evaluated under strain. 5.1 Germanium Hole Mobility Unstrained Ge hole mobility [86, 96] vs vertical electric field and device surface orientation is shown in figure 51. Experimental works give a lot of different mobility values ranging from 70cm2/V sec to over 1000cm2/V sec, depending on what the gate dielectric materials are used [86, 87, 88] and if Si buffer is applied [97, 98] between the Ge (or SiGe) and the gate oxide. With Si buffer, the device acts as a buriedGe channel transistor and normally shows large hole mobility due to the lack of confinement and surface roughness scattering. Due to the bad scalability of buriedchannel devices, only surface channel GepMOS is discussed here. Calculated Ge hole mobility matches the measured data and the mobility is much larger mobility than silicon. (110)oriented device shows higher mobility than (001)oriented device, which is consistent with the results of Si. We shall show that the larger hole mobility of germanium mainly comes from the smaller effective mass of the holes. The relative smaller intersubband phonon scattering rate due to the larger subband splitting (and smaller optical phonon energy) also improves the germanium mobility. 5.1.1 Biaxial Tensile Stress In silicon MOSFETs, biaxial tensile strain is obtained via applying Sil_Ge, sub strate underneath the Si channel. Biaxial tension is not a popular stress type for germa nium devices due to the large lattice constant of germanium. For comparison purpose, 500 o Zimmerman, 06 S400 Ge (110)/<110> S 400 E 300 El 0 0 Ge (001)/<110> 200 g Chui, 02 0 Si (001)/<110> 0 0.2 0.4 0.6 0.8 1 Effective Electric Field / MV/cm Figure 51. Germanium hole mobility vs effective electric field. the biaxial tensile strain effect on germanium hole mobility is calculated and showed in Figure 52. Like silicon, the degradation of the hole mobility at low biaxial tensile stress is due to the subtractive nature of strain effect and transverse electric field effect resulting in the increase of the average effective mass, together with a little increased intersubband phonon scattering. At high stress, the mobility enhancement is obtained due to reduced intersubband optical phonon scattering. 5.1.2 Biaxial Compressive Stress Biaxial compressive stress in germanium MOSFETs channel can be obtained by germanium channel on top of Sil_,Ge, substrate. Silicon transistors can also have biaxial compression with Sil_C, substrate. This is not a favorable stress type for either case, since it does not improve the hole mobility significantly as shown in Figure 53. 300 in=1 xl 013/cm2 250 o (001) Ge 0) 200 E 150 (001) Si S100 S 50 0 0 2 4 6 Biaxial Compressive Stress / GPa Figure 53. Germanium and silicon hole mobility under biaxial compressive stress where the inversion hole concentration is 1 x 1013/cm2. 99 5.1.3 Uniaxial Compressive Stress Uniaxial compressive stress on Si is has been applied to multiple technology nodes because of the maximum mobility enhancement to hole mobility. The hole mobility vs uniaxial compressive stress for Ge is shown in Figure 54 for (001)oriented Ge and Figure 55 for (O110)oriented Ge. For (001)oriented devices, both Si and Ge show large enhancement. One difference between the two curves is that the mobility enhancement for Si saturates at about 3GPa, but it does not saturate until 6GPa of stress is applied to Ge. 1800 pinv=1x1013/cm2 Ge a) 1500 0 > 1200 Si0.25Ge075 900 Si. .Geo . S0.75 0.25 S 600 0 0~  a) 300 S 0 0 2 4 6 Uniaxial Stress / GPa Figure 54. Germanium and silicon hole mobility on (001)oriented device under uniaxial compressive stress where the inversion hole concentration is 1 x 1013/cm2. 5.2 Strain Altered Bulk Ge Valence Band Structure To give the physical insights of the similarity and the difference of the hole mobility enhancement under strain for Ge and Si, strain altered bulk germanium valence band structure is discussed in this section. Strain brings band splitting and effective mass change to semiconductor valence band. Here, we shall focus on the effective mass change with strain and compare the difference between germanium and silicon. In next section, 600 Pin=1 xl 013/cm2 S500 (110)Ge S400 S300 (110) Si >. 200 S 100 0 i4 0 2 4 6 Uniaxial Compressive Stress / GPa Figure 55. Germanium and silicon hole mobility on (1lO)oriented device under uniaxial compressive stress where the inversion hole concentration is 1 x 103/cm2. Ge subband structure in inversion lv. i~ will be discussed and the phonon scattering rate will be calculated. 5.2.1 Ek Diagrams Figure 56 shows the dispersion relation diagrams for (001)Ge under different stress. Like silicon, the heavy hole and light hole bands of relaxed Ge are degenerate at F point as shown in Figure 56(a). The degeneracy is lifted when strain is applied. The band splitting leads to band warping and the change of hole effective mass and phonon scattering rate. The SO band energy is 296 meV lower than the HH and LH bands for relaxed germanium which implies less coupling with HH and LH bands compared with silicon. Under biaxial tensile strain, the top band is LHlike outofplane and HHlike along (110). For both compressive strain in Figure 56(c) and (d), the top band is HHlike outofplane and LHlike along (110). Uniaxial compressive strain brings the most warping on the top valence band. The warping is the smallest under biaxial compressive strain, which ti'i 1 the least mobility enhancement as shown in Figure 53. 5.2.2 Effective Mass Straininduced <110> and outofplane effective mass change at F point are showed in Figure 57 for biaxial tension, 58 for biaxial compression, and 59 for uniaxial compres sion. Compared with silicon, the effective mass for germanium is obviously much smaller along both directions. This ti'. 1; larger hole mobility for germanium than silicon according to Drude's model. One significant difference from Si effective mass is that the hole effective mass change of Ge saturates with stress at much higher stress than silicon. For some of the curves, i.e. "top" band of Figure 57(a) and 59(b), or "bottom band" of Figure 58, the effective mass change does not saturate until the stress goes up to 7 GPa. But for silicon, normally the effective mass change saturates at 2 or 3 GPa. This Ii.; I higher stress for the mobility saturation. The trend of the effective mass change with stress is similar for both silicon and germanium. If we look at the channel direction ((110)) effective mass, the top band Figure 56. Ek diagrams for Ge under (a) no stress; (b) 1 GPa biaxial tensile stress; (c) 1 GPa biaxial compressive stress; and (d) 1 GPa uniaxial compressive stress. Bottom Band Ton Band 1 2 3 4 5 6 7 (b) Figure 57. Conductivity effective mass vs biaxial tensile stress: (a) (<110>) and (b) outofplane direction. ('IC iin,, I direction 0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0.08 0.07 0.06 Top Band 0.05 0 1 2 3 4 5 6 7 (a) 0.22 0.2 0.18 Top Band 0.16 0.14 0.12 bottom Band 0.1 0.08 0.06 0.04 ''' 0 1 2 3 4 5 6 7 (b) Figure 58. Conductivity effective mass vs biaxial compressive stress: (a) C'!, ,iii, I direction (<110>) and (b) outofplane direction. 0.4 ).35 0.3 0.251 0.21 0.15F 0.11 U 0 1 2 3 4 5 6 (a) Figure 59. Conductivity effective mass vs uniaxial compressive stress: direction (<110>) and (b) outofplane direction. (a) C'!i, ,ii, I Bottom Band Top Band effective mass at F point under uniaxial compressive stress is only about 0.0 ,,,,, compared with 0.;:,,,,, for relaxed germanium. The ratio of the change is 9.5 comparing with 5.4 of silicon (0.59mo to 0.11mo). This huge effective mass gain does not result in higher mobility enhancement in Figure 54 because of the much smaller 2D DOS and initial large subband splitting in the inversion Il.i which will be addressed in the next section. Under biaxial tensile stress, the top band has higher channel direction effective mass (increasing with stress) and lower outofplane effective mass which is similar to silicon. This means the stress effect and the transverse electric field effect in the inversion 1iv. r should be subtractive and the hole mobility should be degraded at low stress as shown in Figure 52. Under biaxial compressive stress, the top band has very low conductivity effective mass at F point along (110). As we mentioned before, the band warping is not significant and only happens very close to the F point, which ii.; r; the average effective mass of the system may not decrease much with the stress. 5.2.3 Energy Contours The energy contours of the valence band provide a straightforward picture of the conductivity effective mass and densityofstates of each band. The conductivity and densityofstates effective mass change with strain can also be seen from the shape change of the contours. The 25 meV contours of the unstressed Ge are shown in figure 510. Contours (25 meV) under biaxial compressive and tensile stress are shown in figure 511 and 512. Figure 513 shows the contours under uniaxial compressive stress. The energy contours are similar to those of silicon, but the shape of the contours changes more than Si contours when the same amount of strain is present. Another difference is that under uniaxial compressive stress, the 2D DOS of Ge looks much smaller than Si. From the analysis of Si, lower F point DOS leads to smaller strain induced mobility improvement due to fewer holes are affected by strain. This may explain why the mobility enhancement factor for Ge is not larger than Si, although the effective mass change is much larger at F point. 0.05 0 005 005  i <.... 0.05 0 0 05 005 Figure 510. 25meV energy contours for unstressed Ge: (a) Heavyhole; (b) Lighthole. 0 05  0.05 0 005 0.05 0.05 0 005 0.05 Figure 511. 25meV energy contours for biaxial compressive stressed Ge: (a) Top band; (b) Bottom band. 0.05 005 0 0.05 0.0 005 0.05 (a) 0.05 0 05  0050 0 05 0.05 (b) Figure 512. 25meV energy Bottom band. contours for biaxial tensile stressed Ge: (a) Top band; (b) 0.05  0.05 0 0 05 0.05 0 005 0.05 0.05 0 0 05 0.05 Figure 513. 25meV energy contours for uniaxially compressive stressed Ge: (a) Top band; (b) Bottom band. 5.3 Discussion Of Hole Mobility Enhancement 5.3.1 Straininduced Subband Splitting Based on the triangular potential approximation, subbands with higher outofplane effective mass tend to go closer to the top of the quantum well (lower energy for holes) under vertical electric field. Figure 58 and 59 show that both biaxial compressive stress and uniaxial compressive stress shift up the outofplane HHlike band. This effect is clearly additive to the electric field effect. For biaxial tensile stress, the electric field effect is subtractive to the strain effect and therefore if the electric field is fixed, the subband splitting should decrease at low stress level and at some stress value, the top two subbands would cross over each other just like Si. The subband splitting between the ground and the first excited state of (001) Ge is illustrated in Figure 514. 140 140 (001) Ge, bi. tens. ..  S120 E ,t 100 / 8 8 c 40 4 5 20 \ * SCI 0 0 1 2 3 4 5 6 Stress / GPa Figure 514. Ge subband splitting under different stress. Figure 514 shows that the subband splitting for relaxed Ge pMOSFETs is much larger than that of the Si pMOSFETs. The splitting is larger than the optical phonon energy of Ge(37 meV), while for unstressed (001) Si, the splitting is smaller than optical phonon energy. The difference, together with the fact that Ge has much smaller 2D DOS, ,i. i; that even without the stress, the intersubband optical phonon scattering rate is much smaller compared with silicon. Except biaxial tensile stress, the subband splitting of all compressive stress cases increases with the stress. The amount of increase is much smaller than that of the silicon cases, which indicates smaller phonon scattering rate change with the stress for (001) Ge comparing with Si. This ii, 1' that the strain enhanced hole mobility is mainly because of the effective mass gain. 5.3.2 Biaxial Stress on (001) Ge As we mentioned in C'! lpter 3, with more strain, more region in momentum space is affected. The band which is out of the strainaffected region is normally warped. Figure 5 15 shows the normalized inplane Ek diagram under biaxial compressive stress. On the one hand, the figure shows as the stress increases, more region is near the zone center is warped and has lower DOS. On the other hand, out of the warped area, the band curves up a little as the stress increases, which ii. I the increase of the DOS. The overall effect is that the effective mass gain close to F point due to stress is compromised by the heavier mass of the holes away from the F point. At low stress, the mass change, together with the increasing subband splitting, enhances the hole mobility slightly. Under higher stress, the enhancement is minimal. For Si pMOS under biaxial compressive stress, the Ek diagram is similar to Ge. The difference is that Si has much larger DOS near F point, therefore there is ahv effective mass gain. The different DOS results in the strain enhanced hole mobility difference as in Figure 53. Biaxial tensile stress affects the Ge hole mobility similar to Si devices: the subtractive nature of the strain and transverse electric field effects degrades the hole mobility at low stress, and the decrease of the phonon scattering rate enhances the mobility at high stress. Uniaxial compressive stress on (001)oriented Ge is focused next because it provides the 0.1 0.05 0 0.05 0.1 k X Figure 515. Normalized ground state subband Ek diagram vs biaxial compressive stress. most mobility enhancement and the mobility enhancement mechanism is a little different from Si. 5.3.3 Uniaxial Compression on (001) Ge Ground state 2D DOS of Si and Ge are shown in Figure 516 at different energies. DOS of Ge is much lower than Si. The trend of the DOS change with stress is similar for Si and Ge. Figure 516(a) shows that the DOS of Ge saturates with stress at higher stress than Si, since the effective mass changes with stress at higher stress. This is consistent with the mobility saturation curves in Figure 54. Phonon and surface roughness scattering rate change vs uniaxial stress is shown in Figure 517 and 518. For both Si and Ge, phonon scattering rate does not change much at low stress, and at high stress the phonon scattering rate decreases as the uniaxial stress increases. Ge has lower scattering rate than Si due to the smaller DOS of Ge. For Si, both acoustic phonon and optical phonon scattering rate decreases by 50' when the stress 2.5 014 2.5 2D DOS at Energy=5 meV 2.0 ,' Si (u) cz 1.5 (U) I 0 > 1.0 .i U) > 0.5 "o Ge C) 0.0 0 2 4 6 Uniaxial Compressive Stress / GPa (a) x1014 6 5 )4 0) 3 Si * *  * 0 22 (1 N Ge a) TS 1 C\j0 0 2 4 6 Uniaxial Compressive Stress / GPa (b) Figure 516. Two dimensional densityofstates of the ground state subband for Si and Ge at (a)E5mneV; (b)E 2kT 52meV under uniaxial compressive stress. 113 increases from zero to 6 GPa. For Ge, the phonon scattering rate only decreases ;: .' The surface roughness scattering rate increases with the stress for both Si and Ge due to the hole repopulation under stress as we explained previously. Figure 519 shows the mobility enhancement contribution from effective mass (solid lines) and phonon scattering rate (dashed lines) for Si and Ge. For Si, effective mass gain is the main driving force of the mobility enhancement at low stress, and the scattering rate change is dominant at high stress range (1 GPa3 GPa). From unstressed case to 3 GPa of stress, effective mass gain and phonon scattering rate decrease have comparable enhancement to the hole mobility. For Ge, the phonon scattering only contribute 1.5 times of the enhancement. The effective mass change is dominant in the whole stress range. As we mentioned before, this is because the effective mass change ratio is large under stress (0.;:,,,,, to smaller than 0.0 1,,,). Another observation of the effective mass is that as the stress is over 1 GPa, increasing the stress does not change the hole effective mass for Si, but the effective mass of Ge continue to decrease as the stress increases. This extra effective mass gain contribute to the hole mobility enhancement for Ge at very high stress. 5.3.4 Uniaxial Compression on (110) Ge The confined 2D energy contours are showed in Figure 520 for (001)oriented MOSFETs and Figure 521 for (110)oriented pMOSFETs. For (110) Ge pMOSFETs under uniaxial stress, the strain effect is similar to (110) Si pMOSFETs. The strong quantum confinement warps the subband structure and results in small hole effective mass, which explains the higher unstrained hole mobility than (001) Ge pMOSFETs. As the uniaxial compressive stress is applied, the strain effect is undermined by the strong quantum confinement and only warps the high energy region of each subband. As a result, the hole mobility is not enhanced as significantly as (001)oriented pMOSFETs. 5.4 Summary Germanium hole mobility improvement under biaxial tensile, biaxial compressive and uniaxial compressive stress is analyzed and compared with silicon. The trend of x1012 4 Si r iw ** * 0 2 4 Uniaxial Compressive Stress/ GPa (a) x1012 \ Si Ge I Uniaxial Compressive Stress / GPa (b) Figure 517. Phonon scattering rate vs uniaxial compressive stress: and (b) optical phonon. (a) Acoustic phonon, 0.0 9.0 0) c 0 CO 0 0 O r a5 0 O 6.0 3.0 0.0 x1012 4.5 ) (D * 0. 0 a) to , I t a) C/) 0 Ge 0.0 0 2 4 6 Uniaxial Compressive Stress / GPa Figure 518. Surface roughness scattering rate vs uniaxial compressive stress for Ge and Si. 1 2 Uniaxial Stress / GPa Figure 519. 0.15r Mobility enhancement contribution from effective mass change (solid lines) and phonon scattering rate change (dashed lines) for Si and Ge under uniaxial compressive stress. 0.15 0.15 0 0.15 0.15 Figure 520. Confined 2D energy contours for (001)oriented Ge pMOS with uniaxial compressive stress. Unstressed Ge 1GPa Uniaxial Compression ,  0[ 1GPa Uniaxial Compression 0.15 0 0.15 0.15 0 0.15 k k x x (a) (b) Figure 521. Confined 2D energy contours for (1O0)oriented Ge pMOS with uniaxial compressive stress. each stress type for both germanium and silicon is similaruniaxial compressive stress on (001)oriented transistors has the most hole mobility improvement mainly from the reduced hole conductivity effective mass. Uniaxial compressive stress on (110)oriented devices does not provide as much improvement due to the strong quantum confinement undermining the strain effect. Hole mobility is degraded under low biaxial tensile stress due to the subtractive nature of the strain and vertical electric field effects and hence the increase of the average effective mass. The mobility is enhanced at high stress because of the reduction of the intersubband scattering rate. Biaxial compressive stress does not improve the hole mobility much due to the small DOS after band/subband warping and not much effective mass gain. Unstressed (110) Ge CHAPTER 6 SUMMARY AND SUGGESTIONS TO FUTURE WORK 6.1 Summary In this work, uniaxial stressinduced hole mobility enhancement in (001)oriented Si pMOSFETs is calculated at high stress (up to 3 GPa) and large enhancement factor (4.5x) is obtained. For the first time, coordinates system transformation of Luttinger Kohn's Hamiltonian and KuboGreenwood Equation is performed to investigate the hole mobility in Si and Ge pMOSFETs with surface orientations other than (001). The strong quantum confinement in (110), (111), and (112)oriented pMOSFETs is reported for the first time. The results show that, unlike (001) pMOSFETs, the subband structures of Si and Ge in (110), (111), and (112)oriented pMOSFETs are warped by the confinement. The strong confinement causes smaller hole effective mass and lower phonon scattering rate due to larger subband splitting, which explains the higher hole mobility in those pMOSFETs. To analyze the difference of the stressinduced phonon scattering rate for (001) and (110) pMOSFETs, twodimensional densityofstates (2D DOS) are evaluated at arbitrary energy in the subbands. Comparing with (001) pMOSFETs, (110) p MOSFETs have smaller DOS and DOS does not vary much as the uniaxial stress increases due to the stronger quantum confinement. Under uniaxial stress, the phonon scattering rate for (110) pMOSFETs does not change as much as (001) pMOSFETs. 2D energy contours of the subbands in (001) and (110) pMOSFETs under stress are investigated and smaller effective mass change with stress for (110) pMOSFETs is found which is again due to the stronger quantum confinement. The smaller change of effective mass and phonon scattering rate results in lower mobility enhancement in (110) pMOSFETs. As a result, at high uniaxial stress (3 GPa), (001)/<110>, (110)/<110>, and (110)/<111> pMOSFETs have similar hole mobility. Strain induced hole mobility enhancement is studied theoretically for the first time in ultrathinbody (UTB) nonclassical pMOSFETs, including singlegate (SG) silicon oninsulator (SOI), (001) symmetrical doublegate (SDG) pMOSFETs, and (110) ptype FinFETs. For SG SOI pMOSFETs, the strain effects are as same as traditional Si p MOSFETs. For (001) SDG pMOSFETs and (110) FinFETs, subband modulation is found when the channel thickness is smaller than 20 nm. Due to the interaction of the two surface channels, the subband splitting between the ground state and the first excited state is small (about 3 to 5 meV) as the body thickness is larger than 5 nm. This splitting does not change as the stress increases. Compared with the single gate pMOSFETs, this small splitting is similar to increasing the DOS of the ground state subband. As the stress increases, the average effective mass change is larger than that in single gate pMOSFETs. The low form factors due to the symmetrical structure and low electric field in the channel ii. 1 the phonon scattering rate in double gate pMOSFETs is lower than single gate pMOSFETs, regardless the small subband splitting. For (001) SDG pMOSFETs, the phonon scattering rate change is a little smaller than single gate p MOSFETs as the stress increases. The larger effective mass change and smaller scattering rate change result in similar hole mobility enhancement factor compared with single gate pMOSFETs. For FinFETs, the form factors are much smaller than single gate (110) pMOSFETs and the change with stress is larger which i... i larger scattering rate change. Therefore, the straininduced hole mobility enhancement (3x) is larger than single gate (110) pMOSFETs (2x). Strain effect on hole mobility improvement in (001) and (110) Ge and Sil_,Gex pMOSFETs is calculated for the first time. The mobility enhancement at low stress is similar to Si. At high stress, the maximum mobility enhancement factor for (001) Ge is larger than Si due to the greater effective mass change, especially at high stress. The phonon scattering rate change for Ge pMOSFETs is a little smaller than Si. For (110) Ge pMOSFETs, strong quantum confinement is found and the strain induced mobility enhancement is smaller than (001) Ge. Biaxial compressive stress effect on Ge pMOSFETs is also calculated, and very small enhancement is found. 6.2 Recommendations for Future Work The ...r'essive scaling of silicon C'L\ OS technology has pushed the channel length to nanometer regime. Strain, especially uniaxial compressive strain, can improve the hole mobility of pMOSFETs dramatically and hence enhance the device performance. To further improve the performance of C' \!OS technology, other featureenhanced technology and even new material will be a must have. Nonclassical devices have been seen as possible replacement for simple planar layout single gate bulk silicon devices and have the potential to be scaled down further in the roadmap. Although theoretical calculation shows the performance could be improved by strain, the question still exists how strain can be applied to these devices, especially FinFETs. Germanium is one new material that has been considered to replace silicon in C'\ OS technology. Uniaxial strain even has higher enhancement on germanium pMOS. But the experimental work is still lack for germanium. People are still trying to find out the best layout, proper dielectric and gate materials. It will be a long way but definitely worth working on. How about after all of this? There will be an ultimate limit for the scaling that ballistic transport will take place and the mobility concept will not be valid. Will strain still be useful at that stage? The answer is probably yes, since the strain can reduce the effective mass of the carriers and this will still help the transport. That being said, serious calculation will be necessary to further explain this. APPENDIX A STRESS AND STRAIN Stress a is defined as the force F applied on unit area A. F S= lim (Al) AO A Any stress on an isotropic solid body in a cartesian coordinate system can be expressed as a stress matrix a [13, 99], Oxx T xy T xz Tyx yy c Tyz (A2) TZX Tzy zz where Fi aii lim AiO Ai is called the normal stress on the iface in the idirection and F. T, = lim Ai>O Ai is the shear stress on the iface in the jdirection [100] as shown in Figure Ai. This stress matrix completely characterizes the state of stress at ( i I i For stress S along < 100 >direction, the matrix can be written as 1 0 0 a= S 0 0 0 (A3) For stress S along both <100> and <010>direction (biaxial stress), 1 (A4)0 0 a=S 0 1 0 (A4) o o o Figure Ai. Stress distribution on ( i I i Stress S along <110>direction is a little complicated. The stress is applied on both (100) and (010) planes. If we resolve each component along x and y axes to get both normal and shear terms, each term has the same magnitude of S/2. The stress tensor can be expressed as, S= 1 1 0 (A5) 0 0 0 For stress S along <111>direction, based on the similar analysis, the stress is actually acted on (100), (010), and (001) planes. Each component can be resolved along x, y, and z axes and the stress along each direction is S/3. Therefore the stress tensor is, S= 1 1 1 (A6) 3 1 1 The stress matrix is symmetric where ij = ji, and only 6 components are necessary to represent the stress. Therefore, the 3 x 3 matrix can also be written as a 6 x 1 stress vector. yy zz az (A7) RTz Strain is defined as the distortion of a structure caused by stress. Normal strain is defined as the relative lattice constant change [13, 99], a a (A8) ao where ao and a are lattice constant before and after the strain. However, the deformation of the ( i I I1 cannot be fully represented with the normal strain. It also has shear terms that are defined as change in the interior angles of the unit element. Like stress, strain can also be expressed with a symmetric 3 x 3 tensor or 6 x 1 vector e [100]. ezz C 6 yx g YY y or, C (A9) 2cyz ,zx zy tzz 2 xz 2cxy For most materials the stress is a linear function of strain. The transformation between stress and strain is through a 6 x 6 stiffness matrix C or compliance matrix S [99]. S= C. (A 10) zxx C11 C12 C13 C14 C15 C16 6xx jyy C21 C22 C23 C24 C25 C26 EyY 0zz C31 C32 C33 C34 C35 C36 Czz (A11) TyZ C41 C42 C43 C44 C45 C46 2cy TXz C51 C52 C53 C54 C55 C56 2ez Txy C61 C62 C63 C64 C65 C66 2ecy or, c S'y Cxz S11 S12 S13 S14 S15 S16 xx yy S21 S22 S23 S24 S25 S26 yy Czz S31 S32 S33 S34 S35 S36 zz( (A 12) 2yz S41 S42 S43 544 S45 S46 Tyz 2ez S51l S52 S53 S54 S55 S56 Tz 2cy S61 S62 S63 S64 S65 S66 I Ty For diamond or zincblendetype (i 1I I stiffness matrix and compliance matrix can be simplified as [99] Table Ai. Elastic stiffnesses Ci in units of 1011N/m2 and compliances Sij in units of 1011m2/N C11 Si 1.657 Ge 1.292 S11 0.768 0.964 S12 0.214 0.260 C12 0.639 0.479 C44 0.7956 0.670 C12 C12 C11 0 0 0 S12 S12 S11 0 0 0 S44 1.26 1.49 6xx 2cyz 2c22 2cy 2(Exz oxx oyy Tx XY (A13) (A14) The stiffness and compliance coefficients for following table. silicon and germanium are listed in the Let's go back to the strain tensor. Each strain can be decomposed to two compo nents: hydrostatic term and shear term. The shear term can be further decomposed to shear100 term which only has diagonal elements and shear111 term which only contains nondiagonal elements. 6 hydrostatic + 6shear100 + 6shear111 0 0 0 0 0 0 0 0 0 744 0 0 0 C44 0 0 0 C44 6xx cyy 6zz 2cyz 2cy C12 C11 C12 0 0 0 S12 S11 S12 0 0 0 0 0 0 0 0 0 S44 0 0 S44 0 0 0 0 2cxx + (Cyy +czz 0 0 1 0 0 2czz (Cx + ) /  The hydrostatic term in the strain tensor shifts the energy of all the bands in semiconductors by the same amount simultaneously but does not cause band splitting, since it is actually a constant and in the calculation of the band energy it only acts like adding an additional potential term to the hamiltonian. The semiconductor transport property is independent on the hydrostatic strain term. For two different stress, as long as the shear terms of their strain tensors are equal, their impact to the carrier mobility should be identical. Stress can be applied to semiconductors from any direction. For a silicon MOS FET, only inplane biaxial stress or channel direction uniaxial stress has technological importance. The common silicon wafers that are used in industry are (001)oriented, and normally the channel of the MOSFET is along <110>direction. Biaxial stress here means that the stress is applied in both <100> and <010>directions of the wafer with the same magnitude. Uniaxial stress represents the stress along the <110> channel direc tion. This stress is also called uniaxial longitudinal stress. In the same manner, uniaxial transverse stress normally means the uniaxial stress applied perpendicular to the channel direction. Both of those stresses are applied in the plane of the wafer, therefore they are also "inpl! .ii, stresses. Another kind of uniaxial stress is called "outofpl! i., uniaxial stress which means the stress is applied in the direction perpendicular to the surface of the wafer. For the outofplane uniaxial stress and the inplane biaxial stress on (001) wafer, the strain matrices only have diagonal terms and all nondiagonal terms are zero. The question is, how do these two stresses differ from each other? Let's assume we have out ofplane uniaxial stress a on one sample and inplane biaxial stress a on another sample. For case 1, based on (1.4) and (1.15), the strain tensor can be expressed as, in the form of (1.16), S12 Cu 0 0 S11 + 2S 3 0 0 Sii S12 0 3 0 ? S  +I o 12 0 0 S11 + 2S12 0 0 S11 + 2S12 S/ hydrosatic 0 0 S11 S12 0 0 2(S12 Sn) 8]hear For inplane biaxial stress, S11 + S12 0 0 0 Sn1 + S12 0 (A16) Cb " 0 0 2S12 2(Sil + 2S12) 3 0 0 0 2(SIl + 2S12) 0 0 0 2(S + 2S2) hydrostatic / hydrosatic SI S12 0 0 + 0 S11 S12 0 (A17) 0 0 2(S12 S1l) (1.17) and (1.18) show that the hydrostatic terms of those two strain tensors are different, but the shear terms are identical. This tells us that the biaxial tensile (com pressive) stress should have the same effect as the outofplane uniaxial compressive (or tensile) stress in determining the transport property of the holes. APPENDIX B PIEZORESISTANCE The piezoresistance, or piezoresistive effect, describes the electrical resistance change of materials caused by applied mechanical stress. The first measurement of piezoresistance was performed by Bridgman in 1925 and extensive study on this topic was done ever since. In 1954, Smith measured the piezoresistance effect on Si and Ge [7]. This effect becomes more and more important due to the wide application of Si and Ge on contemporary C'\ OS technology. Similar to stress and strain, the change of resistivity of a material is a symmetrical second rank tensor. The tensor connecting the stress and the piezoresistance is of fourth rank. For Si and Ge, we can simplify the tensor as [7] 11 7r12 712 0 0 0 71l2 rll 712 0 0 0 H 12 712 ri11 0 0 0 n (Bi1) 0 0 0 7T44 0 0 0 0 0 0 7T44 0 0 0 0 0 0 7T44 The most general form of a twodimensional piezoresistance tensor in the inversion 1vr is [8] iT11 712 7T14 I 721 722 7124 (B2) 741 7"42 44 For (001), (110), and (111) surface oriented Si (or Ge), F714 741 = F24 i2 = 0 (principle axis (001) for (001) and (110) surface, (110) for (111) surface). We can further simplify the piezoresistance tensor as [8] For (001) surface oriented Si For (111) surface oriented Si r11 7r12 0 = 12 722 0 0 0 7144 and Ge, 7rn = 722 and / \ll l 0 711 712 0 H 71"12 7i11 0 0 0 7i44 and Ge, 7i44 = T11 712 and 0 0 7in 712 In the piezoresistance tensors, 711 represents the longitudinal piezoresistance coef ficient (along ( 100) for (001) and (110) surface). 2,, is the transverse piezoresistnace coefficient (along ( 010) for (001) and (110) surface). 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Monte carlo study of strained germanium nanoscale bulk pMOSFETs. IEEE Trans. Electr. Dev., 53:533537, 2006. [96] P. Zimmerman. High performance Ge pMOS devices using a Sicompatible process flow. In Int. Electr. Dev. Meeting, pages 261264, 2006. [97] M. L. Lee, C. W. Leitz, Z. C'!, A. J. Pitera, and e. A. Fitzgerald. Strained Ge channel ptype metaloxidesemiconductor fieldeffect transistors grown on Si1lGe1/Si virtual substrates. Appl. Phys. Lett., 79:33443346, 2001. [98] E. A. Fitzgerald. MOSFET channel engineering using strained Si, SiGe, and Ge channels. In ECS Meeting Proceedings, page 923, 2003. [99] P. Y. Yu and M. Cardona. Fundamentals of Semiconductors. Springer, New York, NY, 2001. [100] Nam H. Kim and Bhavani V. Sankar. Finite element analysis and design. Lecture Notes, September 2005. BIOGRAPHICAL SKETCH Guangyu Sun was born in Shandong, China, on June 9th, 1975. In 1992, he was admitted to University of Science and Technology of C('hlii (USTC) in Hefei, ('C!h i From 1992 to 1996 he studied in USTC and received his B.S. degree in applied physics in 1996. He subsequently participated in the master's program and obtained the M.S. degree in 1999. In the fall of 1999, he came to the United States and became a Florida Gator. In the spring of 2004, he entered Prof. Thompson's group and has been studying the strain effects on Si and Ge MOSFETs, pursuing a Ph.D. degree. PAGE 1 1 PAGE 2 2 PAGE 3 3 PAGE 4 IamgratefultoallthepeoplewhomadethisdissertationpossibleandbecauseofwhommygraduateexperiencehasbeenonethatIwillcherishforever.FirstandforemostIthankmyadvisor,Dr.ScottE.Thompson,forgivingmeaninvaluableopportunitytoworkonchallengingandextremelyinterestingprojectsoverthepastfouryears.HehasalwaysmadehimselfavailableforhelpandadviceandtherehasneverbeenanoccasionwhenIhaveknockedonhisdoorandhehasnotgivenmetime.Hetaughtmehowtosolveaproblemstartingfromasimplemodel,andhowtodevelopit.Ithasbeenapleasuretoworkwithandlearnfromsuchanextraordinaryindividual.IthankDr.JerryG.Fossum,Dr.HuikaiXie,Dr.ChristopherStanton,andDr.JingGuoforagreeingtoserveonmydissertationcommitteeandforsparingtheirinvaluabletimereviewingthemanuscript.IalsothankDr.ToshiNishidaforalotofhelpfuldiscussionsandkindhelp.MycolleagueshavegivenmealotofassistanceinthecourseofmyPh.D.studies.Dr.YongkeSunhelpedmegreatlytounderstandthephysicsmodel,andwealwayshadfruitfuldiscussions.Dr.ToshiNumataalsogavemegoodadviceandsomeinsightfulideas.IalsothankJisongLim,SagarSuthram,andallothergroupmemberswhomademylifeheremoreinteresting.Iacknowledgehelpandsupportfromsomeofthestamembers,inparticular,ShannonChillingworth,TeresaStevensandMarcyLee,whogavememuchindispensableassistance.Iowemydeepestthankstomyfamily.Ithankmymotherandfather,andmywife,Anita,whohavealwaysstoodbyme.Ithankthemforalltheirloveandsupport.WordscannotexpressthegratitudeIowethem.Itisimpossibletorememberall,andIapologizetothoseIhaveinadvertentlyleftout. 4 PAGE 5 page ACKNOWLEDGMENTS ................................. 4 LISTOFFIGURES .................................... 7 LISTOFTABLES ..................................... 11 ABSTRACT ........................................ 12 CHAPTER 1INTRODUCTIONANDOVERVIEW ....................... 14 1.1HistoryofStraininSemiconductors ...................... 15 1.2ApplyStraintoATransistor .......................... 17 1.3MainContributionsofMyResearch ...................... 18 1.4BriefDescriptionofTheDissertation ..................... 19 2KPMODELANDHOLEMOBILITY ...................... 21 2.1ThekpMethod ................................ 21 2.1.1IntroductiontokpMethod ...................... 21 2.1.2Kane'sModel .............................. 25 2.1.3LuttingerKohn'sHamiltonian ..................... 28 2.2HoleMobilityinInversionLayers ....................... 32 2.2.1SelfconsistentProcedure ........................ 32 2.2.2HoleMobility .............................. 33 2.3ScatteringMechanisms ............................. 34 2.3.1PhononScattering ............................ 34 2.3.2SurfaceRoughnessScattering ..................... 36 2.4Summary .................................... 38 3STRAINEFFECTSONSILICONPMOSFETS ................. 39 3.1PiezoresistanceCoecientsandHoleMobility ................ 40 3.1.1PiezoresistanceCoecients ....................... 40 3.1.2HoleMobilityvsSurfaceOrientation ................. 41 3.1.3HoleMobilityandVerticalElectricField ............... 42 3.1.4StrainenhancedHoleMobility ..................... 42 3.2BulkSiliconValenceBandStructure ..................... 48 3.2.1DispersionRelation ........................... 48 3.2.2HoleEectiveMasses .......................... 49 3.2.3ValenceBandunderSuperLowStrain ................. 56 3.2.4EnergyContours ............................ 56 3.3StrainEectsonSiliconInversionLayers ................... 60 3.3.1QuantumConnementandSubbandSplitting ............ 60 5 PAGE 6 ......................... 64 3.3.3StraininducedHoleRepopulation ................... 67 3.3.4ScatteringRate ............................. 73 3.3.5MassandScatteringRateContribution ................ 77 3.4Summary .................................... 78 4STRAINEFFECTSONNONCLASSICALDEVICES .............. 80 4.1SingleGateSOIpMOS ............................. 82 4.1.1HoleMobilityvsSiliconThickness ................... 82 4.1.2StrainenhancedHoleMobilityofSOISGpMOS ........... 83 4.2DoublegatepMOSFETs ........................... 85 4.2.1(001)SDGpMOS ............................ 87 4.2.2StrainEectonFinFETs ........................ 89 4.3Summary .................................... 93 5STRAINEFFECTSONGERMANIUMPMOSFETS .............. 95 5.1GermaniumHoleMobility ........................... 96 5.1.1BiaxialTensileStress .......................... 96 5.1.2BiaxialCompressiveStress ....................... 97 5.1.3UniaxialCompressiveStress ...................... 100 5.2StrainAlteredBulkGeValenceBandStructure ............... 100 5.2.1EkDiagrams .............................. 102 5.2.2EectiveMass .............................. 102 5.2.3EnergyContours ............................ 107 5.3DiscussionOfHoleMobilityEnhancement .................. 110 5.3.1StraininducedSubbandSplitting ................... 110 5.3.2BiaxialStresson(001)Ge ....................... 111 5.3.3UniaxialCompressionon(001)Ge ................... 112 5.3.4UniaxialCompressionon(110)Ge ................... 114 5.4Summary .................................... 114 6SUMMARYANDSUGGESTIONSTOFUTUREWORK ............ 119 6.1Summary .................................... 119 6.2RecommendationsforFutureWork ...................... 121 APPENDIX ASTRESSANDSTRAIN ............................... 122 BPIEZORESISTANCE ................................ 130 REFERENCES ....................................... 133 BIOGRAPHICALSKETCH ................................ 141 6 PAGE 7 Figure page 11SchematicdiagramofbiaxialtensilestressedSiMOSFETonrelaxedSi1xGexlayer .......................................... 17 12UniaxialstressedSiMOSFETwithSi1xGexSource/Drainorhighlystressedcappinglayer ..................................... 18 31Holemobilityvsdevicesurfaceorientationforrelaxedsilicon ........... 41 32Holemobilityvsinversionchargedensityforrelaxedsilicon.Bothmeasurementsandsimulationshowlargermobilityon(110)devices. ............... 43 33Holemobilityvsstresswithinversionchargedensity11013=cm2. ....... 44 34Calculatedstraininducedholemobilityenhancementfactorvs.experimentaldatafor(001){orientedpMOS. ........................... 45 35Holemobilityenhancementfactorvsuniaxialstressfordierentchanneldoping. 45 36Calculatedstraininducedholemobilityenhancementfactorvs.stressfor(001){orientedpMOSwithdierentinversionchargedensity. .............. 47 37Ekrelationforsiliconunder(a)nostress;(b)1GPabiaxialtensilestress;and(c)1GPauniaxialcompressivestress. ........................ 50 38NormalizedEkdiagramofthetopbandunderdierentamountofstress.Largerstresswarpsmoreregionoftheband.Theenergyatpointforallcurvesissettozeroonlyforcomparisonpurpose. ...................... 51 39Channeldirectioneectivemassesforbulksiliconunder(a)biaxialtensilestress;and(b)uniaxialcompressivestress. ......................... 52 310Twodimensionaldensityofstateseectivemassesforbulksiliconunder(a)biaxialtensilestress;and(b)uniaxialcompressivestress. .............. 53 311Outofplaneeectivemassesforbulksiliconunder(a)biaxialtensilestress;and(b)uniaxialcompressivestress. ......................... 54 312Holeeectivemasschangeunderverysmallstress.Thechangeinthisstressregionexplainsthe\discontinuity"oftheholeeectivemassbetweentherelaxedandhighlystressedSi. ............................. 57 313The25meVenergycontoursforunstressedSi:(a)Heavyhole;(b)Lighthole. 58 314The25meVenergycontoursforbiaxialtensilestressedSi:(a)Topband;(b)Bottomband. ..................................... 59 7 PAGE 8 ................................... 59 316Quantumwellandsubbandsenergylevelsundertransverseelectriceld. .... 61 317Schematicplotofstraineectonsubbandsplitting,theeldeectisadditivetouniaxialcompressionandsubtractivetobiaxialtension. ............ 64 318Subbandsplittingbetweenthetoptwosubbandsunderdierentstress. ..... 65 319Outofplaneeectivemassesforh110isurfaceorientedbulksiliconunderuniaxialcompressivestress. ............................... 66 320The2Denergycontours(25,50,75,and100meV)forbulk(001)Si.Uniaxialcompressivestresschangesholeeectivemassmoresignicantlythanbiaxialtensilestress. ..................................... 68 321Conned2Denergycontours(25,50,75,and100meV)for(001)Si.Thecontoursareidenticaltothebulkcounterparts. .................... 69 322The2Denergycontours(25,50,75,and100meV)forbulk(110)Siunder(a)nostress;(b)uniaxialstressalongh110i;and(c)uniaxialstressalongh111i. .. 70 323Conned2Denergycontours(25,50,75,and100meV)for(110)Si.Theconnedcontoursaretotallydierentfromtheirbulkcounterpartswhichsuggestssignicantconnementeect. ............................ 71 324Groundstatesubbandholepopulationunderdierentstress. ........... 72 325Stresseectonthe2dimensionaldensityofstatesofthegroundstatesubband. 74 326TwodimensionaldensityofstatesatE=4kT. ................... 75 327Straineecton(a)acousticphonon,and(b)opticalphononscatteringrate. .. 76 328Straineectonsurfaceroughnessscatteringrate. ................. 77 329Holemobilitygaincontributionfrom(a)eectivemassreduction;and(b)phononscatteringratesuppressionforpMOSFETsunderbiaxialanduniaxialstress. 79 41HolemobilityvsSOIthicknessforsinglegateSOIpMOS.Themobilitydecreaseswiththethicknessduetostructuralconnement. ................. 83 42HolemobilityforsinglegateSOIpMOSvsuniaxialstressatchargedensityp=11013=cm2. ..................................... 84 43HolemobilityenhancementfactorofUTBSOISGdevicesvsuniaxialcompressivestressatchargedensityp=11013=cm2. ................... 85 44SubbandsplittingUTBSOISGdevicesvsuniaxialcompressivestressatchargedensityp=11013=cm2. .............................. 86 8 PAGE 9 87 46HolemobilityofSDGdevicesunderuniaxialcompressivestressatchargedensityp=11013=cm2. ................................ 88 47HolemobilityenhancementfactorofSDGMOSFETsvsuniaxialcompressivestressatchargedensityp=11013=cm2. ..................... 89 48HolemobilityofFinFETsunderuniaxialstresscomparedwithbulk(110)orienteddevicesatchargedensityp=11013=cm2. .................... 90 49HolemobilityenhancementfactorofFinFETsunderuniaxialcompressivestressatchargedensityp=11013=cm2. ......................... 91 410Holemobilitygaincontributionfromeectivemassandphononscatteringsuppressionunderuniaxialcompressionfor(110)/h110iFinFETscomparedwithSG(110)/h110ipMOSFETsatchargedensityp=11013=cm2. ........ 92 51Germaniumholemobilityvseectiveelectriceld. ................ 97 52Germaniumandsiliconholemobilityunderbiaxialtensilestresswheretheinversionholeconcentrationis11013=cm2. ..................... 98 53Germaniumandsiliconholemobilityunderbiaxialcompressivestresswheretheinversionholeconcentrationis11013=cm2. .................. 99 54Germaniumandsiliconholemobilityon(001)orienteddeviceunderuniaxialcompressivestresswheretheinversionholeconcentrationis11013=cm2. ... 100 55Germaniumandsiliconholemobilityon(110)orienteddeviceunderuniaxialcompressivestresswheretheinversionholeconcentrationis11013=cm2. ... 101 56E{kdiagramsforGeunder(a)nostress;(b)1GPabiaxialtensilestress;(c)1GPabiaxialcompressivestress;and(d)1GPauniaxialcompressivestress. ... 103 57Conductivityeectivemassvsbiaxialtensilestress:(a)Channeldirection(<110>)and(b)outofplanedirection. ............................ 104 58Conductivityeectivemassvsbiaxialcompressivestress:(a)Channeldirection(<110>)and(b)outofplanedirection. ...................... 105 59Conductivityeectivemassvsuniaxialcompressivestress:(a)Channeldirection(<110>)and(b)outofplanedirection. .................... 106 51025meVenergycontoursforunstressedGe:(a)Heavyhole;(b)Lighthole. .... 108 51125meVenergycontoursforbiaxialcompressivestressedGe:(a)Topband;(b)Bottomband. ..................................... 108 9 PAGE 10 ....................................... 109 51325meVenergycontoursforuniaxiallycompressivestressedGe:(a)Topband;(b)Bottomband. ................................... 109 514Gesubbandsplittingunderdierentstress. ..................... 110 515NormalizedgroundstatesubbandEkdiagramvsbiaxialcompressivestress. .. 112 516TwodimensionaldensityofstatesofthegroundstatesubbandforSiandGeat(a)E=5meV;(b)E=2kT=52meVunderuniaxialcompressivestress. ....... 113 517Phononscatteringratevsuniaxialcompressivestress:(a)Acousticphonon,and(b)opticalphonon. ............................... 115 518SurfaceroughnessscatteringratevsuniaxialcompressivestressforGeandSi. 116 519Mobilityenhancementcontributionfromeectivemasschange(solidlines)andphononscatteringratechange(dashedlines)forSiandGeunderuniaxialcompressivestress. .................................... 117 520Conned2Denergycontoursfor(001){orientedGepMOSwithuniaxialcompressivestress. .................................... 117 521Conned2Denergycontoursfor(110){orientedGepMOSwithuniaxialcompressivestress. .................................... 118 A1Stressdistributiononcrystals. ............................ 123 10 PAGE 11 Table page 21LuttingerKohnparameters,deformationpotentialsandsplitoenergyforsiliconandgermanium. ................................. 31 31CalculatedandmeasuredpiezoresistancecoecientsforSipMOSFETswith(001)or(110)surfaceorientation.Therstvalueofeachpairisfrommeasurementsandthesecondisfromcalculation. ...................... 40 A1ElasticstinessesCijinunitsof1011N=m2andcompliancesSijinunitsof1011m2=N 11 PAGE 12 Myresearchexploresthestrainenhancedholemobilityinsilicon(Si)andgermanium(Ge)ptypemetaloxidesemiconductoreldeecttransistors(pMOSFETs).Thepiezoresistancecoecientsarecalculatedandmeasuredviawaferbendingexperiments.Withgoodagreementinthemeasuredandcalculatedsmallstresspiezoresistancecoecients,kpcalculationsareusedtogivephysicalinsightsintoholemobilityenhancementatlargestress(3GPaforSiand6GPaforGe)forstressesoftechnologicalimportance:inplanebiaxialandchanneldirectionuniaxialstresson(001)and(110)surfaceorientedpMOSFETswithh110iandh111ichannels. Themathematicaldenitionofstrainandstressisintroducedandthetransformationbetweenthestrainandstresstensorisdemonstrated.SelfconsistentcalculationofSchr~odingerEquationandPoissonEquationisappliedtostudythepotentialandsubbandenergylevelsintheinversionlayers.Subbandstructures,twodimensional(2D)densityofstates(DOS),holeeectivemass,phononandsurfaceroughnessscatteringrateareevaluatednumericallyandtheholemobilityisobtainedfromalinearizationofBoltzmannEquation. Theresultsshowthatholemobilitysaturatesatlargestress.Underbiaxialtensilestress,theholemobilityisdegradedatsmallstressduetothesubtractivenatureofthestrainandquantumconnementeects.Atlargestress,holemobilityisimprovedviathesuppressionofthephononscattering.Biaxialcompressivestressimprovesholemobility 12 PAGE 13 Straineectsonnonclassicaldevices(singlegate(SG)silicononinsulator(SOI)anddoublegate(DG)pMOSFETs)arealsoinvestigated.ThecalculationshowsthatthemobilityenhancementforSGSOIandDG(001)pMOSFETsissimilartotraditionalSipMOSFETs.HolemobilityenhancementinFinFETsismorethantraditional(110)pMOSFETsduetothesubbandmodulation. 13 PAGE 14 Metaloxidesemiconductoreldeecttransistors(MOSFETs)havebeenscaleddownaggressivelytoachievedensity,speedandpowerimprovementsince1960s[ 1 ].Asthechannellengthisscaledtosubmicronevennanoscalelevel,thesimplescalingofcomplementarymetaloxidesemiconductor(CMOS)devicesbringssevereshortchanneleects(SCEs)suchasthresholdvoltagerollo,degradedsubthresholdslope,anddraininducedbarrierlowering(DIBL).Oxidethicknesshastobereducedtosub10nm(about1nminthestateofthearttechnology)andchanneldopinghastobeincreasedupto1019=cm3inordertomaintaingoodcontrolofthechannel[ 1 ].Thethinoxideandthehighchanneldopingresultinhighverticalelectriceldinthechannelthatseverelyreducesthecarriermobility.Furtherscalingofthedevicesdoesnotbringperformanceimprovementduetocarriermobilitydegradation. WithnothingtoreplacesiliconCMOSdevicesinthenearfutureandtheneedtomaintainperformanceimprovementsandMoore'slaw,featureenhancedSiCMOStechnologyhasbeenrecognizedasthedriverforthemicroelectronicsindustry.StrainisonekeyfeaturetoenhancetheperformanceofSiMOSFETs.BiaxialtensilestrainhasbeeninvestigatedbothexperimentallyandtheoreticallyinCMOStechnology[ 2 3 4 ].Itimprovestheelectronmobility[ 5 ],butdegradestheholemobilityatlowstressrange(<500MPa)[ 3 ].Recently,uniaxialstresshasbeenappliedtoIntel's90,65,and45{nmtechnologiestoimprovethedrivecurrentwithoutsignicantlyincreasedmanufacturingcomplexity[ 5 6 ]. ThegoalofthisdissertationistoprovidephysicalinsightsintothestrainenhancedholemobilityinSiandGepMOSFETs.Beforeweinvestigatetheholemobility,thehistoryofstraintechnologyandthemethodstoapplystraintoatransistorareisdiscussedinthischapter.Theorganizationofthedissertationisalsointroduced. 14 PAGE 15 7 8 ],andthetheoreticalapproachtothestraineect,i.e.,deformationpotentialtheory,canbetracedbacktothe1950s.Butnotuntilintheearly1980'sdidscientistsandengineersstarttorealizethatstraincouldbeapowerfultooltomodifythebandstructureofsemiconductorsinabenecialandpredictableway[ 9 10 ]. Deformationpotentialtheory,whichdenestheconceptofstraininducedenergyshiftofthesemiconductor,wasrstdevelopedtoaccountforthecouplingbetweentheacousticwavesandelectronsinsolidsbyBardeenandShockley[ 11 ],whostatedthatthelocalshiftofenergybandsbytheacousticphononwouldbeproducedbyanequivalentextrinsicstrain,hencetheenergyshiftsbybothintrinsicandextrinsicstraincanbedescribedinthesamedeformationpotentialframework.ThedeformationpotentialtheorywasappliedbyHerringandVogt[ 12 ]in1955intheirtransportstudiesofsemiconductorconductionbands.Asetofsymbols,,wasusedtolabelthedeformationpotentials.HerringandVogt[ 12 ]alsosummarizedtheindependentdeformationpotentialsconstrainedbysymmetryatdierentconductionbandvalleys.Atthepoint,anothersetofsymbolsarecommonlyused:ac;av;b;andd,whereav;b;anddarethreeindependentvalencebanddeformationpotentialswhichhaveacorrespondencetotheLuttingerparameters[ 13 ]employedinbandcalculations.Thekpmethodweuseinthisworkreliesonthesethreedeformationpotentialstoaccountforthestraineects. Smithmeasuredthepiezoresistancecoecientsforn{andp{typestrainedbulksiliconandgermaniumin1954[ 7 ].Thiswastherstexperimentalworkthatstudiedstraineectsonsemiconductortransport.HerringandVogtusedShockley'sbandmodelandascribedtheelectronmobilitychangetotwostraineects,\electrontransfereect"andintervalleyscatteringratechangecausedbyvalleyenergyshift[ 12 ].This 15 PAGE 16 Piezoresistancecoecientsarewidelyusedintheindustryduetoitssimplicityinrepresentingthesemiconductortransportproperties(mobility,resistance,andetal:)understrain.Itisdenedastherelativeresistancechangewiththestressappliedonthesemiconductor.Piezoresistancecoecient()canbeexpressedas whereistheappliedstressandistheresistivityofthesemiconductor. In1968,Colman[ 8 ]measuredthepiezoresistancecoecientsinp{typeinversionlayers.Thiswasthersttimethatstraineectonholetransportwasinvestigatedintheinversionlayers.Thesimilarityanddierenceofthepiezoresistancecoecientscomparedwiththebulksiliconwasexplainedqualitativelyinthatwork. Therstsiliconn{channelMOSFETwhichusedbiaxialstresstoimprovetheelectronmobilitywasdemonstratedbyWelseretal:[ 14 ]in1992.Theworkshowedthattheelectronmobilitywasimprovedby2.2times.Abiaxialstressedsiliconp{channelMOSFETwasrstreportedbyNayaketal:[ 15 ]in1993wheretheholemobilitywasenhancedby1.5times.In1995,Rim[ 16 ]showedtheholemobilityenhancementinsiliconpMOSFETsontopofSi1xGexsubstratewithdierentgermaniumcomponents.TheideaofusinglongitudinaluniaxialstresstoimprovetheperformanceofMOSFETswasactivatedbyItoetal:[ 17 ]andShimizuetal:[ 18 ]inthelate1990'sthroughtheinvestigationsofintroducinghighstresscappinglayersdepositedonMOSFETstoinducechannelstress.Gannavarametal:[ 19 ]proposedSi1xGexinthesourceanddrainregionforhigherboronactivationandreducedexternalresistancewhichalsofurnishedatechnicallyconvenientmeanstoemployuniaxialchannelstress.ThesestudiesopenedthegatetousestrainasactivefactorinVLSIdevicedesignandresultedinextensiveindustrialapplications. 16 PAGE 17 For(001)wafer,biaxialtensilestressinSiMOSFETsisappliedtothechannelbyusingtheSi1xGexsubstrate.Thelatticemismatchstretchessiliconatomsinbothh100iandh010idirectionswhichisillustratedinFigure 11 .Thepercentageofgermaniumcontentinthesubstratedeterminesthemagnitudeofthestrain.Thisinplanetensilestraincanalsobeachievedbyapplyinguniaxialcompressivestressfromtheoutofplanedirection[ 20 ]withcappinglayer.TheoutofplaneuniaxialcompressionisequivalenttotheinplanebiaxialtensionindeterminingthetransportpropertiesofSi.Thedetailsareshownintheappendix.ForGeMOSFETs,biaxialtensilestressisnotapplicableduetoitslargelatticeconstant.BiaxialcompressivestressisusuallyintroducedbyapplyingSiorSi1xGexsubstrate. Figure11. SchematicdiagramofbiaxialtensilestressedSiMOSFETonrelaxedSi1xGexlayer Uniaxialstresscanbeappliedfromoutofplane,inplanelongitudinal(paralleltothechannel),orinplanetransverse(perpendiculartothechannel)direction.Theinplanelongitudinalstressisappliedtothechannelbyeitherdopinggermaniumtosourceand 17 PAGE 18 12 [ 20 ]. Figure12. UniaxialstressedSiMOSFETwithSi1xGexSource/Drainorhighlystressedcappinglayer Withoutfurtherclarication,uniaxialstressinthisworkrepresentsinplaneuniaxiallongitudinalstress.Itisnormallyalongh110isinceitistheclassicalchanneldirection.Biaxialstressmeansinplanebiaxialstress.For(110){orientedwafer,biaxialstressisemployedinbothparallelandperpendiculardirectiontothechannel(<110>{and<100>{directions).Thestraininthosetwodirectionsarenotassameas(001){orientedwafer(<100>{and<010>{directions)duetothedierentYoung'sModulusin<110>{and<100>{directions. 18 PAGE 19 Straininducedholemobilityenhancementisstudiedtheoreticallyforthersttimeinultrathinbody(UTB)nonclassicalpMOSFETs,includingsinglegate(SG)silicononinsulator(SOI),(001)symmetricaldoublegate(SDG)pMOSFETs,and(110)ptypeFinFETs.ForSGSOIpMOSFETs,thestraineectsareassameastraditionalSipMOSFETs.For(001)SDGpMOSFETsand(110)FinFETs,subbandmodulationisfoundwhenthechannelthicknessissmallerthan20nm.Asthestressincreases,themobilityenhancementin(001)SDGpMOSFETsiscomparabletotraditionalSGpMOSFETs.ForFinFETs,theformfactorsaremuchsmallerthanSG(110)pMOSFETsandthechangewithstressislargerwhichsuggestsmorereductionofthephononscatteringrate.Therefore,thestraininducedholemobilityenhancement(200%)islargerthansinglegate(110)pMOSFETs(100%). 19 PAGE 20 StrainenhancedholemobilityinsinglegateSipMOSFETsisthendiscussed.TheunstrainedSiholemobilityversusdevicesurfaceorientationandverticalelectriceldiscalculated.Holemobilityunderbiaxialanduniaxialstressin(001)and(110)pMOSFETsisshowed.Thebandstructureofbulksiliconunderstrainisdiscussed.IntheSiinversionlayers,theconnedenergycontours,subbandsplitting,holepopulationingroundstatesubband,twodimensional(2D)densityofstates(DOS),phononandsurfaceroughnessscatteringrateareevaluated.Thedierenceofstraininducedholemobilityenhancementin(001)and(110)pMOSFETsunderbiaxialanduniaxialstressisexplained. Uniaxialstrain{inducedholemobilityenhancementiscalculatedforUTBnonclassicalpMOSFETs,includingsinglegateSOI,(001)SDGpMOSFETs,and(110)ptypeFinFETs.ThesimilarityanddierencefromthetraditionalSipMOSFETsarediscussedandphysicalinsightsaregiven. StraininducedholemobilityenhancementinGepMOSFETsisdiscussed.Unstrainedholemobilityin(001)and(110)GepMOSFETsiscalculated.StraineectonholemobilityinSi1xGexwitharbitraryGecomponentsisevaluated.Tounderstandthephysics,thebulkvalencebandstructureandholeeectivemasswithstraineectsarecalculated.Intheinversionlayers,thesubbandstructure,2DDOSandscatteringratearecalculatedandtheirrelationtoholemobilityisanalyzed. WeconcludewiththeresultsthatweobtaininthisdissertationandsuggestpossiblefutureresearchonstrainedSiandGe. 20 PAGE 21 Globaldescriptionsofthedispersionrelationsofbulkmaterialscanbeobtainedviapseudopotentialortightbindingmethods[ 21 ].However,suchglobalsolutionoverthewholeBrillouinzoneisunnecessaryformanyaspectsofsemiconductorelectronicproperties.Whatisneededistheknowledgeofthedispersionrelationsoverasmallkaroundthebandextrema[ 21 ].kpmethodiswidelyusedinnowadaysquantumwellandquantumdotscalculationduetoitssimplicityandaccuracyregardingthepropertiesinthevicinityofconductionbandandvalencebandedgeswhichgovernmostopticalandelectronicphenomena. Tostudytheuniaxialorbiaxialstraineectonholemobilityintheinversionlayers,a6bandkpmodel,Luttinger{Kohn'sHamiltonian[ 13 ],isutilizedinthiswork.Inthischapter,kpmethodandthederivationoftheluttinger{Kohn'sHamiltonianisintroducedrst.Thentheprocedurecalculatingtheholemobilityisexplained.Finally,theevaluationofscatteringmechanisms,mainlythephononandsurfaceroughnessscattering,isdiscussed. Inthecalculationoftheholemobilitywithstraineectintheinversionlayers,SchrodingerEquationandPoissonEquationaresolvedselfconsistentlytosimulatethepotentialenergyinthechannel.Thesubbandstructureandthetwodimensionaldensityofstates(2DDOS)ofeachsubbandarecalculatedandthescatteringrelaxationtimeisevaluatedinkspace.Finally,holemobilityisobtainedfromalinearizationoftheBoltzmannequation. 2.1.1IntroductiontokpMethod 21 22 23 ]isessentiallybasedontheperturbationtheoryandwasrstintroducedbyBardeen[ 24 ]andSeitz[ 25 ].Itisalsoreferredtoaseectivemasstheoryinliteratures.Thekpmethodismostusefulforanalyzingthebandstructure 21 PAGE 22 Foranelectroninaperiodicpotential whereR=n1a1+n2a2+n3a3,anda1,a2,a3arethelatticevectors,andn1,n2,andn3areintegers,theelectronwavefunctioncanbedescribedbytheSchrodingerequation (2{2) wherep=hr=iisthemomentumoperator,m0isthefreeelectronmass,andV(r)representsthepotentialincludingtheeectivelatticeperiodicpotentialcausedbythenuclei,ionsandcoreelectronsorthepotentialduetotheexchangecorrelation,impurities,etc. ThesolutionoftheSchrodingerequation (2{3) satisesthecondition (2{4) (2{5) where andkisthewavevector.Equations 2{4 2{5 and 2{6 istheBlochtheorem,whichgivesthepropertiesofthewavefunctionofanelectroninaperiodicpotentialV(r). TheeigenvaluesforEquation 2{3 canbecategorizedintoaseriesofbandsEn;n=1;2;:::[ 26 ]duetotheperturbationoftheperiodicpotentialattheBrillouinzoneedge. 22 PAGE 23 InsertingtheBlochfunctionEquation 2{5 intoEquation 2{7 ,wehave m0kp+V(r)#unk(r)=En(k)unk(r): Includingthespinorbitinteractionterm h intheHamiltonianandsimplifyingtheequation,Equation 2{8 becomes =En(k)unk(r): wherecisthespeedoflightandisthePaulispinmatrix.hasthecomponents[ 22 ] RewritingtheHamiltonianinEquation 2.1.1 ,wehave [H0+W(k)]unk=Enkunk; where (2{13) and SinceonlyW(k)dependsonwavevectork,Equation 2{13 canbeusedtoevaluatethebandpropertyatk0.IftheHamiltonianH0hasacompletesetoforthonormal 23 PAGE 24 theoreticallyanyfunctionwithlatticeperiodicitycanbeexpandedusingeigenfunctionsun0.Substitutingtheexpression intoEquation 2.1.1 ,multiplyingfromtheleftbyun0,integratingandusingtheorthonormalityofthebasisfunctions,wehave Solvingthismatrixequationgivesusboththeexacteigenstatesandeigenenergies.Aswementionedearlier,onlythedispersionrelationsoverasmallkrangearoundthebandextremaareimportantdescribingtheelectronicpropertiesofthesemiconductor.Onlyenergeticallyadjacentbandsarenormallyconsideredwhenstudyingthekexpansionofonespecicbandforsimpleness.Topursueacceptablesolutionswhenkincreases,onehastoincreasethenumberofthebasisstates,orconsiderhigherorderperturbations,orevenboth. NeglectingthenondiagonaltermsinEquation 2{17 forsmallk,theeigenfunctionisunk=un0,andthecorrespondingeigenvalueisgivenbyEnk=En0+h2k2 where 24 PAGE 25 (2{20) thesecondordereigenenergiescanbewrittenas Equation 2{21 canalsobeexpressedas where 1 2{23 istheeectivemasstensor,and;=x;y;z.Theeectivemassgenerallyisanisotropicandk{dependent.Inthevicinityofthepoint,sometimesmcanbetreatedaskindependent,sinceatthislevelofapproximation,theeigenenergiesclosetothepointonlydependquadraticallyonk[ 22 23 ]. 2{17 givesexactsolutionsofbotheigenenergiesandeigenfunctions.Inreality,itisalmostimpossibletoincludeacompletesetofbasisstates,thereforeonlystronglycoupledbandsareincludedinusualkpformalism,andtheinuenceoftheenergeticallydistantbandsistreatedasperturbation. InKane'smodelforSi,Ge,orIIIVsemiconductors,fourbandsareconsideredasstronglycouplesbands{theconduction,heavyhole(HH),lighthole(LH),andtespinorbitsplito(SO)bandsareconsidered,whichhavedoubledegeneracywiththeirspincounterparts.Therestbandsaretreatedasperturbationandcanbeanalyzedwiththesecondorderperturbationtheory. 25 PAGE 26 2.1.1 witheigenfunction (2{24) Thebandedgefunctionsun0(r)are Conductionband:jS"i,jS#iforeigenenergyEs(stype), Valenceband:jX"i,jY"i,jZ"i,jX#i,jY#i,jZ#iforeigenenergyEp(ptype). Normallythefollowingeightbasisfunctionsarechosen 2;1 2i=jS"i=jS"i;u2=j3 2;3 2i=jHH"i=1 2;1 2i=jLH"i=1 3jZ"i;u4=j1 2;1 2i=jSO"i=1 2;1 2i=jS#i=jS#i;u6=j3 2;3 2i=jHH#i=1 2;1 2i=jLH#i=1 3Z#i;u8=j1 2;1 2i=jSO#i=1 ThissetofbasisstatesisaunitarytransformationofthebasisfunctionsandtheeigenfunctionsoftheHamiltonian 2{13 .TheeigenenergiesforjSi,jHHi,jLHiandjSOiatk=0areEg,0,0,,respectively,whereEgisthebandgap,andtheenergyofthetop 26 PAGE 27 Atthislevelofapproximation,thebandsarestillatbecausetheHamiltonian 2{13 iskindependent.IncludingW(k)inEquation 2{14 intotheHamiltonian,anddeningKane'sparameteras m0hSjzjZi; weobtainamatrixexpressionfortheHamiltonianH=H0+W(k),i.e., 3Pk00q 3Pkz1 3Pkz000q 3Pk+00+h2k2 3Pkz1 3Pk+0000Pk+h2k2 3Pkz0001 3Pk00+h2k2 wherek+=kx+iky,k=kxiky,andkx,ky,kzarethecartesiancomponentsofk.TheHamiltonian 2{27 iseasytodiagonalizetondtheeigenenergiesandeigenstatesasfunctionsofk.Wehaveeighteigenenergies,butduetospindegeneracy,thereareonlyfourdierenteigenenergieslistedbelow.Fortheconductionband, Forthelightholeandsplitobands, (2{29) 27 PAGE 28 Theseresultsarenotcompletesincetheeectsofhigherbandshavenotbeenincluded.TheywillbetakenintoaccountnextwhendiscussingtheLuttingerKohnmodel. 27 ]wherethesixvalencebandsaretreatedinclassAandtherestbandsareputinclassB. WelabelclassAwithsubscriptnandclassBwithsubscript.Wavefunctionuk(r)canbeexpandedas ChoosetheeigenstatesforclassA,wehave 2;3 2i=jHH"i=1 2;1 2i=jLH"i=1 3jZ"i;u3=j3 2;1 2i=jLH#i=1 3Z#i;u4=j3 2;3 2i=jHH#i=1 2;1 2i=jSO"i=1 2;1 2i=jSO#i=1 WithLowdin'smethodweonlyneedtosolvetheeigenequation 28 PAGE 29 (2{34) where (2{36) m0kju0i=Xhk m0pj(j2A;62A) (2{37) LetUAjn=Djn,Djncanbeexpressedas whereDjnisdenedas (2{39) ToexpressDjnexplicitly,wedine ThendenetheLuttingerparameters1,2,and3as 29 PAGE 30 3(a0+2B0)h2 6(a0B0)h2 (2{41) FinallyweobtaintheLuttingerKohnHamiltonian 2SR+0PQSq 2S+p 2Sp 2S+p 2;3 2ij3 2;1 2ij1 2;1 2ij3 2;1 2ij3 2;3 2ij1 2;1 2ij1 2;1 2i where, Whenstrainispresentinthesemiconductor,P,Q,R,andSinEquation 2{43 canberesolvedtotwoparts:kpterms(Pk,Qk,Rk,andSk)andstrainterms(P,Q,R,andS).Theycanbeexpressedas[ 13 ] 30 PAGE 31 2b(xxyy)idxy;S=d(zxiyz); 28 ]. Table21. LuttingerKohnparameters,deformationpotentialsandsplitoenergyforsiliconandgermanium. Si4.220.391.442.462.355.30.044 Ge13.354.255.692.092.555.30.296 31 PAGE 32 2.2.1SelfconsistentProcedure @zinEquation 2{45 .Coordinatesystemtransformationisneededtocalculatecaseswithothersurfaceorientation. SubbandenergycanbeevaluatedbysolvingSchrodingerEquation, [H(k;z)+V(z)]k(z)=E(k)k(z) (2{45) whereV(z)denesthepotentialenergyinthequantumwell.Triangularpotentialapproximationiswidelyusedinsimulationsforsimplicity.Stern[ 29 ]statedthatitshouldnotbeusedwhenmobilechargesarepresent.Inordertoaccuratelysimulatethepotentialinthequantumwell,SchrodingerEquation 2{45 issolvedselfconsistentlywithPoissonEquation (2{46) wherep(z)andn(z)aremobileholeandelectrondensity,N+D(z)andNA(z)arespacechargedensity. TonumericallyevaluateSchrodingerEquationandPoissonEquation,FiniteDierenceMethodisutilized.TheequationsareevaluatedonazmeshofNzpointsintheinterval(0;zmax)[ 3 30 31 ],wherezmaxhereisthesumofthethicknessofsiliconlayerandoxidelayer.Thisyieldsa6Nz6Nzeigenvalueproblemofthetridiagonalblockform[ 3 ].SchrodingerEquationbecomes 32 PAGE 33 whereeachi=(zi)isasixcomponentcolumnvectorj(zi),theindexjrunningoverthekpbasis,and^H,^Hi,^H+=^H+are66blockdiagonaldierenceoperators,functionsoftheinplanewavevectork. Inprinciple,thepotentialV(z)resultsfromthreeterms:animageterm,vimg(z);anexchangeandcorrelationpotential,Vxc(z);andtheHartreeterm,VH(z)[ 3 30 ].Fischetti[ 3 ]suggeststhattheimagepotentialcancelsthemanybodycorrectionsgivenbytheexchangeandcorrelationtermandtheHartreetermisfocussedasthesolutionoftheselfconsistentcalculationofSchrodingerEquation 2{45 andPoissonEquation 2{46 3 ] (2{48) whereps=Ppisthetotalholeconcentrationintheinversionlayer,pistheholedensityofsubband,()x(K;)isthexcomponentofthemomentumrelaxationtimeinsubband,and 1+expEEF 33 PAGE 34 Theevaluationofdensityofstates(DOS)and@E @ktermneedfurtherconsideration.InenergyspaceamaximumkineticenergyEmaxforeachsubbandisselectedinordertoaccountcorrectlyforthethermaloccupationofthetopmostsubband.Inourcalculation,weassumedEmax=120meVanddividedtheenergyspaceto1200uniformparts,thenevaluatedDOSand@E @kineachpart. 3 ],hencetheyareneglectedinourcalculation.Onlyphononscatteringandsurfaceroughnessscatteringareinvestigated. 32 ].Vibrationoftheionsabouttheirequilibriumpositionsintroducesinteractionbetweenelectronsandtheions.Thisinteractioninducestransitionsbetweendierentstates.Andthisprocessiscalledphononscattering. Phononscatteringcanbecategorizedtoacousticphononscatteringandopticalphononscatteringbasedonthephaseofthevibrationofthe2dierentatomsinoneprimitivecell.Bothcontributetothemomentumrelaxationtime.Acousticphononenergyisnegligiblecomparedwithcarrierenergy,whileopticalphononenergyisabout61.3meVforsiliconand37meVforgermaniumatlongwavelengthlimit.Whenstrainisappliedtothecrystal,theHHandLHdegeneracyisliftedatpoint,aswementionedpreviously.Therefore,theinterbandopticalphononscatteringwillbelimitedduetobandsplittingandmobilityisenhanced.Infact,thisisonlysignicantwhenstrainishighandthebandsplittingisbeyondtheopticalphononenergy.Thereasoningwillbeshowninthefollowingsection.Oneshouldalsonoticethattheanisotropicnatureof 34 PAGE 35 32 ]isusedwherewereplacetheanisotropicholephononmatrixelementwithappropriateangleaveragedquantities. Firstforacousticphonon,relaxationtimecanbeexpressedas[ 3 ] 1 (2{50) whereeff=7:18eVistheeectiveacousticdeformationpotentialofthevalenceband,isthe2dimensionaldensityofstatesofsubbandwhichisdenedas (2)2Z20dK(E;) Thetwodimensionalcarrierscatteringrateforthephononassistedtransitionsofacarrierfromaninitialstateinthethsubbandandanalstateinthethsubbandisproportionaltotheformfactor 2W=1 2Z+1jI(qz)j2dqz; where TheformfactorFillustratestheinteractionbetweeninitialstateandnalstateduetothewavefunctionoverlapping.where()k(z)or()k(z)istheenvolopefunctionatkforsubbandor,respectively.zisthecoordinateperpendiculartotheSi=SiO2interface,andqzisthechangeinthecomponentperpenticulartotheinterfacesofthecarriermomentuminatransitionfromthethsubbandtothethsubband. FollowingPrice'spioneeringwork,Wcanbeexpressedas 35 PAGE 36 Ifthenalstateisalsothsubband,Wrepresentstheeectivequantumwellwidthforthethsubband. SincetheacousticphononenergyissmallcomparedwithsubbandsplittingoreventhethermalenergykT,acousticphononscatteringisanequalenergyscatteringprocess[ 32 ].Thescatteringratesolelydependsonthedensityofstatesofthenalstates.Straineectonacousticphononscatteringissmallerthanthatonopticalphononscatteringwhichisshowninoursimulation. Second,theopticalphononscatteringrelaxationtimeisexpressedas[ 3 ] 1 1f0[E(K)]nop+1 21 2 Forabsorptionandemission,respectively,whereDop=13:24108eV=cmistheopticaldeformationpotentialconstantofthevalenceband,h!op=61:3eVisthesiliconopticalphononenergy.Opticalphononscatteringisnotsignicantlyreducedforstress<1GPasincethesubbandsplittingislessthantheopticalphononenergy. 36 PAGE 37 33 3 ].TheearlyformulationbyPrangeandNee,Saitoh,andAndoisstillthebestmodelavailable[ 3 ].Dierentroughnessparametersareusedindierentreferences.Here,we'lluseGamiz'modelandcorrespondingparameters[ 34 ]. Asweknow,thesurfaceroughnessscatteringiscausedbytheroughnessofthesurfaceandhencetheabruptpotentialchangeatSi=SiO2interface.2assumptionsareneededinthesimplicationoftheproblem[ 34 ].Therstassumptionistoconsidertheinterfacebetweensiliconandoxideisanabruptboundarywhichrandomlyvariesaccordingtoafunctionoftheparallelcoordinate,r,(r).AnotherassumptionisthatthepotentialV(z)closetotheinterfacecanbeexpressedby Thescatteringratecanbeexpressedas[ 34 ], 1 m(z)dz2mL2Z20d InthisequationVm(z) misapproximatelyequaltotheeectiveelectriceld,whichmeansthescatteringrateisproportionaltothesquareoftheelectriceld.Therefore,surfaceroughnessscatteringbecomesmoresignicantwhenelectriceldreacheshigherlevel. DierentvaluesforLandaretakenbydierentresearcherstoexplaintheexperimentaldata.Here,weuseL=20:4nmand=4nm[ 3 ]forsiliconassuggestedbyFischetti.n=0:5[ 34 ]ischoseninthiswork. 37 PAGE 38 Inthiswork,MATLABandCcodesarewrittentocalculatetheholeeectivemass,bandandsubbandstructures,andholemobilityinSiandGepMOSFETs.Tocalculateholemobilitydependenceondevicesurfaceorientation,coordinatetransformationisperformedtocalculateholemobilityin(110),(111),and(112)orientedSiandGetoaccountforthedierentquantumconnementconditions.Fordierentsurfaces,dierentsurfaceroughnessparametersareutilizedtottheinterfaceroughnesscondition.DOSandformfactorsarecalculatedinthewholekspace. 38 PAGE 39 HoletransportintheinversionlayerofsiliconpMOSFETsunderarbitrarystressanddevicesurfaceorientationisdiscussedinthischapter.Piezoresistancecoecientsarecalculatedandmeasuredatstressupto300MPaviawaferbendingexperimentsforstressesoftechnologicalimportance:uniaxialcompressiveandbiaxialtensilestresson(001)and(110)surfaceorienteddevices.Withgoodagreementinthemeasuredvscalculatedlowstresspiezoresistancecoecients,kpcalculationareusedtogiveinsightathighstress(1{3GPa).Theresultsshowthatbiaxialtensilestressdegradestheholemobilityatlowstressduetothequantumconnementosettingthestraineect.Uniaxialstresson(001)/<110>,(110)/<110>,and(110)/<111>devicesimprovestheholemobilitymonotonically.Unstressed(110)orienteddeviceshavesuperiormobilityover(001)orienteddevicesduetothestrongquantumconnementcausingsmallerconductivityeectivemassoftheholes.Whenthestressispresent,theconnementof(110)orienteddevicesunderminesthestresseect,hencetheenhancementfactorfor(110)orienteddevicesislessthan(001)orienteddevices.Holemobilityenhancementsaturatesasthestressincreases.Athighstress,themaximumholemobilityfor(001)/<110>,(110)/<110>,and(110)/<111>devicesiscomparable. Physicalinsightsaregiventoexplainthedierencebetweenbiaxialanduniaxialstress,andthedierenceof(110)and(001)pMOSFETs.Thebulksiliconvalencebandstructureunderuniaxialcompressiveorbiaxialtensilestrainisshownandthedierenceineectivemasschangeiscalculated.Thedierenceoftheverticalelectriceld(quantumconenment)eecton(001)and(110)orientedpMOSFETsisexplained.Subbandsplitting,groundstatesubbandholepopulation,andtwodimensional(2D)densityofstates(DOS)ofsubbandsarecalculatedunderstress.Scatteringratechangewithstressisalsodiscussed. 39 PAGE 40 7 8 ].Table 31 comparedmeasuredandcalculatedpiezoresistancecoecients.Inthemeasurements,thestressisappliedusing4pointorconcentricringbendingofthewafers.Thepiezoresistancecoecientsareobtainedthroughthelinearregressionofthemeasuredresistanceversusstress.Theactualstraininthedevicesismeasuredthroughtheresistancechangeofastraingaugemountedonthesample,andviathelaserdetectedcurvaturechangeofbentwafer.InTable 31 ,L,T,andBiaxialrepresentlongitudinal,transverse,andbiaxialpiezoresistancecoecients,respectively. Table31. CalculatedandmeasuredpiezoresistancecoecientsforSipMOSFETswith(001)or(110)surfaceorientation.Therstvalueofeachpairisfrommeasurementsandthesecondisfromcalculation. Substrate(001)(110) Channel<110><110><110> L71.7[ Bothmeasuredandcalculatedresultsintable 31 showthatunderuniaxiallongitudinalstress,(110)/<111>deviceshavethelargetpiezoresistancecoecient,followedbythe(001)/<110>devices.Thepiezoresistancecoecientof(110)/<110>devicesisthelowest.Underuniaxialtransversestress,thepiezoresistancecoecientsaresmallerthanlongitudialstressforallpMOSFETs.Thetablealsoshowsthatthebiaxialtensilestrainincreasesthechannelresistanceandhencedegradestheholemobilityatlowstress. 40 PAGE 41 36 ]reportedthatforptypedeviceswith<110>channel,themobilityisthehighestin(110)orientedandlowestin(001)orientedpMOSFETs.TheholemobilityonafewsurfaceorientationsissimulatedandcomparedwithSato'sexperimentalresults[ 36 37 ]inFigure 31 .Goodagreementisfoundbetweenthecalculationandtheexperimentaldata.Twodierentsurfaceroughnessmodels[ 3 34 ]areusedinthecalculation.Bothmodelsarequiteaccurateandinthefollowingresults,Gamiz'surfaceroughnessmodelisutilized. Figure31. Holemobilityvsdevicesurfaceorientationforrelaxedsiliconwith<110>channel.Theholemobilityishigheston(110)andloweston(001)devices.Dierentsurfaceroughnessscatteringmodelsareusedinthesimulation(solid:Gamiz1999;dotted:Fischetti2003). 41 PAGE 42 38 39 40 ]inFigure 32 .Theagreementbetweenthecalculationandtheexperimentalresultssuggeststhisworkusereasonablescatteringmechanisms.Normally(110){Sihassmootherinterfacewiththegatedielectricmaterials[ 41 42 ],hencethesurfaceroughnessscatteringrateislowerthan(001){orienteddevices.Lee[ 43 ]evensuggestedthattheeectiveeldin(110){orienteddevicesissmallerthan(001){orienteddevices,whichalsoindicatessmallersurfaceroughnessscatteringrateconsideringthatthescatteringrateisinverselyproportionaltotheeectiveelectriceld[ 3 34 ].Thesmallersurfaceroughnessscatteringrateispartlyresponsibleforthehigherholemobilityonunstressed(110){orienteddeicesthanthatofthe(001){orienteddevices.Tottheappropriatesurfaceroughnesscondition,theroughnessparametersusedareL=2:6nm;=0:4nmfor(001){orientedpMOSFETsandL=1:03nm;=0:27nmfor(110){orientedpMOSFETsinthiswork.Thesamesurfaceroughnessscatteringmodelisutilizedinthemobilitycalculationevenwhenthestrainispresent,assumingthattheprocessinducedstrain(uniaxialstrain)doesnotchangetheSi=SiO2interfaceproperties[ 3 44 ]. 33 showstheholemobilityversus(upto3GPa)stressatinversionchargedensitypinv=11013=cm2andchanneldopingdensityND=11017=cm3)for(001)/<110>,(110)/<110>,and(110)/<111>pMOSFETs.Uniaxialcompressivestressimprovestheholemobilitymonotonicallyasthestressincreases.Theholemobilityenhancementsaturatesatlargestress(3GPa).Underuniaxiallongitudinalcompressivestress,themaximumholemobilityenhancementfactoris350%for(001)/<110>pMOSFETs,150%for(110)/<111>pMOSFETs,and100%for(110)/<110>pMOSFETs.At3GPauniaxialstress,(001)and(110)pMOSFETshavecomparablehole 42 PAGE 43 Holemobilityvsinversionchargedensityforrelaxedsilicon.Bothmeasurementsandsimulationshowlargermobilityon(110)devices. 43 PAGE 44 Figure33. Holemobilityvsstresswithinversionchargedensity11013=cm2.Theenhancementfactoristhehighestfor(001)/<110>devicesandlowestfor(110)/<110>devices.Athighstress(3GPa),threeuniaxialstresscaseshavesimilarholemobility. Calculatedstraininducedholemobilityenhancementfactorof(001){orientedpMOSFETsisshowninFigure 34 comparingwithexperimentaldata[ 45 46 47 48 49 5 50 51 ].Goodagreementisfoundbetweenthecalculatedandmeasureddata. InFigure 33 and 34 ,thechanneldopingdensityissettobe11017=cm3inthecalculation.Theinversionchargedensityis11013=cm2.Incontemporarytechnology,theactualchanneldopingisupto11019=cm3.Themobilityenhancementfactoriscalculatedwithdierentchanneldopingdensityatinversionchargedensityof11013=cm2inFigure 35 .Theenhancementfactorsaresimilarforallthreedopinglevels.Forsimplicity,therestoftheworkwillusechanneldoping11017=cm3. 44 PAGE 45 Calculatedstraininducedholemobilityenhancementfactorvs.experimentaldatafor(001){orientedpMOS. Figure35. Holemobilityenhancementfactorvsuniaxialstressfordierentchanneldoping. 45 PAGE 46 36 comparestheholemobilityenhancementfactorfordierentinversionchargedensity.Thegureshowsthattheenhancementfactordecreasesastheinversionchargedensityincreases.Thisisbecausewithmoreinversioncharge,holesarepopulatedtothehigherenergylevelsinthevalenceband,whilethestressonlyaectsthevicinitiesofpoint.Thiscausestheaveragechangeoftheholeeectivemassdecrease.Moreinversionchargesincreasestheelectriceldinthechannelwhichunderminesthestraineect.Thedetailwillbeaddressedlaterinthischapter. Withthestraininducedholemobilitychangeasweshowedhere,physicalinsightsofthedierenceofbiaxialanduniaxialstress,andthedierencebetween(001){and(110){orientedpMOSFETsisgiveninthenextsections.Straininducedsiliconvalencebandstructurechange,subbandstructurecausedbythetransverseelectriceld,andtheholeeectivemassandscatteringratechangewiththestrainareanalyzed. 46 PAGE 47 (b) Calculatedstraininducedholemobilityenhancementfactorvs.stressfor(001){orientedpMOSwithdierentinversionchargedensity. 47 PAGE 48 m whereisthecarriermomentumrelaxationtimethatisinverselyproportionaltoscatteringrateandmisthecarrierconductivityeectivemass.Insiliconinversionlayers,carriersareconnedinapotentialsuchthattheirmotioninonedirection(perpendiculartothesiliconoxideinterface)isrestrictedandtheelectronicbehaviorofthesecarriersistypicallytwodimensional(2D).Themobilityofthe2Dholegasisdierentfromthe3Dholesinbulksilicon.Butthesimplicityofthebulkbandstructurecalculationcangiveusinsightstohowtheeectivemassesoftheholeschangewiththestressandhelpunderstandhowthequantumconnementmodiesthesubbandpositionandsplittingwhichisimportantto2Dholemobility.Therefore,bulkvalencebandstructureisdiscussedinthissectionbeforewemovetotheSipMOSFETs. 37 .Fortheunstressedsilicon,theHeavyhole(HH)andLighthole(LH)bandsaredegenerateatpoint.Thisis4folddegeneracytakingintoaccountthespin.TheSpinorbitalSplito(SO)bandis44meVbelowHHandLHbands.Whenstressisapplied,thedegeneracyofHHandLHbandsisliftedasshowningure 37 (b)and(c).Thesetwobandsarealsoreferredtoasthetopandthesecondbandindicatingthesplitenergylevels.Thebandsplittingresultsinthebandwarpingwhichchangestheeectivemassoftheholes.Inthemeantime,thesplittingcausestherepopulationoftheholesinthesystem.Whenthestressislargeandthesplittingishigh,mostholeswilllocateinthetopbandbasedonFermiDiracdistributionfunctionaslongasthedensityofstates(DOS)ofthetopmostbandisnot 48 PAGE 49 Figure 37 showsthatthestressonlyaectthebandpropertyclosetopoint.Theguresshowthatawayfromthezonecenter,thebandstructureisalmostidenticaltotheunstressedsilicon.Figure 38 illustratesthatmoreregionaroundthezonecenterandmorecarriersareaectedbythestresswhenthestressincreases.Therefore,thestraineectcannotbeexplainedonlybythepropertiesatthepoint.Instead,thestatisticsofthewholesystemshouldbeconsidered.Figure 38 suggeststhatasthestressincreasesfrom500MPato1.5GPa,thebandwarpingandeectivemassatpointchangeverylittle.Thenextsubsectionwillalsoshowthis.Intheprocess,moreholesareaectedbythestress,thereforetheaverageholebehaviorswillstillchange.WeshowedinFigure 36 thatthemobilityenhancementfactordecreasesastheamountofinversionchargesincreases.Thiscanbeunderstoodasfollows.Fordeviceswithmoreinversioncharges,moreholesoccupythehigherenergystateswhentheinversionchargedensityincreases.Atthesamestress,theaveragechangeinducedbystressissmallerthanthecaseswithfewerinversioncharges. 39 310 and 311 .Figure 39 showsthe<110>{directioneectivemasses,gure 310 showsthe2dimensionaldensityofstateeectivemasses,andtheoutofplane<001>{directioneectivemassesareillustratedingure 311 Figure 39 310 and 311 alsosuggestthatwithstrain,theHHandLHbandsarenolonger\pure"HHorLHanymoreduetostrongcouplingofthewavefunctions.Thepropertyofeachbanddependsheavilyonthecrystalorientation.AsinglebandcanbeHHlikealongonedirection,butLHlikealonganother.Ingeneral,ifthecrystalshowscompressivestrainalongonedirection,thetopbandisLHlikealongthisspecic 49 PAGE 50 (b) (c) Figure37. Ekrelationforsiliconunder(a)nostress;(b)1GPabiaxialtensilestress;and(c)1GPauniaxialcompressivestress. 50 PAGE 51 (b) NormalizedEkdiagramofthetopbandunderdierentamountofstress.Largerstresswarpsmoreregionoftheband.Theenergyatpointforallcurvesissettozeroonlyforcomparisonpurpose. 51 PAGE 52 (b) Channeldirectioneectivemassesforbulksiliconunder(a)biaxialtensilestress;and(b)uniaxialcompressivestress. 52 PAGE 53 (b) Twodimensionaldensityofstateseectivemassesforbulksiliconunder(a)biaxialtensilestress;and(b)uniaxialcompressivestress. 53 PAGE 54 (b) Outofplaneeectivemassesforbulksiliconunder(a)biaxialtensilestress;and(b)uniaxialcompressivestress. 54 PAGE 55 39 .Alongz{direction(outofplane),thesampleshowstensilestrainasweshoedinChapter1.ThetopbandisLHlikealongthisdirectionasshowninFigure 311 .Thisisaveryimportantissueforbiaxialtensilestress.Aswewillshowinthefollowingsection,thetransverseelectriceldeectosetsthebiaxialstresseectandcausestheholemobilitydegradationatlowstress.Similaranalysiscanbeappliedtouniaxialcompressivestress.Underuniaxialcompression,the<110>channeldirectionexperiencescompressivestrain,thereforethetopbandisLHlikealongthechannel.Atthesametime,theoutofplanedirectionexperiencestensilestrain,thetopbandisHHlikeoutofplane. Thespinorbitalsplito(SO)bandisalsocoupledwithHHandLHbandwhenstrainispresent.ThisbandisnotasimportantduetothelargeenergyseparationfromHHandLHbandsandhenceveryfewholeslocateinthisband. AsstatedpreviouslythatnormalMOSFETshave<110>directionasthechanneldirection,conductivityeectivemassalongthisdirectionaectstheholemobilitydirectlyaccordingtoDrude'smodel,a.k.aequation 3{1 .Figure 39 tellsusthatcomparedwiththebiaxialtensilestress,theuniaxialcompressivestressinducesmuchsmallertopbandeectivemasswhichsuggestsgreaterholemobilityimprovementisexpectedforuniaxialcompressivestress. TwodimensionaldensityofstateseectivemassesasshowninFigure 310 givesaqualitativeestimationofthe2Ddensityofstatesoftheholesineachband.The2DDOSisnotdirectlyrelatedtothebulkelectronicpropertiesofsemiconductors.Intheinversionlayers,large2DDOSofthegroundstatesubbandsuggestsmostholeslocatinginthissubband.Thisreducesintersubbandphononscatteringpossibility.Inthemeantime,ifthegroundstatesubbandhasverylowconductivityeectivemass,thelargeDOS 55 PAGE 56 39 310 and 311 ,asignicantdiscontinuitycanbefoundatlowstrain(stress<1MPa).Aswementionedbefore,HHbandbecomesLHlikealong<110>directionunderuniaxialcompressionandalongoutofplanedirectionunderbiaxialtension.Intheholemobilitycalculation,thediscontinuityoftheholeeectivemassisalsoaconfusingquestion,althoughitisnotimportantinindustriessinceanysingletransistorwouldhavemuchlargerstraininthechannelintheprocess.Tounderstandthe\discontinuity",holeeectivemassatpointiscalculatedforsuperlowstress[ 52 ]asshownin 312 Theguresshowthatunderuniaxialcompressivestress,theHHbandisalwaysHHlikeandtheLHbandisalwaysLHlikeoutofplane.Alongthe<110>direction,theeectivemasscurvescrossoveratabout3kPawhereHHbandbecomesLHlikeandLHbandbecomesHHlike.Biaxialtensilestressactsdierently.TheinplaneHHandLHbandsarestillHHlikeandLHlike,respectively.Intheoutofplanedirection,theHHbandbecomesLHlikeandLHbandbecomesHHlikeasthestressisgreaterthan1kPa. Asthestressincreasesbeyond100kPa,theconductivityeectivemassdoesnotchangeatpoint.Theaverageeectivemasschangeofthesystemcomesfromthefactthatmoreregionofthebandsisaectedbythestress. 313 forunstressedbulksilicon.TheanisotropicnatureoftheSivalencebandisclearlyshown.UsingthesimpleparabolicapproximationE=h2k2 56 PAGE 57 (b) Holeeectivemasschangeunderverysmallstress.Thechangeinthisstressregionexplainsthe\discontinuity"oftheholeeectivemassbetweentherelaxedandhighlystressedSi. 57 PAGE 58 314 for1GPabiaxialtensilestressandgure 315 for1GPauniaxialcompressivestress.Thecontours,aswellasEkrelationcurves,showstraininduceslowerconductivityeectivemassalong<110>directionforthetopband.Theeectivemassesforunstressedbulksiliconare0:59m0forHHand0:15m0forLHbandwherem0isthefreeelectronmass.Thosetwonumbersbecome0:28m0=0:22m0for1GPabiaxialtensilestressand0:11m0=0:2m0for1GPauniaxialcompressivestress.ThebottombandeectivemassesdonotshowenhancementcomparedwiththeLHbandmassoftheunstressedsilicon.Again,forbulkelectronictransport,uniaxialcompressivestressshouldenhancetheholemobilityasstressincreases,sincethetopbandisLHlikealong<110>direction.BiaxialtensilestressdoesnothavethemassadvantagesincethetopbandisHHlikeinplane.Thepossiblemobilityenhancementcomesonlyfrombandsplittingcausingphononscatteringratereduction.Forholesintheinversionlayers,thestatementisstilltrueaswewillshownext. (b) The25meVenergycontoursforunstressedSi:(a)Heavyhole;(b)Lighthole. 58 PAGE 59 (b) The25meVenergycontoursforbiaxialtensilestressedSi:(a)Topband;(b)Bottomband. (b) The25meVenergycontoursforuniaxiallycompressivestressedSi:(a)Topband;(b)Bottomband. 59 PAGE 60 29 ].Thequantumconnementleavesasetoftwodimensionalsubbandsinkspace(kx;ky).Thesubbandstructuresareaectedbyboththestressandthetransverseelectriceld.InpMOSFETs,thetopmosttwosubbands(4countingthespin),thegroundstateandtherstexcitedstatesubbands,containmostoftheholesandanalyzingthosetwosubbandsgivesusqualitativeunderstandingoftheholetransportproperties.Therefore,thosetwosubbandswillbefocusedinthefollowingdiscussionstoexplainthestraineects,althoughupto12subbandsareactuallytakenintoaccountintheholemobilitycalculation. Inthissection,weshallexplainwhythebiaxialtensilestressanduniaxialcompressivestressaectthesubbandstructureandtheholemobilitydierentlyunderthetransverseelectriceld.Thedierenceof(001)and(110){orienteddevicesunderuniaxialstresswillalsobestudied. 1 ].Takingholes(pMOS)asanexample,theconductionandvalencebandsbendup(benddownfornMOS)towardsthesurfaceduetotheappliednegativegatebiasatstronginversionregion.Thismeansholemotioninzdirectionthatisperpendiculartothesiliconsurfaceisrestrictedandthusisquantized,leavingonlya2dimensionalmomentumorkvectorwhichcharacterizesmotioninaplanenormaltotheconningpotential.Therefore,theinversionlayerholes(orelectrons)mustbetreatedquantummechanicallyas2dimensional(2D).Figure 316 illustratesthequantumwellandquantizedsubbands[ 51 ], 60 PAGE 61 Figure316. Quantumwellandsubbandsenergylevelsundertransverseelectriceld. Thecomplexcalculationproceduresomehowpreventspeopleunderstandingthephysicsbehindstressandelectriceldeect.Togivethephysicalinsightsintotherelationbetweenthosetwoeects,triangularpotentialapproximationisutilizedtoestimatethesubbandenergylevels.Thetriangularpotentialapproximationstatesthatthebandbendingsolelydependsondepletionchargesundersubthresholdconditionwhenthemobilechargedensityisnegligible.ThepotentialV(z)isreplacedbyeEeffz,whereEeffistheeectiveelectriceldinthedepletionlayer.Triangularpotentialapproximation 61 PAGE 62 SolvingSchrodinger'sequation, [H(k;z)+V(z)]k(z)=E(k)k(z) (3{2) onewillgetthesubbandenergies.Theenergyofsubbandicanbeexpressedas[ 1 ], 4#2=3i=0;1;2;::: wherehisplankconstant,eistheelectroncharge,andmzistheoutofplaneholeeectivemass,alsoknownasconnementeectivemass.ThiseectiveeldisdenedastheaverageelectriceldperpendiculartotheSiSiO2interfaceexperiencedbythecarriersinthechannel.Itcanbeexpressedintermsofthedepletionandinversionchargedensities: (3{4) where=1 2forelectronsand1 3forholes[ 1 53 ].WefocusontheinversionregionofMOSFETswheretheeectiveeldisover0.5MV/cmthroughoutthiswork.Thisequationfortheeectiveelectriceldisanempiricalequation.Itmaynotbeaccuratetomodelthecarriertransportfordeviceswithsurfaceorientationotherthan(001)orotherdevicestructuressuchassilicononinsulator(SOI)devicesordouble{gated(DG)devices. Equation 3{3 showsthatthesubbandenergyofholesisinverselyproportionaltotheoutofplaneeectivemassoftheholes.Withthetransverseelectriceld,thesubbandthatisHHlikeoutofplaneisshiftedup(lowerenergyforholes)andthesubbandthatisLHlikeoutofplaneisshifteddown(higherenergy).Figure 311 and 39 showthatin(001){orienteddevices,biaxialtensilestrainshiftstheoutofplaneLHlikebandupwhichistheinplaneHHlikeband.Theelectriceldeectosetsthebiaxialtensile 62 PAGE 63 317 schematically.Ifthestraincontinuesincreasing,thestrainbecomesdominantdeterminingthesubbandenergiesandstructures.Duringtheprocess,theaverageholeeectivemassincreasessinceholestransferfromtheinplaneLHlikesubbandtotheHHlikesubband.Thisincreasingeectivemassisresponsibletotheinitialmobilitydegradationunderbiaxialtensilestrainwhichisobservedbothinexperimentsandourcalculation.ThemobilityenhancementshowninFigure 33 comesfromthesuppressedintersubbandphononscatteringrateduetothehighsubbandsplittingaswillbeshownlater.Underuniaxialcompressivestrain,thetopbandisHHlikeoutofplaneandLHlikealongthechannel,whichsuggeststhestrainandtheelectriceldeectsareadditive.Basedonthesimilaranalysis,boththeuniaxialcompressivestrainandthequantumconnementeectsshiftuptheoutofplaneHHlikebandwhichisLHlikealongthechannel.ThereforethegroundstatesubbandisalwaysLHlikealongthechannelandtheaverageeectivemassdecreasesmonotonicallyasthestressincreases. ThecalculatedsubbandsplittingbetweenthegroundstateandtherstexcitedstateisshowedinFigure 318 fordierentstressandsurfaceorientation.Forbiaxialstress,thesplittingiszeroat500MPawhichsuggeststhecrossingoveroftheHHlikeandLHlikesubbands.Foralluniaxialstresscases,thesubbandsplittingincreaseswiththestress.Like(001)/<110>devices,thegroundstatesubbandofboth(110)/<110>and(110)/<111>devicesisHHlikeoutofplaneandLHlikealongthechannelunderuniaxialcompressivestress.Thedierenceistattheoutofplaneeectivemassofthegroundstatesubbandin(110){orienteddevicesismuchlarger 319 thanthatofthe 63 PAGE 64 Schematicplotofstraineectonsubbandsplitting,theeldeectisadditivetouniaxialcompressionandsubtractivetobiaxialtension. (001){orienteddevices,whichresultsinmuchlargersubbandsplittingatlowstress.Thesplittingfor(110){orienteddevicesdoesnotchangeasmuchas(001){orienteddevices,andthesplittingsaturatesmuchfasterwiththestresscomparedwith(001){devices.Thisisduetothestrongquantumconnementundermingingthestraineect,whichisnotobservedin(001){orienteddevices. Ingeneral,inplanecompressivestressisdesirableforpMOS,sinceitcausesthesilicontopvalencebandtobeHHlikeoutofplaneandLHlikeinplane,whichisadditivetotheelectriceldeect.<110>uniaxialcompressivestressisthebestchoicebecauseitgivesverysmallconductivityeectivemass. 318 showsthedierenceofthesubbandsplittingbetween(001)and(110){orienteddevices.Figure 33 showsthatthemaximumenhancementfactorat3GPastressfor(001){orienteddevicesunderuniaxialstressismuchlargerthan(110){orienteddevices. 64 PAGE 65 Subbandsplittingbetweenthetoptwosubbandsunderdierentstress. Toexplainthephysics,thebulkandconned2Denergycontoursofthegroundstatesubbandfor(001)and(110){orientedSiareshowninFigure 320 321 322 ,and 323 .Theguresshowthatfor(001)/<110>devices,thegroundstateholeeectivemassdecreaseswithuniaxialcompressivestress(LHlike)alongthechannel),butthereductionisnotasnotableunderbiaxialstress.ComparedwiththebulkSienergycontours,theelectricelddoesnotmodifythesubbandstructureinkxkyplanefor(001){orienteddevices(itdoesaectthesubbandsplittingthough).Theconductivityeectivemassesalongthechanneldirectionarealmostidenticaltothoseofbulkcounterparts.Theconnementeectismuchmoresignicanton(110){orienteddevices.Theconnedeectivemassofthegroundstatesubbandisverylowalong<110>and<111>directionevenforunstressedSi,whichexplainswhyunstressed(110){orienteddeviceshavesuperiorholemobilityover(001){orienteddevices(theconnementeectisalsosignicantin(111)and(112)pMOSFETs( 31 ),thoughtheholeeectivemassislargerthanthatin(110) 65 PAGE 66 Outofplaneeectivemassesforh110isurfaceorientedbulksiliconunderuniaxialcompressivestress. 66 PAGE 67 324 .For(001)devicesunderuniaxialstress,theinitialdecreaseoftheholepopulationisduetothedecreasedDOSnearpoint.Asstressincreases,theincreasingsubbandsplittingcausestheholepopulationincreasingandtheaverageconductivityeectivemasskeepsdecreasingsincethegroundstatesubbandisLHlikealongthechannelunderuniaxialcompressviestress.Forbiaxialstress,thedecreaseoftheholepopulationatlowstressagainreectstheinitialconnementeectliftingtheinplaneLHlikesubbandandreducingthesubbandsplitting( 318 ).ThisinplaneLHlikesubbandisshifteddownasthestressincreasesandtheinplaneHHlikesubbandisshiftedup.Afterthecrossingoverofthetwosubbands,thegroundstatesubbandpopulationstartsincreasingwiththestress.For(110){orienteddevicesunderuniaxialcompressivestress,thegroundstateholepopulationincreaseswiththestress,butitsaturatesatmuchlowerstresscomparedwith(001){orienteddeviceswhichisconsistentwiththesubbandsplittingchange.Theholepopulationof(110){orienteddevicesisalwayslowerthan(001){orienteddevicesunderuniaxialcompressivestress,althoughthesubbandsplittingismuchlarger.Thesubbandsplittingandholepopulationdierenceof(001)and(110){orienteddevicescanbeexplainedbythegroundstatesubband2DDOSasshowninFigure 325 .DOSdierencealsosuggeststhedierentstraininducedmobilitychange.Bothguresshowthat(001){orienteddeviceshavelargerDOSthan(110){orienteddevices.For(001)/<110>devicesunderuniaxialcompressivestress,althoughtherstexcitedstatesubbandisHHlike 67 PAGE 68 (b) (c) The2Denergycontours(25,50,75,and100meV)forbulk(001)Si.Uniaxialcompressivestresschangesholeeectivemassmoresignicantlythanbiaxialtensilestress. 68 PAGE 69 (b) (c) Conned2Denergycontours(25,50,75,and100meV)for(001)Si.Thecontoursareidenticaltothebulkcounterparts. 69 PAGE 70 (b) (c) The2Denergycontours(25,50,75,and100meV)forbulk(110)Siunder(a)nostress;(b)uniaxialstressalongh110i;and(c)uniaxialstressalongh111i. 70 PAGE 71 (b) (c) Conned2Denergycontours(25,50,75,and100meV)for(110)Si.Theconnedcontoursaretotallydierentfromtheirbulkcounterpartswhichsuggestssignicantconnementeect. 71 PAGE 72 324 .ThedecreasingDOSinFigure 325 (b)forbothbiaxialanduniaxialstressof(001){orienteddevicesalsosuggeststhatthephononscatteringratedecreaseswithtestress.TheDOSof(110){orienteddevicesdoesnotchangewiththestressespeciallyathighstressregion(1{3GPa)whichsuggeststhephononscatteringrateshouldnotchangemuch. Figure324. Groundstatesubbandholepopulationunderdierentstress. Aswementionedintheprevioussection,thestressdoesnotwarpthebandstructureevenlyinthewholekspace.ThiscanalsobeseenfromtheDOSchangeinFigure 325 (b)wheretheDOSatEnergyE=52meV(2kTwhereT=300k)isshown.Takinguniaxialstresson(001)devicesasanexample,whenthestressislow,onlyasmallregionclosetopointisaectedandbecomesLHlikealong<110>direction(stillHHlikealongtransverseandoutofplanedirection),whiletherestofthebandwithhigherenergy(includingtheenergylevelshowedhere)doesnotrespondtothestressyet.Asthestress 72 PAGE 73 325 (b)suggestswhenthestressislowerthanabout500MPa,thestressistoosmalltowarpthebandatthisenergylevel.Whenthestressincreases,DOSstartsdecreasingbecausethestressstartswarpingthebandatthisenergyandthe<110>directionbecomesLHlike.TheDOScurvebecomesatagainwhenthestresseectsaturatesforthisenergylevel.For(001)pMOSFETsunderbiaxialstress,Figure 325 (b)doesnotshowaDOSpeaklikeFigure 325 (a)whichmeansthepositioncrossingoverofthetoptwosubbandsonlyhappensclosetopoint,andtheHHlikebandisalwaysontopoutofthatregion. For(110)pMOSFETs,theDOSisconstantwiththestress,whichisduetothestrongquantumconnementeect.Todiscoverthestraineect,2DDOSat4kT(102meVatT=300K)isshowninFigure 326 .For(001)pMOSFETs,thecurveshavethesimilartrendcomparedwiththeDOScurvesat2kT.TheonlydierenceisthattheDOSstartstodecreaseathigherstress.For(110)pMOSFETs,DOSdecreasesatlowstressandthechangeisnotassignicantlyas(001)pMOSFETs.Figure 325 and 326 suggestthatthestrainin(110)pMOSFETsonlywarpsthesubbandathighenergyregionduetothestrongquantumconnement.Thestraininducedmobilitychangeshouldbelessthan(001)pMOSFETs,sincesmallerportionofholeslocateathighenergycomparedwithpoint. 3 1 ]. Figure 327 showsthatfor(001){orienteddevices,thephononscatteringratedoesnotchangemuchwhenthestressislowerthan500MPa.Thisindicatesthatatlowstress,theholemobilityenhancement(ordegradation)isalmostpurelycausedbytheeectivemass 73 PAGE 74 (b) Stresseectonthe2dimensionaldensityofstatesofthegroundstatesubbandat(a)thetopofthesubband(E=0);(b)E=2kT.(110){deviceshavemuchsmaller2DDOSwhichlimitsthegroundstateholepopulation(largerintersubbandphononscattering).AnotherobservationisthatDOSof(110){devicesdoesnotchangewithstress. 74 PAGE 75 TwodimensionaldensityofstatesatE=4kT. change.Whenthestressincreasesfrom500MPato3GPa,thephononscatteringratedecreasesby50%forbothacousticphononandopticalphononscattering.thephononscatteringratereductionoverweighstheeectivemasschangetobecomethemaindrivingforcetoimprovetheholemobilityinthisstressrange,especiallyforbiaxialstress. Unlike(001){orienteddevices,phononscatteringratechangesmoreatlowstressregionratherthanhighstressregionfor(110){orienteddevicesunderuniaxialcompressivestress.ThisisconsistentwithFigure 318 and 324 thatthesubbandsplittingandthegroundstatesubbandholepopulationonlyincreaseatlowstress.Theconstantphononscatteringrateathighstressexplainswhytheholemobilityof(110)/<111>devicesat3GPaisnotsignicantlylargerthan(001)/<110>or(110)/<110>devices,regardlessofthelargestpiezoresistancecoecientatlowstress. Figure 328 showsthatthesurfaceroughnessscatteringrateincreaseswithstressfor(001){orienteddevices.Thisisduetotheincreasingholepopulationintheground 75 PAGE 76 (b) Straineecton(a)acousticphonon,and(b)opticalphononscatteringrate.Opticalphononscatteringisthedominantscatteringmechanismimprovingthemobility.Phononscatteringratechangesmainlyinhighstressregionfor(001)devicesandlowstressregionfor(110)devices. 76 PAGE 77 Straineectonsurfaceroughnessscatteringrateofholesintheinversionlayer.Asstressincreases,thescatteringrateincreasesfor(001)devicesduetotheincreasingoccupationinthegroundstatesubbandwhichbringsthecentroidsoftheholesclosertotheSi=SiO2interface. statesubbandwhichbringsthecentroidsoftheholesclosertotheSi=SiO2interface.Themagnitudeofthesurfaceroughnessscatteringrateismuchsmallerthanthephononscatteringrateandthereforetheincreasingsurfaceroughnessscatteringdoesnotaecttheholemobilityasmuch.Thesurfaceroughnessscatteringratefor(110){orienteddevicesdoesnotchangemuchwiththestress,whichisconsistentwiththefactthatthegroundstatesubbandholepopulationisrelativelyconstantwiththestress. 329 illustratesthestress{inducedholemobilityenhancementcontributionfromholeeectivemassandphononscatteringratereduction,respectively.Underuniaxialcompression,(001)/h110ipMOSFETshavethelargestmobilityimprovementfrombothaspects.Comparedwith(110)/h111ipMOSFETs,(110)/h110ipMOSFETs 77 PAGE 78 Uniaxialstresson(110)devicesimprovestheholemobilitytoo.Buttheimprovementisnotasmuchas(001)orienteddevices.Thisisduetothestrongconnementeecton(110)orienteddevicesunderminingthestresseect.Whennostressispresent,theconnementeectswapsthesubbandstructureandreducestheholeeectivemassaroundthe{point.Thiseectivemassadvantageoverthe(001)orientedunstressedpMOScausesthattheholemobilityismuchlarger.Whenthestressisapplied,theeectivemasschangeisnotassignicant,neitherdoesthesubbandsplitting.Therefore,themobilityenhancementwiththestressisnotsupposedtobeasmuchasthe(001)orientedpMOS. Itisalsonoticedthatthesubbandsplittingsaturateswhenthestressreaches2or3GPa,sodoestheeectivemass.Thisleadstothesaturationofthestressenhancedholemobility. 78 PAGE 79 (b) Holemobilitygaincontributionfrom(a)eectivemassreduction;and(b)phononscatteringratesuppressionforpMOSFETsunderbiaxialanduniaxialstress. 79 PAGE 80 AsthesiliconCMOStechnologyisscaledtosub{100nm,evensub{50nmscale,furthersimplescalingoftheclassicalbulkdevicesislimitedbytheshortchanneleects(SCEs)anddoesnotbringperformanceimprovement.Theultrathinbody(UTB)silicononinsulator(SOI)transistorarchitecture[ 54 55 56 57 58 ]hasbeenconsideredpossiblereplacementforthebulkMOSFETs.ThebasicideaofSOICMOSfabrication[ 54 56 ]istobuildtraditionaltransistorstructureonaverythinlayerofcrystallineSiwhichisseparatedfromthesubstratebyathickburiedoxidelayer(BOX).ComparedwiththebulkCMOS,UTBSOItechnologybringsbenetssuchasreducedjunctioncapacitancewhichincreasesswitchingspeed,nobodyeectsincethebodypotentialisnottiedtothegroundorVddbutcanrisetothesamepotentialasthesource,lowsubsurfaceleakagecurrent,andetal.. SOIMOSFETsareoftendistinguishedaspartiallydepleted(PD)transistorsthattheSithicknessislargerthanthemaximumdepletionwidthandfullydepleted(FD)SOItransistorsthattheSiisthinnerthanthemaximumdepletionwidth.FDSOItechnology[ 1 ]addadditionalperformanceenhancementsoverPDSOIincludinglowverticalelectriceldinthechannel(highermobility)duetothefactthatmostFDSOItransistorshaveundopedchannel,furtherreductionofthejunctioncapacitance,andbetterscalability.AlthoughFDSOItechnologyhasbetterscalabilitythanclassicaldevicestructures,itisstilldiculttoscalethedevicetosub{20nmscale.InshortchannelFDSOIMOSFETs,thethickBOXactslikeawidegatedepletionregionandisvulnerabletosourcedraineldpenetrationandresultsinsevereshortchanneleects[ 1 59 60 ].Tobettercontrolthechannel,doublegate(DG)transistors,especiallyFinFETs,havebeeninvestigatedtheoreticallyandexperimentally[ 61 62 63 64 ].DGMOSFETshavebetterscalabilitythansinglegate(SG)SOItransistorsandareconsideredpromisingcandidatesforsub20nmtechnologies[ 62 ].Overall,SOISGdevicesandDGdeviceshavebeenshown 80 PAGE 81 Withtheresearchofstraineectsonbulksilicondevices,strainedsiliconUTBFETsdrawtheattentionofresearchersassuchdevicesmaycombinethestraininducedtransportpropertyenhancementswiththeirscalingadvantages.StressenhancedholemobilityinSOI{deviceshasbeeninvestigatedexperimentallyinrecentyears[ 65 66 67 68 69 70 ].In2003,Rim[ 45 ]reportedthebiaxialtensilestressedSOI{pMOSholemobilitywithdependenceofstrainandinversionchargedensity.Zhang[ 71 ]showedtheholemobilityenhancementunderlowuniaxiallongitudinalandtransversestress.(110)surfaceSOIdeviceswithstraineectsarealsoinvestigated[ 72 ].ThoseresultsareconsistentwiththemeasuredandcalculatedresultsforbulkSidevicesthatareshowedinthelastchapter. StrainresearchondoublegatedeviceslagsthatonbulkdevicesandevensinglegateSOIdevicespartlyduetothedicultyemployingstresstothechannelwithoutdamagingthepropertiesofthechannelandSi=SiO2interfaces.Duetothebetterscalabilityandhigherholemobility,moreattentionhasbeendrawnto(110){orientedFinFETsoverplanarDGFETs.Collaert[ 73 ]investigatedstraineectonelectronandholemobilityenhancementonFinFETs.Shin[ 74 ]andhiscolleaguesinvestigatedmultiplestresseectsonptypeFinFETsusingwaferbendingmethod.Verheyen[ 75 ]reported25%drivecurrentimprovementofptypemultiplegateFETdeviceswithgermaniumdopedsourceanddrain.Althoughholemobilityenhancementisobservedinthoseexperiments,theactualstressinthenisunknown.Theoretically,straineectsonFinFETsaremuchlessunderstood.Withtheknowledgeofstressenhancingholemobilityinbulkdevices,it'simportanttounderstandhowthatstressalterstheholemobilityinFinFETs.Uniaxialcompressivestresswillbefocusedinthisworksinceitprovidesthegreatestholemobilityimprovementthanotherstressonbulkdevices.Anotherreasonisthatfor(110){orientedFinFETs,thestressinthechannelisnormallyuniaxiallongitudinalstressevenifSiGe 81 PAGE 82 41 [ 67 76 ]showsthattheholemobilityisalmostindependentofthesiliconthicknesswhenSOIthicknessisover10nm.Ifthesiliconthicknessissmallerthan10nm,holemobilitydecreasesastheSOIthicknessdecreases.Themainreasonisthattheincreaseintheformfactor(/1=ph)causestheincreasingphononscatteringrate[ 77 ]duetothestructuralconnement.Anotherreasonistheincreasingsurfaceroughnessscatteringcausingsignicantloweringofthesurfaceroughnesslimitedmobility,sinceholesaremucheasierinvolvedinthesurfaceroughnessscatteringasthesiliconthicknessdecreases. SubbandsplittingiscalculatedforSOIpMOSandcomparedwiththebulkpMOS.Withthesameinversionchargeanddopingdensity,thesplittingisverysimilarforbothcases.IftheSOIthicknessdecreasesfrom20nmto5nm,thechangeofthesubbandsplittingislessthan5%.Thestructureofeachsubbandisalsoidenticaltothebulkdevices.IftheSOIthicknessissmallerthan5nm,subbandsplittingincreasesasthe 82 PAGE 83 HolemobilityvsSOIthicknessforsinglegateSOIpMOS.Themobilitydecreaseswiththethicknessduetostructuralconnement. SOIthicknessdecreases.Thisdoesnotbringsmallerintersubbandscatteringrate.Therapidlyincreasingformfactoractuallykeepsthescatteringrateincreasing. Anotherissuerelatedtothesiliconthicknessissubbandmodulation.BothmeasurementsandMonte{CarlosimulationshowthatthephononlimitedmobilityincreasesatverythinSOIthickness[ 67 69 77 ].ThisissueonlyhappenstonMOS.Uchida'smeasurementsshowthereisnosuchmobilitypeakinptypeUTBSOIFETs[ 67 ],whichisconsistentwithourcalculation. 45 ]reportedthatbiaxialtensilestrainimproves(ordegrades)theholemobilityassameasitdoestothebulkdevices,whichissupportedbyourcalculation.Uniaxialcompressivestrainisfocusedinthischapterduetoitsmuchlargermobilityenhancementfactorthanbiaxialtensilestrain. Figure 42 showsthesinglegateSOIpMOSholemobilityvsuniaxialcompressivestresscomparingwithbulkSidevices.CalculatedcurvesforSOIthicknessof3nmand5 83 PAGE 84 Figure42. HolemobilityforsinglegateSOIpMOSvsuniaxialstressatchargedensityp=11013=cm2. TheholemobilityenhancementfactorforSOIpMOSwithSOIthicknessof3nmisshowninFigure 43 .TheenhancementfactorforSOIdevicesissimilartothecaseofbulkdevicesatlowstress,butlargerthanbulkFETsathighstress.AswementionedinChapter3thatfor(001){orientedSipMOS,themobilityisenhancedmainlyduetothedecreasedholeeectivemassatlowstress.Athighstress,phononscatteringratereductionduetotheincreasingsubbandsplittingisthemaindrivingforcetoimprovethemobility.Theoverlappingcurvesatlowstresssuggesttheeectivemassgainshouldbesimilarforbothcases.CalculationshowsthatthestructureofeachsubbandinSOIpMOSisassameasthebulkcounterpartwhichalsosuggeststheeectivemasschangeforbothcasesshouldbethesame.Figure 44 showsthesubbandsplittingofthegroundstateandtherstexcitedstatesubbandsforSOIandbulkFETs.ThelargersplittingforSOI 84 PAGE 85 Figure43. HolemobilityenhancementfactorofUTBSOISGdevicesvsuniaxialcompressivestressatchargedensityp=11013=cm2. UchidareportedthatastheSOIthicknessreducesdownto2{3nm,theuctuationoftheSi=SiO2interfaceisthemainfactortolimitthecarriermobility[ 67 69 ].Therefore,thelargeholemobilityenhancementasshowninFigure 43 cannotbeobtainedinrealdevices.Anewsurfaceroughnessmodelisneededtosolvethisproblem.InourdiscussionofthedoublegatedevicesincludingFinFETslaterinthischapter,thesmallestSithicknessweconsiderwouldbe5nm. 85 PAGE 86 SubbandsplittingUTBSOISGdevicesvsuniaxialcompressivestressatchargedensityp=11013=cm2. 86 PAGE 87 45 .ThesubbandsplittingforSDGMOSFETsandFinFETsisverysmallwhentheSithicknessisover5nm(5meVwhentSi=5nm,3meVwhentSi=15nm).IftheSithicknessisbelow5nm,thestronginteractionofthetwosurfacechannelcausesthesubbandsplittingincreasingdrastically(i.e.18meVfortSi=3nm): ComparisonofthesubbandsplittingofdoublegateandsinglegateMOSFETs. 46 and 47 ,respectively.Doublegatedeviceshavehighermobilitythantraditionalbulktransistorsmainlyduetotheundopedbody,muchsmallerchanneleectiveelectriceldandbulkinversion[ 1 ].Figure 46 showsthattheholemobilitydecreasesasthesiliconthicknessdecreases.ThereasonisassameassinglegateSOIdevicesandhasbeenexplainedinlastsection. ThemobilityenhancementfactorofSDGpMOSinFigure 47 isverysimilartothebulkcase,butthemechanismsarealittledierent.Therstexcitedsubband(veryclosetothegroundstate)providessmalleraverageeectivemasstohelpthemobility 87 PAGE 88 HolemobilityofSDGdevicesunderuniaxialcompressivestressatchargedensityp=11013=cm2. 88 PAGE 89 Figure47. HolemobilityenhancementfactorofSDGMOSFETsvsuniaxialcompressivestressatchargedensityp=11013=cm2. 48 ,comparingwiththesinglegate(110)and(001)orientedptypedevicesattheinversionchargedensityof11013=cm22.Inthecalculationofthesinglegatedevices,thedopingdensityistakentobe11017=cm3.ThisisalowdopingdensitycomparedwiththecontemporaryCMOStechnology.Evenso,theFinFETshowssignicantlygreatermobilitythanthebulkdevices.Iflargerdopingdensityisapplied,themobilityadvantageoftheFinFETwouldbeevenlarger.When3GPauniaxialcompressivestressisapplied 89 PAGE 90 49 .Eventhoughthe(001)orientedpMOSshowsgreaterrelativeenhancement(over400%),theabsolutemobilityisstilllowerthanthatoftheFinFETduetoitslowmobilitywithnostress. Figure48. HolemobilityofFinFETsunderuniaxialstresscomparedwithbulk(110)orienteddevicesatchargedensityp=11013=cm2. Wementionedinthelastchapterthat2DDOSofthetopmostsubbandin(110)orienteddevicesisverysmallnearpointnomatterifthestressispresentandthestressdoesnotwarpthesubbandsmuch.Thereforetheaverageeectivemassdoesnotchangeasmuchasstandard(001){orienteddeviceswhenuniaxialstressispresent.RegardingFinFETs,strongsubbandmodulationisobservedwherethetopmost2subbandsareclosetoeachother(like(001)SDGpMOSFETs)asweillustratedinFigure 45 .ThisextrasubbandissoclosetothegroundstatesubbandanditactslikeincreasingtheDOSofthegroundstatesubband.Moreimportantly,thebandbendingattheSi=SiO2interfaceis 90 PAGE 91 HolemobilityenhancementfactorofFinFETsunderuniaxialcompressivestressatchargedensityp=11013=cm2. 91 PAGE 92 Figure410. Holemobilitygaincontributionfromeectivemassandphononscatteringsuppressionunderuniaxialcompressionfor(110)/h110iFinFETscomparedwithSG(110)/h110ipMOSFETsatchargedensityp=11013=cm2. TounderstandtheholemobilitydierencebetweenFinFETsandtraditionalsinglegate(110)/h110i,theholemobilitygaincontributionfromeectivemasschangeandphononscatteringratechangeisshowninFigure 410 .ItshowsthatphononscatteringratechangeisthemainfactortoimprovetheholemobilityforbothFinFETsandbulk 92 PAGE 93 Thecalculationalsoshowstheenhancementisnotastrongfunctionofthesiliconthicknessofthenasthenthicknessisabove5nm.Ifthenisthinnerthanthat,moresubbandsplittingisobserved(about18meVfor3nmofthenthickness).Sincethesplittingisstillnottoolarge,ouranalysisabouttheeectivemassstaystrue.Surfaceroughnessscatteringrateismuchlargerandtheholemobilityenhancementwouldnotbeaslargeasthatforthickern.Anaccuratesurfaceroughnessmodelforsuchdeviceswouldbenecessarytoevaluatethemobilitychangenumerically. Fordoublegatedevices,subbandsplittingisdrasticallysmallerthanthebulkdevicesduetotheinteractionofthequantumstatesofthetwosurfacechannels.For(001){orientedplanarsymmetricalDGpMOS,thestructureofeachsubbandisstillidenticaltothecounterpartinthebulkdevices.TheextraeectivemassgainiscanceledbytheintersubbandphononscatteringandthetotalholemobilityenhancementissimilartothebulkFETsatlowstress.Butwhenthestressisover2GPa,theeectivemassgainissaturate.ThemobilitygainislessthanthatofbulkFETsduetothelargerintersubbandopticalphononscattering.Thiseectisnotthatsignicantduetothesmallerformfactors. 93 PAGE 94 94 PAGE 95 Asshortchanneleects(SCEs)preventthesimplescalingoftraditionalSiMOSFETsachievinghistoricalperformanceimprovement,newmaterial,aswellasfeatureenhancedtechnology(straintechnology),attractattentionoftheresearchers.Germaniumisoneofthosenewmaterialsduetoitslargeelectronandholemobility.Withthestrainedsilicontechnologyintheindustry,it'sainterestingtopictodiscoverhowthestrainaectstheelectronandholemobilityingermaniumMOSFETs. GermaniumhasbeenofspecialinterestinhighspeedCMOStechnologyforyears[ 78 79 ].Thebulkgermaniumholemobilityislargerthanthatofothersemiconductormaterials,anditselectronandholemobilityaremuchlessdisparatethanothermaterials.In1989,germaniumholemobilityof770cm2=VsecinapMOSFETwasexhibitedbyMartin[ 80 ]andhisco{workersusingSiO2asthegateinsulator.Sincethen,moreandmorework[ 81 82 ]hasbeendoneongermaniumorSiGechannelpMOS[ 83 84 85 ].Inordertoreducethesurfaceroughnessandlimittheband{to{bandtunnelingissue,silicon{germaniumorSi{SiGedualchannelisalsousedinsomeapplications.Dierentgatedielectricmaterials[ 86 87 88 ]havebeenutilizedtondthebestmaterialtolimitthesurfaceroughnessattheinterfacebetweengateinsulatorandgermaniumchannel.Duetotheuncertaintyinthesurfaceroughnessandthesurfacestates,dierentholemobilityvalueshavebeenreportedinthosepublications.Inrecentyears,withthestraintechnologyappliedtosiliconCMOS,straineectisalsoinvestigatedongermaniumMOSFETs[ 87 89 90 91 92 ].ThestrainisnormallyachievedbyapplyingSiGesubstrateunderneaththegermaniumorSiGechannel.Butmostoftheworkstaysonlyinexperiments,thephysicalinsightsofthestraineectongermaniumMOSFETshavenotbeendiscussedcarefully.TheonlyavailabletheoreticalworksaresomeMonteCarlosimulations[ 93 94 95 ].Thegoalofthischapteristogivephysicalinsightsofstraineectsongermaniumutilizingkpcalculation. 95 PAGE 96 86 96 ]vsverticalelectriceldanddevicesurfaceorientationisshowningure 51 .Experimentalworksgivealotofdierentmobilityvaluesrangingfrom70cm2=Vsectoover1000cm2=Vsec,dependingonwhatthegatedielectricmaterialsareused[ 86 87 88 ]andifSibuerisapplied[ 97 98 ]betweentheGe(orSiGe)andthegateoxide.WithSibuer,thedeviceactsasaburiedGechanneltransistorandnormallyshowslargeholemobilityduetothelackofconnementandsurfaceroughnessscattering.Duetothebadscalabilityofburiedchanneldevices,onlysurfacechannelGepMOSisdiscussedhere.CalculatedGeholemobilitymatchesthemeasureddataandthemobilityismuchlargermobilitythansilicon.(110)orienteddeviceshowshighermobilitythan(001)orienteddevice,whichisconsistentwiththeresultsofSi.Weshallshowthatthelargerholemobilityofgermaniummainlycomesfromthesmallereectivemassoftheholes.Therelativesmallerintersubbandphononscatteringrateduetothelargersubbandsplitting(andsmalleropticalphononenergy)alsoimprovesthegermaniummobility. 96 PAGE 97 Germaniumholemobilityvseectiveelectriceld. thebiaxialtensilestraineectongermaniumholemobilityiscalculatedandshowedinFigure 52 Likesilicon,thedegradationoftheholemobilityatlowbiaxialtensilestressisduetothesubtractivenatureofstraineectandtransverseelectriceldeectresultingintheincreaseoftheaverageeectivemass,togetherwithalittleincreasedintersubbandphononscattering.Athighstress,themobilityenhancementisobtainedduetoreducedintersubbandopticalphononscattering. 53 97 PAGE 98 Germaniumandsiliconholemobilityunderbiaxialtensilestresswheretheinversionholeconcentrationis11013=cm2. 98 PAGE 99 Germaniumandsiliconholemobilityunderbiaxialcompressivestresswheretheinversionholeconcentrationis11013=cm2. 99 PAGE 100 54 for(001){orientedGeandFigure 55 for(110){orientedGe.For(001){orienteddevices,bothSiandGeshowlargeenhancement.OnedierencebetweenthetwocurvesisthatthemobilityenhancementforSisaturatesatabout3GPa,butitdoesnotsaturateuntil6GPaofstressisappliedtoGe. Figure54. Germaniumandsiliconholemobilityon(001)orienteddeviceunderuniaxialcompressivestresswheretheinversionholeconcentrationis11013=cm2. 100 PAGE 101 Germaniumandsiliconholemobilityon(110)orienteddeviceunderuniaxialcompressivestresswheretheinversionholeconcentrationis11013=cm2. 101 PAGE 102 56 showsthedispersionrelationdiagramsfor(001)Geunderdierentstress.Likesilicon,theheavyholeandlightholebandsofrelaxedGearedegenerateatpointasshowninFigure 56 (a).Thedegeneracyisliftedwhenstrainisapplied.Thebandsplittingleadstobandwarpingandthechangeofholeeectivemassandphononscatteringrate.TheSObandenergyis296meVlowerthantheHHandLHbandsforrelaxedgermaniumwhichimplieslesscouplingwithHHandLHbandscomparedwithsilicon.Underbiaxialtensilestrain,thetopbandisLHlikeoutofplaneandHHlikealongh110i.ForbothcompressivestraininFigure 56 (c)and(d),thetopbandisHHlikeoutofplaneandLHlikealongh110i.Uniaxialcompressivestrainbringsthemostwarpingonthetopvalenceband.Thewarpingisthesmallestunderbiaxialcompressivestrain,whichsuggeststheleastmobilityenhancementasshowninFigure 53 57 forbiaxialtension, 58 forbiaxialcompression,and 59 foruniaxialcompression.Comparedwithsilicon,theeectivemassforgermaniumisobviouslymuchsmalleralongbothdirections.ThissuggestslargerholemobilityforgermaniumthansiliconaccordingtoDrude'smodel.OnesignicantdierencefromSieectivemassisthattheholeeectivemasschangeofGesaturateswithstressatmuchhigherstressthansilicon.Forsomeofthecurves,i.e.\top"bandofFigure 57 (a)and 59 (b),or\bottomband"ofFigure 58 ,theeectivemasschangedoesnotsaturateuntilthestressgoesupto7GPa.Butforsilicon,normallytheeectivemasschangesaturatesat2or3GPa.Thissuggestshigherstressforthemobilitysaturation. Thetrendoftheeectivemasschangewithstressissimilarforbothsiliconandgermanium.Ifwelookatthechanneldirection(h110i)eectivemass,thetopband 102 PAGE 103 (b) (c) (d) E{kdiagramsforGeunder(a)nostress;(b)1GPabiaxialtensilestress;(c)1GPabiaxialcompressivestress;and(d)1GPauniaxialcompressivestress. 103 PAGE 104 (b) Conductivityeectivemassvsbiaxialtensilestress:(a)Channeldirection(<110>)and(b)outofplanedirection. 104 PAGE 105 (b) Conductivityeectivemassvsbiaxialcompressivestress:(a)Channeldirection(<110>)and(b)outofplanedirection. 105 PAGE 106 (b) Conductivityeectivemassvsuniaxialcompressivestress:(a)Channeldirection(<110>)and(b)outofplanedirection. 106 PAGE 107 54 becauseofthemuchsmaller2DDOSandinitiallargesubbandsplittingintheinversionlayers,whichwillbeaddressedinthenextsection.Underbiaxialtensilestress,thetopbandhashigherchanneldirectioneectivemass(increasingwithstress)andloweroutofplaneeectivemasswhichissimilartosilicon.ThismeansthestresseectandthetransverseelectriceldeectintheinversionlayershouldbesubtractiveandtheholemobilityshouldbedegradedatlowstressasshowninFigure 52 .Underbiaxialcompressivestress,thetopbandhasverylowconductivityeectivemassatpointalongh110i.Aswementionedbefore,thebandwarpingisnotsignicantandonlyhappensveryclosetothepoint,whichsuggeststheaverageeectivemassofthesystemmaynotdecreasemuchwiththestress. 510 .Contours(25meV)underbiaxialcompressiveandtensilestressareshowningure 511 and 512 .Figure 513 showsthecontoursunderuniaxialcompressivestress.Theenergycontoursaresimilartothoseofsilicon,buttheshapeofthecontourschangesmorethanSicontourswhenthesameamountofstrainispresent.Anotherdierenceisthatunderuniaxialcompressivestress,the2DDOSofGelooksmuchsmallerthanSi.FromtheanalysisofSi,lowerpointDOSleadstosmallerstraininducedmobilityimprovementduetofewerholesareaectedbystrain.ThismayexplainwhythemobilityenhancementfactorforGeisnotlargerthanSi,althoughtheeectivemasschangeismuchlargeratpoint. 107 PAGE 108 (b) 25meVenergycontoursforunstressedGe:(a)Heavyhole;(b)Lighthole. (b) 25meVenergycontoursforbiaxialcompressivestressedGe:(a)Topband;(b)Bottomband. 108 PAGE 109 (b) 25meVenergycontoursforbiaxialtensilestressedGe:(a)Topband;(b)Bottomband. (b) 25meVenergycontoursforuniaxiallycompressivestressedGe:(a)Topband;(b)Bottomband. 109 PAGE 110 5.3.1StraininducedSubbandSplitting 58 and 59 showthatbothbiaxialcompressivestressanduniaxialcompressivestressshiftuptheoutofplaneHHlikeband.Thiseectisclearlyadditivetotheelectriceldeect.Forbiaxialtensilestress,theelectriceldeectissubtractivetothestraineectandthereforeiftheelectriceldisxed,thesubbandsplittingshoulddecreaseatlowstresslevelandatsomestressvalue,thetoptwosubbandswouldcrossovereachotherjustlikeSi.Thesubbandsplittingbetweenthegroundandtherstexcitedstateof(001)GeisillustratedinFigure 514 Figure514. Gesubbandsplittingunderdierentstress. Figure 514 showsthatthesubbandsplittingforrelaxedGepMOSFETsismuchlargerthanthatoftheSipMOSFETs.Thesplittingislargerthantheopticalphonon 110 PAGE 111 515 showsthenormalizedinplaneEkdiagramunderbiaxialcompressivestress.Ontheonehand,thegureshowsasthestressincreases,moreregionisnearthezonecenteriswarpedandhaslowerDOS.Ontheotherhand,outofthewarpedarea,thebandcurvesupalittleasthestressincreases,whichsuggeststheincreaseoftheDOS.Theoveralleectisthattheeectivemassgainclosetopointduetostressiscompromisedbytheheaviermassoftheholesawayfromthepoint.Atlowstress,themasschange,togetherwiththeincreasingsubbandsplitting,enhancestheholemobilityslightly.Underhigherstress,theenhancementisminimal.ForSipMOSunderbiaxialcompressivestress,theEkdiagramissimilartoGe.ThedierenceisthatSihasmuchlargerDOSnearpoint,thereforethereisalwayseectivemassgain.ThedierentDOSresultsinthestrainenhancedholemobilitydierenceasinFigure 53 BiaxialtensilestressaectstheGeholemobilitysimilartoSidevices:thesubtractivenatureofthestrainandtransverseelectriceldeectsdegradestheholemobilityatlowstress,andthedecreaseofthephononscatteringrateenhancesthemobilityathighstress.Uniaxialcompressivestresson(001){orientedGeisfocusednextbecauseitprovidesthe 111 PAGE 112 NormalizedgroundstatesubbandEkdiagramvsbiaxialcompressivestress. mostmobilityenhancementandthemobilityenhancementmechanismisalittledierentfromSi. 516 atdierentenergies.DOSofGeismuchlowerthanSi.ThetrendoftheDOSchangewithstressissimilarforSiandGe.Figure 516 (a)showsthattheDOSofGesaturateswithstressathigherstressthanSi,sincetheeectivemasschangeswithstressathigherstress.ThisisconsistentwiththemobilitysaturationcurvesinFigure 54 PhononandsurfaceroughnessscatteringratechangevsuniaxialstressisshowninFigure 517 and 518 .ForbothSiandGe,phononscatteringratedoesnotchangemuchatlowstress,andathighstressthephononscatteringratedecreasesastheuniaxialstressincreases.GehaslowerscatteringratethanSiduetothesmallerDOSofGe.ForSi,bothacousticphononandopticalphononscatteringratedecreasesby50%whenthestress 112 PAGE 113 (b) TwodimensionaldensityofstatesofthegroundstatesubbandforSiandGeat(a)E=5meV;(b)E=2kT=52meVunderuniaxialcompressivestress. 113 PAGE 114 Figure 519 showsthemobilityenhancementcontributionfromeectivemass(solidlines)andphononscatteringrate(dashedlines)forSiandGe.ForSi,eectivemassgainisthemaindrivingforceofthemobilityenhancementatlowstress,andthescatteringratechangeisdominantathighstressrange(1GPa{3GPa).Fromunstressedcaseto3GPaofstress,eectivemassgainandphononscatteringratedecreasehavecomparableenhancementtotheholemobility.ForGe,thephononscatteringonlycontribute1.5timesoftheenhancement.Theeectivemasschangeisdominantinthewholestressrange.Aswementionedbefore,thisisbecausetheeectivemasschangeratioislargeunderstress(0:38m0tosmallerthan0:04m0).Anotherobservationoftheeectivemassisthatasthestressisover1GPa,increasingthestressdoesnotchangetheholeeectivemassforSi,buttheeectivemassofGecontinuetodecreaseasthestressincreases.ThisextraeectivemassgaincontributetotheholemobilityenhancementforGeatveryhighstress. 520 for(001){orientedMOSFETsandFigure 521 for(110){orientedpMOSFETs.For(110)GepMOSFETsunderuniaxialstress,thestraineectissimilarto(110)SipMOSFETs.Thestrongquantumconnementwarpsthesubbandstructureandresultsinsmallholeeectivemass,whichexplainsthehigherunstrainedholemobilitythan(001)GepMOSFETs.Astheuniaxialcompressivestressisapplied,thestraineectisunderminedbythestrongquantumconnementandonlywarpsthehighenergyregionofeachsubband.Asaresult,theholemobilityisnotenhancedassignicantlyas(001){orientedpMOSFETs. 114 PAGE 115 (b) Phononscatteringratevsuniaxialcompressivestress:(a)Acousticphonon,and(b)opticalphonon. 115 PAGE 116 SurfaceroughnessscatteringratevsuniaxialcompressivestressforGeandSi. 116 PAGE 117 Mobilityenhancementcontributionfromeectivemasschange(solidlines)andphononscatteringratechange(dashedlines)forSiandGeunderuniaxialcompressivestress. (b) Conned2Denergycontoursfor(001){orientedGepMOSwithuniaxialcompressivestress. 117 PAGE 118 (b) Conned2Denergycontoursfor(110){orientedGepMOSwithuniaxialcompressivestress. eachstresstypeforbothgermaniumandsiliconissimilaruniaxialcompressivestresson(001)orientedtransistorshasthemostholemobilityimprovementmainlyfromthereducedholeconductivityeectivemass.Uniaxialcompressivestresson(110)orienteddevicesdoesnotprovideasmuchimprovementduetothestrongquantumconnementunderminingthestraineect.Holemobilityisdegradedunderlowbiaxialtensilestressduetothesubtractivenatureofthestrainandverticalelectriceldeectsandhencetheincreaseoftheaverageeectivemass.Themobilityisenhancedathighstressbecauseofthereductionoftheintersubbandscatteringrate.BiaxialcompressivestressdoesnotimprovetheholemobilitymuchduetothesmallDOSafterband/subbandwarpingandnotmucheectivemassgain. 118 PAGE 119 119 PAGE 120 Straineectonholemobilityimprovementin(001)and(110)GeandSi1xGexpMOSFETsiscalculatedforthersttime.ThemobilityenhancementatlowstressissimilartoSi.Athighstress,themaximummobilityenhancementfactorfor(001)GeislargerthanSiduetothegreatereectivemasschange,especiallyathighstress.ThephononscatteringratechangeforGepMOSFETsisalittlesmallerthanSi.For(110)GepMOSFETs,strongquantumconnementisfoundandthestraininduced 120 PAGE 121 GermaniumisonenewmaterialthathasbeenconsideredtoreplacesiliconinCMOStechnology.UniaxialstrainevenhashigherenhancementongermaniumpMOS.Buttheexperimentalworkisstilllackforgermanium.Peoplearestilltryingtondoutthebestlayout,properdielectricandgatematerials.Itwillbealongwaybutdenitelyworthworkingon. Howaboutafterallofthis?Therewillbeanultimatelimitforthescalingthatballistictransportwilltakeplaceandthemobilityconceptwillnotbevalid.Willstrainstillbeusefulatthatstage?Theanswerisprobablyyes,sincethestraincanreducetheeectivemassofthecarriersandthiswillstillhelpthetransport.Thatbeingsaid,seriouscalculationwillbenecessarytofurtherexplainthis. 121 PAGE 122 StressisdenedastheforceFappliedonunitareaA. A Anystressonanisotropicsolidbodyinacartesiancoordinatesystemcanbeexpressedasastressmatrix[ 13 99 ], whereii=limAi!0Fi 100 ]asshowninFigure A1 .Thisstressmatrixcompletelycharacterizesthestateofstressatcrystals. ForstressSalong<100>direction,thematrixcanbewrittenas ForstressSalongboth<100>and<010>{direction(biaxialstress), 122 PAGE 123 Stressdistributiononcrystals. StressSalong<110>directionisalittlecomplicated.Thestressisappliedonboth(100)and(010)planes.Ifweresolveeachcomponentalongxandyaxestogetbothnormalandshearterms,eachtermhasthesamemagnitudeofS=2.Thestresstensorcanbeexpressedas, ForstressSalong<111>direction,basedonthesimilaranalysis,thestressisactuallyactedon(100),(010),and(001)planes.Eachcomponentcanberesolvedalongx,y,andzaxesandthestressalongeachdirectionisS=3.Thereforethestresstensoris, 123 PAGE 124 Strainisdenedasthedistortionofastructurecausedbystress.Normalstrainisdenedastherelativelatticeconstantchange[ 13 99 ], wherea0andaarelatticeconstantbeforeandafterthestrain. However,thedeformationofthecrystalcannotbefullyrepresentedwiththenormalstrain.Italsohassheartermsthataredenedaschangeintheinterioranglesoftheunitelement.Likestress,straincanalsobeexpressedwithasymmetric33tensoror61vector[ 100 ]. 124 PAGE 125 99 ]. or, Fordiamondorzincblendetypecrystal,stinessmatrixandcompliancematrixcanbesimpliedas[ 99 ] 125 PAGE 126 ElasticstinessesCijinunitsof1011N=m2andcompliancesSijinunitsof1011m2=N C11C12C44S11S12S44 Ge1.2920.4790.6700.9640.2601.49 Thestinessandcompliancecoecientsforsiliconandgermaniumarelistedinthefollowingtable. Let'sgobacktothestraintensor.Eachstraincanbedecomposedtotwocomponents:hydrostatictermandshearterm.Thesheartermcanbefurtherdecomposedtoshear{100termwhichonlyhasdiagonalelementsandshear{111termwhichonlycontainsnondiagonalelements. PAGE 127 30BBBBBB@xx+yy+zz000xx+yy+zz000xx+yy+zz1CCCCCCAhydrosatic+1 30BBBBBB@2xx(yy+zz)0002yy(xx+zz)0002zz(xx+yy)1CCCCCCAshear100+0BBBBBB@0xyxzyx0yzzxzy01CCCCCCAshear111 Thehydrostaticterminthestraintensorshiftstheenergyofallthebandsinsemiconductorsbythesameamountsimultaneouslybutdoesnotcausebandsplitting,sinceitisactuallyaconstantandinthecalculationofthebandenergyitonlyactslikeaddinganadditionalpotentialtermtothehamiltonian.Thesemiconductortransportpropertyisindependentonthehydrostaticstrainterm.Fortwodierentstress,aslongasthesheartermsoftheirstraintensorsareequal,theirimpacttothecarriermobilityshouldbeidentical. Stresscanbeappliedtosemiconductorsfromanydirection.ForasiliconMOSFET,onlyinplanebiaxialstressorchanneldirectionuniaxialstresshastechnologicalimportance.Thecommonsiliconwafersthatareusedinindustryare(001){oriented,andnormallythechanneloftheMOSFETisalong<110>{direction.Biaxialstressheremeansthatthestressisappliedinboth<100>{and<010>{directionsofthewaferwiththesamemagnitude.Uniaxialstressrepresentsthestressalongthe<110>channeldirection.Thisstressisalsocalleduniaxiallongitudinalstress.Inthesamemanner,uniaxialtransversestressnormallymeanstheuniaxialstressappliedperpendiculartothechanneldirection.Bothofthosestressesareappliedintheplaneofthewafer,thereforetheyarealso\inplane"stresses.Anotherkindofuniaxialstressiscalled\outofplane"uniaxial 127 PAGE 128 Fortheoutofplaneuniaxialstressandtheinplanebiaxialstresson(001)wafer,thestrainmatricesonlyhavediagonaltermsandallnondiagonaltermsarezero.Thequestionis,howdothesetwostressesdierfromeachother?Let'sassumewehaveoutofplaneuniaxialstressononesampleandinplanebiaxialstressonanothersample.Forcase1,basedon(1.4)and(1.15),thestraintensorcanbeexpressedas,intheformof(1.16), Forinplanebiaxialstress, PAGE 129 (1.17)and(1.18)showthatthehydrostatictermsofthosetwostraintensorsaredierent,butthesheartermsareidentical.Thistellsusthatthebiaxialtensile(compressive)stressshouldhavethesameeectastheoutofplaneuniaxialcompressive(ortensile)stressindeterminingthetransportpropertyoftheholes. 129 PAGE 130 Thepiezoresistance,orpiezoresistiveeect,describestheelectricalresistancechangeofmaterialscausedbyappliedmechanicalstress.TherstmeasurementofpiezoresistancewasperformedbyBridgmanin1925andextensivestudyonthistopicwasdoneeversince.In1954,SmithmeasuredthepiezoresistanceeectonSiandGe[ 7 ].ThiseectbecomesmoreandmoreimportantduetothewideapplicationofSiandGeoncontemporaryCMOStechnology. Similartostressandstrain,thechangeofresistivityofamaterialisasymmetricalsecondranktensor.Thetensorconnectingthestressandthepiezoresistanceisoffourthrank.ForSiandGe,wecansimplifythetensoras[ 7 ] Themostgeneralformofatwodimensionalpiezoresistancetensorintheinversionlayeris[ 8 ] For(001),(110),and(111)surfaceorientedSi(orGe),14=41=24=42=0(principleaxish001ifor(001)and(110)surface,h110ifor(111)surface).Wecanfurthersimplifythepiezoresistancetensoras[ 8 ] 130 PAGE 131 For(001)surfaceorientedSiandGe,11=22and For(111)surfaceorientedSiandGe,44=1112and Inthepiezoresistancetensors,11representsthelongitudinalpiezoresistancecoefcient(alongh100ifor(001)and(110)surface).12isthetransversepiezoresistnacecoecient(alongh010ifor(001)and(110)surface).InstandardMOSFETs,thechanneldirectionisalongh110iandtheuniaxialstressisappliedeitheralongh110iorh110i.Byrotationaltransformationofthetensorthenewlongitudinalandtransversepiezoresistancecoecientsbecome[ 8 ] 2(11+12+44) (B{6) and 2(11+1244) (B{7) 131 PAGE 132 . 132 PAGE 133 [1] Y.TaurandT.H.Ning.FundamentalsofModernVLSIDevices.CambridgeUniversityPress,Cambridge,UK,1998. 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