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- Permanent Link:
- https://ufdc.ufl.edu/UFE0021757/00001
## Material Information- Title:
- Strain Effects on Silicon CMOS Transistors Threshold Voltage, Gate Tunneling Current, and 1/f Noise Characteristics
- Creator:
- Lim, Jisong
- Place of Publication:
- [Gainesville, Fla.]
- Publisher:
- University of Florida
- Publication Date:
- 2007
- Language:
- english
- Physical Description:
- 1 online resource (128 p.)
## Thesis/Dissertation Information- Degree:
- Doctorate ( Ph.D.)
- Degree Grantor:
- University of Florida
- Degree Disciplines:
- Electrical and Computer Engineering
- Committee Chair:
- Thompson, Scott
- Committee Co-Chair:
- Nishida, Toshikazu
- Committee Members:
- Ural, Ant
Srivastava, Ramakant Fan, Zhonghui H. - Graduation Date:
- 12/14/2007
## Subjects- Subjects / Keywords:
- Charge carriers ( jstor )
Compressive stress ( jstor ) Drains ( jstor ) Electric current ( jstor ) Electrons ( jstor ) Energy levels ( jstor ) Noise measurement ( jstor ) Noise spectra ( jstor ) Oxides ( jstor ) Tensile stress ( jstor ) Electrical and Computer Engineering -- Dissertations, Academic -- UF biaxial, compressive, deformation, drain, gate, hamiltonian, hydrostatic, jig, mechanical, noise, oxide, shear, source, strain, stress, substrate, tensile, threshold, trap, tunneling, uniaxial - Genre:
- Electronic Thesis or Dissertation
born-digital ( sobekcm ) Electrical and Computer Engineering thesis, Ph.D.
## Notes- Abstract:
- The study of strain effects on CMOS (complementary-metal-oxide-semiconductor) transistors has been mainly focused on drive current enhancements. In computer central processing unit (CPU) chips, strained CMOS transistors play an important role for high speed computer operations since the CPU speed is directly related to the current drive capabilities of the transistors controlling the CMOS logic circuitry in the chips. In addition to this strain effect on channel carrier mobilities, strain also affects other important physical properties in MOSFETs. This dissertation investigates such strain effects on the MOSFET operation as threshold voltage, gate tunneling current, and low-frequency 1/f noise characteristics. Strain engineering of the MOSFET channel alters the inversion subband energy levels which, in turn, influences these physical properties as well as channel carrier mobilities. More specifically, shear strain reduces symmetry in silicon and lifts the degeneracy of both conduction and valence bands. As a result, the constant energy surfaces are severely warped for the valence band whereas the shapes of the energy surfaces are unchanged to the first order in stress for the conduction band. The strain effects on silicon MOSFETs are then classified as an effect of band splitting and shifts due to the hydrostatic and shear strain components for both conduction and valence bands, and an additional effect of subband (heavy- and light-hole) effective mass change due to the band warping for the valence band. Based on these key strain effects on MOSFETs, the calculated values for threshold voltage shifts are quite consistent with the measured data for n-channel MOSFETs under uniaxial tensile stress as well as the published experimental data for biaxially tensile-strained n-channel MOSFETs. Gate tunneling currents are also well predicted by this strain model including band splitting, shifts and warping. Furthermore, conduction deformation potential constants are determined through this gate tunneling current measurements on n-channel MOSFETs under mechanical stress. Strain effects on 1/f noise are also studied in conjunction with applications of strained devices to high performance RF or high speed CMOS circuits. Detailed physical mechanisms of the strain effects on 1/f noise power spectral density (PSD) are identified and the contribution of each mechanism to the resultant change in 1/f noise PSD is estimated on the basis of the measured data. ( en )
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- In the series University of Florida Digital Collections.
- General Note:
- Includes vita.
- Bibliography:
- Includes bibliographical references.
- Source of Description:
- Description based on online resource; title from PDF title page.
- Source of Description:
- This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
- Thesis:
- Thesis (Ph.D.)--University of Florida, 2007.
- Local:
- Adviser: Thompson, Scott.
- Local:
- Co-adviser: Nishida, Toshikazu.
- Electronic Access:
- RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2009-12-31
- Statement of Responsibility:
- by Jisong Lim.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright Lim, Jisong. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Embargo Date:
- 12/31/2009
- Classification:
- LD1780 2007 ( lcc )
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PAGE 1 1 STRAIN EFFECTS ON SILICON CMOS TRANSISTORS: THRESHOLD VOLTAGE, GATE TUNNELING CURRENT, AND 1/ f NOISE CHARACTERISTICS By JI-SONG LIM A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007 PAGE 2 2 2007 Ji-Song Lim PAGE 3 3 To my family whose encouragement and support have made this possible PAGE 4 4 ACKNOWLEDGMENTS I would lik e to express my sincere gratitude to all those who gave me the opportunity to complete this thesis. Special thanks are due to my principal advisor, Dr. Scott E. Thompson, for his constant encouragement and expert guidance. He has introduced me to the research area of strained silicon CMOS transist ors, and has provided valuable advice and direction throughout my doctorate studies. I would also like to thank my co-advisor, Dr. Toshikazu Nishida, for encouraging my research work, for his consistent excellent tec hnical advice, and for teaching me to write a good research paper. Thanks also go to Drs., Ramaka nt Srivastava, Ant Ural, and Hugh Fan for their interest and participation in serving on my committee and th eir suggestions and comments. All the former and current members of the strained silicon group have contributed immeasurably to my research work through inter active discussions. I have been privileged to work with Kehuey Wu, Guangyu Sun, Andy Koeh ler, Sagar Suthram, Uma Aghoram, Nirav Shah, Younsung Choi, Nidhi Mohata, Xiaodong Yang, Srivatsan Parthasarathy, Min Chu, Xiaoliang Lu, Yongke Sun, Toshi Numata, Hyunwoo Park, Tony Acosta. These people have made me working in the lab and office with lots of fun, and ma ny have provided useful advice for me. I am especially thankful to Kehuey Wu for his help with the proper use of a wafer bending jig. I dedicate this dissertation to my family, without whose s upport and patience so long time, my achievement could not have been complete. Last but not least, I especi ally like to thank my loving mother, brother, sisters and relatives in Ko rea for their everlasting love and trust in my efforts, and for standing by me wh enever I needed them the most. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........7 LIST OF FIGURES.........................................................................................................................8 ABSTRACT...................................................................................................................................11 CHAP TER 1 INTRODUCTION..................................................................................................................13 Historical Overview of Research on Strain Effects on Sem iconductors................................ 13 Motivation...............................................................................................................................16 Scope and Organization......................................................................................................... .17 2 STRAIN EFFECT ON SILICON ENERGY BAND............................................................. 19 Representation of Strain Compone nt with S-m atrix Element................................................ 19 Strain Effect on Conduction and Valence Band.....................................................................23 Strain Effect on Conduction Band...................................................................................24 Strain Effect on Valence Band........................................................................................ 27 3 STRAIN EFFECT ON THRESHOLD VOLTAGE...............................................................36 Wafer Bending Experiment.................................................................................................... 36 Strain Effect on Threshold Voltage........................................................................................43 Strain-Induced Fermi Energy Level Shift....................................................................... 43 Strain-Induced Threshold Voltage Shift.......................................................................... 48 Results and Discussion......................................................................................................... ..52 Summary.................................................................................................................................59 4 STRAIN EFFECT ON GATE TUNNELING CURRENT....................................................62 Measurement of Direct Tunneling Current.............................................................................62 Direct Tunneling Current Mode l from Inversion Electron ..................................................... 63 Explanation for Direct Tunneling Current Change with Stain ........................................63 Physical Model for Direct T unneling Current in n-MOSFET ......................................... 65 Extraction of Conduction Band Deformation Potential Constant................................... 68 Summary.................................................................................................................................72 PAGE 6 6 5 STRAIN EFFECT ON LOW-FREQUENCY 1/F NOISE CHARACTERI STICS............... 73 Introduction................................................................................................................... ..........73 Conventional Charge Trapping Model...................................................................................74 Wafer Bending Experiment on 1/f Noise ................................................................................82 Measurements of dc Currents and Drain Current Noise PSD ......................................... 82 Measuremen t Results ....................................................................................................... 84 n-MOSFET under tensile stress............................................................................... 84 n-MOSFET under compressive stress...................................................................... 85 p-MOSFET under compressive stress...................................................................... 86 Data analysis............................................................................................................87 Charge Trapping Model under Strain..................................................................................... 93 Mechanism for Change in Noise PSD under Strain........................................................ 93 Charge Trapping Model under Strain.............................................................................. 94 Strain effect on the exponent in 1/ f noise power spectrum................................. 98 Strain effect on the noise PSD magnitude.............................................................. 100 Summary...............................................................................................................................105 6 SUMMARY AND RECOMMENDATIO NS FOR FUTURE WORK ................................ 107 Summary...............................................................................................................................107 Recommendations for Future Work..................................................................................... 108 APPENDIX A CONDUCTION BAND DEFORMATI ON POTENTIALS For GE ................................... 110 B YOUNGS MODULUS IN A (110)-SI W AFER................................................................. 114 C STRAINED-SI MOFETS ON A (110) wafer...................................................................... 116 LIST OF REFERENCES.............................................................................................................122 BIOGRAPHICAL SKETCH.......................................................................................................128 PAGE 7 7 LIST OF TABLES Table page 2-1 Strain components for three principal uniaxial stresses and an in-plane biaxial stress,and elastic com pliance constant values for Si.......................................................... 23 2-2 Conduction band splittings along the [001] direction for [100], [001], [110] and [111] uniaxial stresses and an in-plane biaxial stress. .......................................................28 2-3 Effective masses of heavyand light -holes under stress, cited from [27].........................29 2-4 Valence band splitting for [100], [110], and [111] uniaxial stresses and an in-plane biaxial stress. ......................................................................................................................32 3-1 Electron population in three lowest ener gy states at the threshold voltage for a uniform substrate doping density of NA............................................................................ 53 5-1 Relative change in noise PSD paramete rs for an applied stress of 100MPa. ....................93 5-2 Comparison of the measured and calculat ed relative m agnitude changes in drain current noise PSD, SID(1 Hz ; )/ SID(1 Hz ;0), at a stress of 100MPa................................ 105 PAGE 8 8 LIST OF FIGURES Figure page 1-1 Uniaxiallyand biaxially-strained Si MOSFETs............................................................... 15 2-1 Three uniaxial stresses applied along [100], [110], and [ 111] directions. .........................22 2-2 Energy band structure of silicon for the [100] and [111] direction. ..................................24 2-3 Six constant energy ellipsoids in k -space near the conduction band edges....................... 25 2-4 Average energy level shift and band splitting in the conduction band along the kz direction under in-pla ne biaxial stress............................................................................... 28 2-5 Stress effects on the valence energy bands near k = 0 (a) Unstressed. (b) Under [111] uniaxial com pression................................................................................................ 30 2-6 Stress effects on the valence energy bands near k = 0 (Adapted from [27])(a) Under in-plane biaxial tension. (b) U nder [110] uniaxial compression........................................ 30 2-7 Valence band edge shift (average energy level shift plus band splitting) vs. in-plane biaxia l stress with (6 6 Hamiltonian) and without spin-orbit interaction (4 4 Hamiltonian)......................................................................................................................33 2-8 Band splitting in the valence band v s. in-plane biaxial stress with (6 6 Hamiltonian) and without spin-o rbit interaction (4 4 Hamiltonian)....................................................... 34 2-9 Conduction and valence band edge shifts under in-plane biaxial (uniaxial) stress along the out-of-plane direction. ........................................................................................ 35 3-1 Two types of fixtures to simulate uni axially-strained and biaxially-strained MOSFETs.. ........................................................................................................................37 3-2 Illustration of a uniaxial wafer bending jig........................................................................ 38 3-3 Youngs modulus of Si versus direction in the (001) plane........................................... 40 3-4 Illustration of a ring-type biaxial stress. ............................................................................ 41 3-5 Fermi energy level shift vs in-plane biaxial stress. ........................................................... 47 3-6 Energy band diagram of the n+-polygate and p-type Si substrate...................................... 50 3-7 Lowest two energy levels of inversio n electrons at the threshold voltage. ....................... 50 3-8 Illustration of each component constituti ng the th reshold voltage shift formulas............. 52 3-9 Threshold voltage shifts vs. stress..................................................................................... 54 PAGE 9 9 3-10 Illustration of 2-D and 3-D repopulation processes occurred in the conduction band edges under uniaxial [110] or in-plane tensile stress. ........................................................ 58 3-11 Two types of strained p-MOSFETs commonl y adopted in the indus try as a standard...... 61 4-1 Change in nand p-MOSFET gate tunneling current versus stress [20]. ..........................63 4-2 Ground and second lowest energy states of p-MOSFETs in the inversion potential well.. ...................................................................................................................................65 4-3 (a) Schematic band diagram for the gate direct tunnel current in an n-MO SFET on a (100)-wafer. (b) 2 and 4 energy level shifts unde r compressive stress and MOSFET inversion layer confinement.............................................................................. 66 4-4 (a) Relative direct tunnel current change [ IG( )/ IG(0)] versus applied compressive stress at different gate voltages. Data (squares) were measured on industrial MOSFETs. The solid lines are our model. (b)-(d) Breakout of the various contribution to IG( )/ IG(0) at different gate voltages:..................................................... 69 4-5 Stress-induced repopulation between 2 and 4 ground state electrons............................70 4-6 Change in slopes (d[ IG( )/ IG(0)]/ d ) versus gate voltage with 95% confidence error bars.....................................................................................................................................71 5-1 Illustration of 1/f noise generation at the SiO2/Si interface, adapted from [64]................ 74 5-2 Discrete modulation of the channel ch arge current due to a single trap. ........................... 75 5-3 Contribution to noise PSD by oxide traps only within a few kT around f nE.....................77 5-4 Shifts in trap location due to oxide band bending.............................................................. 81 5-5 Schematic block diagram for 1/f noise and dc current measurements............................... 83 5-6 Noise power spectrums for an n-channel MOSFET with and without the setup noise. .... 84 5-7 Measurements of an n-MO SFET under tensile stress........................................................ 85 5-8 Measurements of an n-MOSFET under compressive stress.............................................. 86 5-9 Measurements of a p-MOSFET under compressive stress................................................ 87 5-10 Schematic illustration for extracting a loca l average value of noise PSD at a specific frequency of interest from the m easured raw data............................................................. 89 5-11 Analysis of p-channel MOSFET data under compressive stress....................................... 90 5-12 Analysis of n-channel MOSFET data under tensile stress................................................. 91 PAGE 10 10 5-13 Analysis of n-channel MOSFET data under compressive stress....................................... 92 5-14 Schematic band diagram of an n-channel MOSFET under mechanical stress.................. 96 5-15 Trap redistribution for an n-cha nnel MOSFET under m echanical stress........................ 100 5-16 Relative change in trapping probability vs. tunneling channel car rier energy for an applied stress of 100MPa. ................................................................................................ 101 A-1 Eight-fold degenerate -valleys in the Ge conduction band. ..........................................113 B-1 Youngs modulus of Si as a function of direction in the (110) plane......................... 115 C-1 (a) Representation of an arbitrary stress in a spherical coordinate. (b) New direction specified in a (110)-wafer.............................................................................................117 C-2 (a) Six ellipsoids of -valleys. (b) Diagram for cal culating the effective band splitting along the [110]-direction................................................................................... 120 C-3 (a) Effective band splitting vs. stress direction. (b) Conductivit y effective m ass of 4 valley electrons vs. channel direction.............................................................................. 121 PAGE 11 11 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy STRAIN EFFECTS ON SILICON CMOS TR ANSISTORS: THRESHOLD VOLTAGE, GATE TUNNELING CURRENT, AND 1/ f NOISE CHARACTERISTICS By Ji-Song Lim December 2007 Chair: Scott E. Thompson Cochair: Toshikazu Nishida Major: Electrical and Computer Engineering The study of strain effects on CMOS (c omplementary-metal-oxide-semiconductor) transistors has been mainly focused on drive current enhancements. In computer central processing unit (CPU) chips, strained CMOS transi stors play an important role for high speed computer operations since the CPU speed is direc tly related to the current drive capabilities of the transistors controlling the CMOS logic circuitry in the chips. In addition to this strain effect on channel carrier mobilities, strain also affect s other important physical properties in MOSFETs. This dissertation investigates such strain eff ects on the MOSFET operation as threshold voltage, gate tunneling current, and low-frequency 1/f noise characteristics. St rain engineering of the MOSFET channel alters the inve rsion subband energy levels whic h, in turn, influences these physical properties as well as channel carrier mobilities. More specifically, shear strain reduces symmetry in silicon and lifts th e degeneracy of both conduction and valence bands. As a result, the constant energy surfaces are severely warped for the valence band whereas the shapes of the energy surfaces are unchanged to the first orde r in stress for the conduction band. The strain effects on silicon MOSFETs are then classified as an effect of band splittin g and shifts due to the hydrostatic and shear strain components fo r both conduction and valence bands, and an PAGE 12 12 additional effect of subband (heavyand light-h ole) effective mass change due to the band warping for the valence band. Based on these key strain effects on MOSFETs, the calculated values for threshold voltage shifts are quite consistent with the measured data for n-chan nel MOSFETs under uniaxial tensile stress as well as the published experimental data for biaxially tensile-strained n-channel MOSFETs. Gate tunneling currents are also well pr edicted by this strain model including band splitting, shifts and warping. Furthermore, conduction deformation potential constants are determined through this gate tunneling curre nt measurements on n-channel MOSFETs under mechanical stress. Strain effects on 1/f noise are also studied in conj unction with applications of strained devices to high performance RF or high speed CMOS circuits. Detailed physical mechanisms of the strain effects on 1/f noise power spectral density (PSD) are identified and the contribution of each mechanism to the resultant change in 1/f noise PSD is estimated on the basis of the measured data. PAGE 13 13 CHAPTER 1 INTRODUCTION Historical Overview of Research on Strain Effects on Semiconductors Over the past 50 years, strain effects on se m iconductors such as Si and Ge have been extensively studied both theoretically and experimentally. Ba rdeen and Shockley first introduced a deformation potential theory in 1950 to expl ain phonon scattering in semi conductors [1]. It was shown in their paper that the el ectronor hole-phonon interaction cau ses a static displacement of the atoms, thus resulting in the conduction or valence band energy shifts Later in 1956, Herring and Vogt further generalized the deforma tion potential method [2] and formulated the conduction band energy shift as a function of strain together with a couple of deformation potential constants. Strain effects on valence ba nds were also quantified by Pikus and Bir [3]. They constructed a strain Hamiltonian for valen ce bands based on an invariance property of the angular momentum under symmetry operations of a crystal. In 1963, Hasegawa added the spinorbit split-off (SO) band effects to this Hamilt onian [4], which has become a currently-used typical strain Hamiltonian form of valence bands. Experimentally, numerous endeavors have been made to determine deformation potentials in strained Si, Ge and other materials. In 1954, Smith first observed the piezoresistance change in strained Si and Ge [5]. Since th en, piezoresistance measurements have been used as an important experimental method for determining the deformation potential constants [6]. Hensel and Feher conducted a cyclotron resonance experiment in 1 963 [7] to determine deformation potentials in Ge. By the photoluminescence technique, Balslev de termined deformation potentials for Si and Ge [8], and Bhargave and Nathan for GaAs [9 ]. Besides, Pollak and Cardona used a piezoelectroreflectance technique [10] to measure deformation potentials for Si, Ge and GaAs. In all these experiments mentioned above, however, only strain-induced relative energy level shifts could be measured. As a result, the dilation de formation potential for Si (commonly denoted by PAGE 14 14 symbol, d), which is related to the absolute energy level shift in the cond uction band, has been reported over a very wide range including opposite signs, 1.13 to -10.7 eV [11, 12]. Recently in our lab, an experimental met hod to determine a value of d for Si has been proposed [13]. The method is based on the change in the gate di rect tunneling currents of Si n-MOSFETs under externally applied mechanical stress, from whic h the obtained value is quite consistent with theoretical works. While the earlier studies on strain were ma inly focused on bulk semiconductor properties, practical studies for CMOS device applicati ons began in the 1980s with the advance of fabrication process technologies [14, 15]. In order to increase the drive current (or charge carrier mobility) in Si MOSFETs, typically two types of process techniques were developed for introducing strain into the Si surface channel. As shown in Fig.1-1, one type of technique is to use epitaxial technology to form a thin layer of Si on top of the relaxed silicon germanium (Si1xGex) layer [16, 17]. Since the Si1-xGex layer has a larger lattice constant, the thin Si channel layer is permanently under biaxial tensile strain. The other type is source/drain engineering and thin film process techniques, which were deve loped by Intel in the 1990 s, such as high-stress nitride capping layers around the ga te and selective epitaxial Si1-xGex in the source/drain regions [18]. The nitride capping layer a pproach is applied to n-MOSFETs for uniaxial tensile strain and the embedded Si1-xGex approach to p-MOSFETs for uniaxia l compressive strain [18]. Strain has been re cognized as one of the key technology fe atures for scaled MOSFET devices [19, 20]. Over th e past 30 years, the downscali ng of MOSFET dimensions has continued to improve device performances [21], a nd actually great benefits have been achieved in terms of the transistor speed, de nsity, cost, and power consumption. PAGE 15 15 Figure 1-1. Uniaxiallyand biaxia lly-strained Si MOSFETs, adapte d from [18, 27]. (a) Nitride capping layered n-MOSFET. (b) p-MOSFET with a Si1-xGex source and drain. (c) Biaxially-strained Si MOSFET on a relaxed Si1-xGex layer. However, as the dimension scaling approaches physical limitations (for example, ~30 nm gate length and ~1 nm gate oxide thickness), a number of difficulties such as short channel effects and increased gate leakage currents arise, thus making the device scaling an increasingly challenging task. To maintain th is historical trend of perfor mance improvements through scaling, the industry requires noble soluti ons. One solution under active st udy to reduce the gate leakage current is the use of high-k materials for the gate dielectric [21]. The introduction of highdielectric materials, however, results in drive current loss due to carrier mobility degradation (a) (b) (c) PAGE 16 16 [19]. Then currently, strained-Si MOSFETs are drawing researchers attention again as an alternative to compensating for this drive current loss. Motivation Typically, biaxial tens ile stress is introduced via a thin epitaxial Si channel grown on a relaxed Si1-xGex substrate. Alternatively, carrier mob ility enhancement can be obtained using longitudinal uniaxial tensile stress typically introduced with a nitride capping layer or through an embedded Si1-xGex source/drain. Since the strain capping layer or the embedded Si1-xGex approach requires negligible alte rations to a standard CMOS pr ocess flow, uniaxial strain was widely adopted at the 90-nm ge neration [18]. Biaxial and uniaxial tensile strain-enhanced channel carrier mobility has been well studied, and shown to primarily result from lower conductivity effective mass under gate bias and stra in and reduced intervalley scattering [14-21]. On the other hand, less attention has been paid to other important physic al properties such as threshold voltage, gate tunneli ng current and low frequency 1/f noise characteristics. Understanding the threshold voltage shift is important when determining the performance gain of strained Si. Since performance benchm arking needs to be done at constant off-state leakage, an adjustment to compensate for the strain-induced threshold vo ltage shift is required. This adjustment is typically accomplished by increasing the well doping concentration which degrades mobility and increases junction capacitance. The study on threshold voltage is also important in relation to device lifetime and reliability. In the design of strained silicon devices, it is essential to accurately quantify strain-induced energy level shifts and splitting which are m odeled by deformation potentials. Strain-induced band splitting removes conduction and valence band degeneracy and is primarily responsible for the engineered mobility enhancement. While the shear deformation potentials used to calculate energy level splitting are well known with good a ccuracy from piezoresistance measurements PAGE 17 17 (u = 9.16 eV), the hydrostatic deformation potential is difficult to directly measure using the conventional optical techniques. As a result, a very wide rang e of values with opposite signs have been reported (1.13 to -10.7 eV) [11, 12]. This results since the optical experimental techniques directly measure differences in energy levels (for example, strain-induced band gap narrowing) but not the absolute pos ition of the energy levels which leaves large uncertainty in important band parameters such as the electron affinity of strained-Si. Recently, low frequency 1/f noise has drawn researchers attention in c onjunction with applications of strained devices to analog circuits. Since 1/f noise often acts as a critical limiting factor in analog circuit desi gn, it is essential to unders tand the effect of strain on 1/f noise and quantify its magnitude. Although some studies exist on strain effects on 1/f noise, they mainly focus on processing aspects in strained SiGe MOSFETs []. None has elucidated fundamental strain effects on 1/f noise yet. Scope and Organization This dissertation deals with strain effects on MOSFET operations; threshold voltage, gate tunneling current a nd low frequency 1/f noise characteristics. The outline is as follows. Chapter 2 gives a brief introduction of the st rain-stress relation and a representative method of strain components in terms of elasti c compliance constants. Based on this strain representation, the deformation potentials of the conduction and valence bands are calculated, and the band edge shifts and splitting are discussed in detail as strain effects. In Chapter 3, we first examine our stress-appl ying apparatuses of the uniaxial and biaxial jigs and then review the st rain-induced threshold voltage model for n-MOSFETs. Each component of the model is also analyzed thor oughly in conjunction with its underlying physical mechanism. PAGE 18 18 In Chapter 4, strain effects on the gate tunneling current are dealt with. Based on experimental observations, qualitative analyses are made first for both nand p-MOSFETs. The detailed model for n-MOSFETs is then presented. Next, deformation potential constants are extracted from the measured data on the basis of this model prediction. In Chap. 5, strain effects on 1/f noise are dealt with. The experimental method with an electrical measurement setup is first introduced. The measurement results and data analysis for both nand p-MOSFETs are then discussed. Detaile d mechanisms of strain effects on noise PSD are identified and the contribution of each mechanism to the total noise PSD is estimated. Finally, Chap. 6 is the summary and the recommendation for the future work. PAGE 19 19 CHAPTER 2 STRAIN EFFECT ON SILICON ENERGY BAND In this chapter, we will b riefly review the basics of strain effects on both conduction and valence energy bands which are re levant to our work. The magnitude of the energy band shift with strain is typically calculated by introducing elastic compliance constants (S-matrix elements) or stiffness constants (C-matrix elements) to represent strain components. Of these two representations, the S-matrix representation will be us ed throughout this dissertation. Representation of Strain Component with S-matrix Element A solid under stress is generally charact erized by a second order stress tensor whose elements ij are defined as the j-component of the force transferre d per unit area perpendicular to the i-direction. The sign is positive for tensile stress and negative fo r compressive stress. Simple equilibrium considerations yield that ij = ji. Similarly, the deformation can be described by a second order strain tensor : 1 2j i ijji jiu u x x (2-1) where u is the displacement of a solid. The relati on between stress and strain can be also given by the fourth order elastic compliance tensor S : .ijijklklS (2-2) This constitutes a sequence of nine equations, since each strain component of ij is a linear combination of all the stress components of ij, that is, ij = Sij1111 + Sij1212 +..+ Sij3333. In the constitutive relation, tota lly there are 9 components each for both strain and stress tensors, and 81 independent components for th e compliance tensor. However, both and are symmetric, PAGE 20 20 with 6 rather than 9 components each. As a resu lt, we can express Eq. (2-2) in a reduced matrix form where 81 compliance components reduc e to 36 as shown in reference [22]: 11 111213141516 22 212223242526 33 313233343536 414243444546 44 515253545556 55 616263646566 66, SSSSSS SSSSSS SSSSSS SSSSSS SSSSSS SSSSSS (2-3) where the 6 1 column matrices of both strain and st ress are defined, respectively, as 111 222 333 42 3 5 13 6 12, 2 2 2 111 222 333 423 5 13 6 12 (2-4) This resulting matrix relation of Eq. (2-3) is no longer a tensor form because each strain component does not follow the coordinate transformation rule as in Eq. (2-2). Further, in cubic crystals such as Si and Ge, the compliance matrix is simplified to [22] 111212 121112 121211 44 44 44000 000 000 00000 00000 00000SSS SSS SSS S S S S (2-5) Now, let us calculate strain components in a cubic crystal for uniaxial stresses with magnitude applied along crystallographi c axes such as [100], [110], and [111] directions. As shown in Eq. (2-3), since strain components ar e represented as a product of the S-matrix and PAGE 21 21 stress matrix, we need to first get stress component s for each type of stress. For a [100] stress, it is trivial; all stress comp onents are zero except that xx = For the other two directions of stresses, [110] and [111], a littl e care need to be taken so that all stress components can be obtained as second order stress tensor elements. A simple a nd straightforward way is to introduce a dyad notation which is e quivalent to a second order tens or representati on. A dyad is simply a pair of vectors and its oper ation (namely a dyadic) is defined as () () ,.xyzxyz xxxyxzyxyyyz zxzyzzABAiAjAkBiBjBk ABiiABijABikABjiABjjABjk ABkiABkjABkkforanytwovectorsAandB (2-6) The nine elements above correspond directly to e ach element of a second order tensor. Further, in consideration of the properties of a stress tensor (the symmetry condition ij = ji, and rotation-invariant quantity Tr(ij) = magnitude of stress), we can obtain the following stress tensor element in terms of the dyad notation, .ij ij (2-7) Based on Eq. (2-7), we can easily express any st ress components for any stress directions, and consequently obtain strain components through the st rain-stress matrix rela tion. Fig. 2-1 shows three crystallographic stress direct ions and their vector elements. For a [110] stress, the stress vector is given by, (1/2,1/2,0). A simple calculation using Eq. (2-7) leads to xx = xy = /2, xy = yx = /2, with the rest components 0s. Now, using Eq. (2-3) through (2-5) with the stress components obtained above, we can write down the strain-s tress matrix relation for a [110] stress as, PAGE 22 22 111212 121112 121211 44 44 44000 /2 000 /2 000 0 2 00000 0 00000 0 2 00000 /2 2xx yy zz yz xz xySSS SSS SSS S S S (2-8) which yields the following strain components; xx = yy = (S11 + S12)/2, zz = S12, yz = xz = 0, and xy = S44/4. It should be noted that some authors use a different shear strain component, xy = S44/2, for a [110] stress [4, 7, 23]. This discrepanc y arises from the fact that they directly use the tensor notation of Eq. (2-2) to obtain strain components. In our case, however, the contracted matrix notation (or conventional not ation), which is not a tensor fo rm rigorously, is used since it is more popular and simpler [12, 24]. Strain co mponents for a [111] stress can be also calculated in the same way. In Table 2-1, strain component s are listed for three prin cipal uniaxial stresses and an in-plane biaxial stress (xx = yy = with no other components) together with elastic compliance constant values for Si. Figure 2-1. Three uniaxial st resses applied along [100], [110], and [111] directions. [100] [010] [001] [100]1,0,0 [110]/21,1,0 [111]/31,1,1 PAGE 23 23 Table 2-1. Strain components for three principal uniaxial stresses and an in-plane biaxial stress,and elastic compliance constant values for Si. Stress type [100] [110] [111] In-plane biaxial Strain components xx = S11 yy = zz = S12 the rest = 0s xx = yy = (S11+S12)/2 zz = S12 xy = yx = S44/4 the rest = 0s (xy = yx = S44/2) xx = yy = zz = (S11+2S12)/3 xy = yx= yz = zy = zx = xz = S44/6 (xy = yx= yz = zy = zx = xz = S44/3) xx = yy = (S11+S12) zz = 2S12 the rest = 0s S11 S12 S44 Elastic compliance constants 7.68 10-12 [m2/N] -2.14 10-12 [m2/N] 1.26 10-11 [m2/N] *Strain components in the parenthe ses are obtained from the strain -stress tensor relation of Eq. (2-2). Strain Effect on Conduction and Valence Band Silicon is an indirect energy bandgap semiconductor. In the E-k diagram, as shown in Fig. 2-2, its conduction band e dges (the lowest points in the conduc tion band) are located close to the Brillouin zone boundaries along the [100] and its e quivalent directions, while the valence band edge (the highest point in the valenc e band) lies at the origin (called a -point) in k-space. The regions around conduction band edges are called -valleys, and modeled as six ellipsoids since the constant energy surface in E-k relation around the edges is ellipsoidal in shape. A -valley along the [111] direction is al so shown in the figure. It is a higher energy state (EL>Eg), and thus less important for Si The valence band edge comp rises six bands includin g spin degeneracy; its upper bands are four-fold degenera te with heavyand light-hole states mixed with each other, and its lower bands (called a sp in-orbit split-off band) are doubl et with a splitting energy of 44meV. These extrema of the conduction and va lence energy bands are critical to determine most physical properties (e.g., electronic, optical) in semiconductor devices, so we can understand most device characteristics simply by looking at the behaviors of a small portion of the band structure near band edges. PAGE 24 24 Figure 2-2. Energy band st ructure of silicon for the [100] and [111] direction. Strain Effect on Conduction Band In the region around conduction band edges, eac h located at six symmetry axes of [100], the energy dispersion (E-k) relation is given by 22 2 0 **() () 2tl tlkkk Ek mm (2-9) where the subscripts, t and l, represent transverse and longitu dinal directions respectively, and k0 is the position of conduction band edges along th e longitudinal direction. Fig. 2-3 shows six constant energy ellipsoids in k-space near th e conduction band edges. When stress effects on the conduction energy band are concerned, we of ten refer to in-plane or out-of-plane effective masses. In this diagram, in-plane means a kx-ky plane which contains four valleys, and out-of-plane is a plane formed along the kz-direction, which includes two valleys. The inplane and out-of-plane effective masses for these two valleys in the kz-direction are calculated, respectively, as k // [100] k // [111] Eg = 1.12eV E = 3.4eV EL = 2.0eV Eso = 44meV Light-hole band Heavy-holes band Split-off band E( k) PAGE 25 25 22 2*2* 00 22() () 0.19, 0.92,tl tlEk Ek mmmm kk (2-10) where 0m is the free electron mass. For the four valleys lying on the kx-ky plane, the outof-plane effective mass is 0(0.19).tmm In this dissertation, the out-of-plane direction in MOSFETs is always referred to as a gate bi asing direction, unless stated otherwise. Figure 2-3. Six constant energy ellipsoid s in k-space near the conduction band edges. As mentioned in Chapter I, the strain e ffects on the conduction energy band were first quantified by Herring and Vogt. They found that the energy shift ()()CiE for the ith valley can be expressed in terms of strain components [2, 8], ()()[()](),i CdijuiiETr (2-11) where d and u are deformation potential constants named a dilation (o r a dilatation) and a shear (or a uniaxial) deformation potential constant, respectively. Here Tr[ij] is a sum of the diagonal components of the strain matrix, or Tr[ij] xx + yy + zz. The two constant values are ky // [010] kx // [100] kz // [001] PAGE 26 26 theoretically known to be, d = 1.1 eV and u = 10.5 eV [12] for bulk Si, and recently confirmed by our gate leakage measurements on n-type MOSFETs in which d = 1.0 0.1 eV and u = 9.6 1.0 eV [13]. Note that Eq. (2-11) is applied to -valleys and a more general case will be dealt with in Appendix A. Typically Eq. (2-11) can be rewritten as a sum of hydrostatic and shear strain components ()[()] () [()](). 33ij i u CdijuiiTr ET r (2-12) The first term represents a whole average en ergy level shift due to a hydrostatic strain component which corresponds to a volume change in a deformed solid. The second is a band splitting term due to a shear stra in component which is responsibl e for the twisted deformation of a solid, and it causes the six-fold degenerate conduction bands ( 6) to split into two-fold degenerate ( 2) and four-fold degenerate ( 4) subbands. As an example, let us consider bulk Si under in-plane biaxial stress. First, usi ng Table 2-1 and the constant values (d = 1.1 eV and d = 10.5 eV), we can calculate the av erage energy level shif t due to a hydrostatic strain component: 1112 8() [()]2 2 33 3.1310[],Hydro uu CdijdET rS S meV (2-13) with in a unit of Pascal. Unde r hydrostatic strain, the aver age energy level of the sixfold degenerate conduction bands is up-shifted fo r in-plane tension, wh ile down-shifted for inplane compression. In Fig. 2-4, the average en ergy level shift and band splitting are illustrated under in-plane biaxial stress. Next, the band splitting energy due to a shear strain component is calculated as PAGE 27 27 () _1 1 1 2 8[()] 2 ()() 2 33 8.3710[],ij z CShear uzz uTr ES S meV (2-14) which corresponds to the 2 band splitting along the kz-direction, with referenced to the shifted average energy level due to a hydrostatic strain. Further, since the average energy level is not changed with shear strain as depicted in the figure, the following relation must hold: 24() () () ___()2()4()0,zzz CAve Spl SplEEE (2-15) where the factors, 2 and 4, are attributed to the numbers of 2and 4-valleys, respectively. From Eq. (2-14) and (215), we also obtain the 4 band splitting energy, 4() 8 _1 1 1 21 ()24.1910[]. 3z Splu E SS meV (2-16) The general rule for the splitting directions is that 2 (downward) and 4 (upward) when compressive stress is applied along the splitting direction, while 2 and 4 with tensile stress applied along the splitting direction. As shown in the example of the in-plane tensile stress, 2 and 4 along the kz-direction since the direc tion is under compressive stress, but 2 and 4 along the kxor ky-direction. Using the same method so far, we can also obtain conduction band edge shifts for the ot her directions of stress. In table 2-2, conduction band splittings along the [001] direction are listed for [100], [ 110] and [111] uniaxial stresses and an in-plane biaxial stress. Strain Effect on Valence Band Near the band edge (0), k the energy dispersion (E-k) rela tion for heavyand light-hole bands can be determined by the application of kp perturbation to the band edge [7, 25] as PAGE 28 28 Figure 2-4. Average energy level shift and band splitting in the conduction band along the kz direction under in-pla ne biaxial stress. Table 2-2. Conduction band sp littings along the [ 001] direction for [ 100], [001], [110] and [111] uniaxial stresses and an in-plane biaxial stress. Stress type Splitting direction Band splitting [100] 2 4 11121 3uSS 11121 6uSS [001] 2 4 11122 3uSS 11121 3uSS [110] 2 4 11121 3uSS 11121 6uSS [111] No band splitting 0 In-plane biaxial 2 4 11122 2, 3uSS 11121 2 3uSS *The splitting directions (arrows) are based on tensile stresses. Hydrostatic Shear 6 4 2 2 4 4() z SplE 2() z SplE Compressive Strain Tensile Strain Unstrained ()()z CE Hydrostatic Shear PAGE 29 29 2242222222() ( )(''),(''),xyyzzxEkAkBkCkkkkkkwithHHLH (2-17) and for the split-off band we have ()2(),SOEkAk (2-18) where A, B, and C are inverse mass parameters, and their values measured by a cyclotron resonance experiment [26] are A -4.27, B -0.63, and C 4.93 with units of 0/2. m Fig. 2-5 and 2-6 show E-k diagrams and constant energy su rfaces for the heavy-hole, light-hole, and splitoff bands near 0, k with and without stress. Without stress, the constant en ergy surfaces of the heavyand light-hole bands are warped due to th eir strong interaction with each other, while the split-off band has a spherical energy surface. When stress (a uniaxial [111] tension in Fig. 2-5) is applied, the warped energy surfaces develop in to ellipsoids, a prolate ellipsoid for the heavyhole band and an oblate ellipsoid for the light-hole band. For these ellipsoids of heavyand light-hole bands, we can calculate the inand outof-plane effective masses of each hole as in the case of electrons [4, 7, 27]. Table 2-3 is cited from [27] and some values of the effective masses will be used in Chapter 3 and 4. Table 2-3. Effective masses of heavya nd light-holes under stress, cited from [27]. Stress type Direction Effective mass In-plane ** 00()0.21;()0.26HH LHmmmm In-plane biaxial Out-of-plane: [001] ** 00()0.28;()0.20HH LHmmmm [001] ** 00()0.28;()0.20HH LHmmmm [001] uniaxial [100]; [010] ** 00()0.21;()0.26HH LHmmmm [110] ** 00()0.54;()0.15HH LHmmmm [110] ** 00()0.16;()0.44HH LHmmmm [110] uniaxial [001] ** 00()0.21;()0.26HH LHmmmm [111] ** 00()0.86;()0.14HH LHmmmm [111] uniaxial [110];[110] ** 00()0.17;()0.37HH LHmmmm PAGE 30 30 Figure 2-5. Stress effects on the valence energy bands near0. k (Adapted from [27])(a) Unstressed. (b) Under [111] uniaxial compression. Figure 2-6. Stress effects on the valence energy bands near0. k (Adapted from [27])(a) Under in-plane biaxial tension. (b) U nder [110] uniaxial compression. (a) (b) PAGE 31 31 Based on the Pikus and Bir strain Hamiltonian (in the absence of spin-orbit interaction), the valence band edge shift (avera ge energy level shift plus band splitting) under stress is given by [3, 8] 2 2 () 22()[()](1)()()..().., 2j Vijxxyyxyb EaTr cpdcp (2-19) where and signs correspond to the heavyand light-hole bands, respectively, and the constants a, b, and d are deformation potentials and c.p. stands for cyclic permutation with respect to the indices x y and z The deformation potential s are shown in [12] that a = 2.1 eV, b = -2.33 eV, and d = -4.75 eV. Note that the light and heavy-hole bands split upward and downward, respectively, under out-of-plane uniaxial compression (or, in-p lane biaxial tension) and reversely under out-of-plane uniaxial tension (or, in-pla ne biaxial compression). In connection with this opposite splitting for the compression and tension, we need a (1)j term in the equation where j = 0 for the uniaxial tension (or, biaxial compression) and j = 1 for the uniaxial compression (or, biaxial tension). In Eq. (2-19), off-diagonal strain components (xy, yz, yz) are used for [110] and [111] uni axial stresses. As previously pointed out, since off-diagonal components are different for the two methods (c onventional notation and tensor notation), the d2 term in Eq (2-19) must be changed to d2/4 [23, 28] to use the strain components obtained from the tensor notation. The valence band splitting is listed in table 4 fo r [100], [110] and [111] uniaxial stresses and an in-plane biaxial stress. Basically, Eq (2-19) is valid for small magnit udes of stresses since it has been derived with spin-orbit interaction neglected. Actually in our stress measurements, the applied stress levels are not so high (< 300 MPa) that we can use Eq (2-19) to analyze our measurement data. However, in biaxially-strained Si1-xGex MOSFETs the internally applied stress level to the channel is as high as ~1.0 GPa for 20% Ge contents For this amount of high stress, Eq (2-19) is PAGE 32 32 not valid any more as shown in Fig. 2-7. Then, we need a more accurate expression in which the spin-orbit interaction effect is included. The following is the expression for the valence band edge shift based on the results of Hasegawas 6 6 Hamiltonian including the spin orbit band [4, 8, 29, 30]: Table 2-4. Valence band splitti ng for [100], [110], and [111] uni axial stresses and an in-plane biaxial stress. Stress type Splitting direction Band splitting [100] LH HH 1112() bSS [110] LH HH 2 2 2 1112 4422 d bSSS [111] LH HH 4423 d S In-plane biaxial LH HH 1112() bSS *The splitting directions (arrows) are based on tensile stresses. 22 221 ()[()] 29, 22 1 29 22Vi jforHH EaTr forLH f orSO (2-20) where the spin-orbit splitting energy = 44 meV, and = b(zz xx) for both [001] uniaxial and in-plane biaxial stresses, and 2 3 x yd for a [111] uniaxial stress. For a high level of [110] stress, its effects on the valence band edge are to o complicated and controversial to obtain an analytic expression [7, 10, 29, 30]. Instead, we can use Eq. (2-19) for a low level of [110] stress. Note that calculation results for [100] and [010] uniaxial stresses are the same as that of a [001] stress since these three directions are symmetric in a cubic crystal and the valence band edge shift is observed same from everywhere due to the center location of the edge in k-space. In Fig. 2-7, the valence band edge shift (average energy level shift plus band splitting) is plotted in PAGE 33 33 terms of the applied in-plane biaxial stress using Eq. (2-19) and (2-20). In the presence of the spin-orbit interaction, the heavy-hole band is still a pure state but the light-hole and split-off -1 -0.5 0 0.5 1 -100 -80 -60 -40 -20 0 20 40 60 Stress / GPaEnergy / meVLight Hole 6x6 Light Hole 4x4 Heavy Hole 4x4 & 6x6 Split-off 4x4 Split-off 6x6 Figure 2-7. Valence band edge shift (average en ergy level shift plus band splitting) vs. in-plane biaxial stress with (6 6 Hamiltonian) and without spin-orbit interaction (4 4 Hamiltonian). At zero stress the heavyand light-hole energies are chosen to be zero and accordingly the split-off energy to be -44 meV. bands are mixed with each other [29]. Because of this band mixing, the light-hole and split-off bands are quite different for with and without spin-orbit interaction at a high stress level. In Fig. PAGE 34 34 2-8, only the band splitting term is plotted, which exhibits the same result as in [27]. A total sum of band splittings (()()()Shear Shear Shear HH LH SOEEE ) in the valence band is also zero like in the conduction band, each band with the same weighting factor. Figure 2-8. Band splitting in the valence band vs. in-plane biaxial stress with (6 6 Hamiltonian) and without spin-o rbit interaction (4 4 Hamiltonian). Note that ()()()()0,Shear Shear Shear Shear VH HL HS OEEEE each band with the same weighting factor. PAGE 35 35 Fig. 2-9 shows a schematic diagram of the vale nce band edge shift for an in-plane biaxial stress together with that of the conduction band. The energy bandgap is shown to be changed with strain because of the conduction and valence band edge shifts. HH + LH SO LH ( HH) SO HH ( LH ) HH ( LH ) LH ( HH) SOUnstrainedTensile Strain Compressive Strain Eg(0) Eg( ) Eg( ) Figure 2-9. Conduction and valence band edge shifts under in-plane biaxial (uniaxial) stress along the out-of-plane direction. Th e new energy bandgap is determined by 2 and light-hole (heavy-hole) subbands for biaxial (uniaxial) tension, and 4 and heavy-hole (light-hole) subbands for biax ial (uniaxial) compression. PAGE 36 36 CHAPTER 3 STRAIN EFFECT ON THRESHOLD VOLTAGE In the previous chapter, we have presente d how to express strain com ponents for a different type of stresses and calculated deformation potential energies on the conduction and valence bands in terms of strain. Based on th is calculation method for deformation potentials, we can obtain key band parameters (e.g., energy bandgap, electron affinity, valence band offset, and DOS effective masses) to affect the threshold vo ltage as a function of strain. In this chapter, we start with an experimental technique in whic h a Si wafer is bent to introduce stress into a MOSFET channel. Wafer Bending Experiment Two types of fixtures have been designed to simulate uniaxiallyand biaxiallystrained MOSFETs as shown in Fig.1-1. Fig. 3-1 shows these fixtures; a uniaxial and a biaxial jig. The uniaxial jig used in applying stress is a four poi nt bending fixture. Such a bending structure has been well studied and a relation between the app lied force and stress under uniform stress is given by [31-34] 23() FLD wt (3-1) where F is the applied force, D and L are the inner and outer support distances respectively, and w and t are the samples width and thickness as s hown in Fig. 3-2. This formula is accurate when the sample is not severely bent to the applied forces and the dimensions w and t are small enough compared with D and L [32]. Under these conditions, the stress directions applied on the both surfaces of the sample can be approximated to be tangential, and the magnitude of stress applied everywhere between the inner supports can be treated as a constant. A detailed diagram is shown in Fig. 3-2. Eq. (3-1) is a useful fo rmula in calibrating stress sensors [32, 34]. PAGE 37 37 However, we can not use it to directly relate the jig parameters with the measured physical quantities. Figure 3-1. Two types of fixtur es to simulate uniaxially-stra ined and biaxially-strained MOSFETs. (a) For a uniaxial stress, two pa irs of cylindrical rods are used and a sample is inserted between the pairs. (b) Two rings with different diameters are used for a biaxial stress. (a) (b) PAGE 38 38 Figure 3-2. Illustration of a uniaxia l wafer bending jig. The displacement ( d) is defined as d di df. (a) an unstressed sample (b) a stressed sample. Another form of Eq. (3-1) fit for our experi ments is found in some literatures [35, 36]: 2 2 23 td YY La a (3-2) Here, and are the stress and strain values at the center of the sample respectively, Y is Youngs modulus of Si along the stress direction, 2 LD a and the deflection d is the vertical displacement between the uppe r and lower plates of th e uniaxial jig when we a pply stress. In Fig. 3-2, d is defined as d di df, and actually measured by the change in micrometer graduations. By analogy with the uniaxial jig, our ring-type biaxial jig has the same stra in-deflection relation D w(a) (b) PAGE 39 39 if the inner ( D ) and outer support distances ( L ) are replaced by the diameters of the inner and outer concentric rings respectively, that is 2 2 3ring outin outinouttd RR RRR (3-3) where ring is the strain value of the sample at the center of the concentric rings, and Rin and Rout are the radii of the inner and outer rings, respectivel y. Note that the stress ring depends on its direction because Youngs modulus of Si is not isotropic. Fig. 33 is a plot of Youngs modulus as a function of direction in the (001) plane. The relation between Youngs modulus and direction is given by [37] 1 44 44 111112() cossin1. 2 S YSSS (3-4) In general, the measured physical quantities by this ring-type biaxia l jig are not directly converted to the values of an in-plane (xand y-di rection) biaxial stress. In other words, there is some conversion factor between these two types of stress. Let us consid er two orthogonal stress vectors rotated by an angle of about the [001] axis as shown in Fig 3-4. This pair of stress vectors forms a new in-plane biaxial stress and the components of these two vectors are written as ()()cos,sin,0, (/2)()sin,cos,0. (3-5) Using Eq. (2-7), we obtain second order stress tensor elements for each stress vector as follows: 2 2cossincos0 ()()sincossin0, 000ij (3-6) PAGE 40 40 50 100 150 200GPa 30 210 60 240 90 270 120 300 150 330 180 0 [100] [010] Figure 3-3. Youngs modulus of Si versus direction in the (001) plane. The contours of the concentric circles correspond to a Youngs modulus of 50, 100, 150, and 200 GPa. 2 2sinsincos0 (/2)()sincoscos0. 000ij (3-7) As previously done in Eq. (2-8), second order strain tensor components for each stress are expressed as, 22 11 12 44 22 44 12 11 12cossinsincos/20 ()()sincos/2cossin0, 00ijSSS SSS S (3-8) PAGE 41 41 22 11 12 44 22 44 12 11 12sincossincos/20 (/2)()sincos/2sincos0. 00ijSSS SSS S (3-9) Figure 3-4. Illustration of a ring-ty pe biaxial stress. The coordinate origin lies at the center of the concentric rings. Two orthogonal stress vector s form a new pair of biaxial stress. Combining these two matrices of Eq. (3-8) and (3-9) results in 1112 1112 1200 ()(/2)()0 0 002ij ijSS SS S (3-10) Finally, we obtain the strain components for a new biaxial stress, which have the same form as those of the in-plane (x and y-direction) biaxia l stress as we expected, 1112 12()()()(),()2().bi bi bi xxyy zzSS S (3-11) [100] [010] [001] () ()/2 0 PAGE 42 42 Here note that ()(0)bi bi ii ii for i = x y and z Now, we can calculate the band edge shift in the conduction (or, valence) band usi ng the expressions of Eq. (3-11) to compare the in-plane (xand y-direction) biaxial stress. The band edge shift for 2 valleys, from Eq. (2-12), is calculated as 2() 1112 12 1112 12()2(2)()2() 2(2)2()().z ud udESSS SSSY (3-12) Since the strain is independent of stre ss directions as shown in Eq. (3-3), ()(0) and Y (0) = Y (/2) = 1/ S11 from Eq. (3-4). As a result, Eq. (3-12) can be rewritten as 2 2() 1112 12 1 44 1244 1111 1 () 44 1244 1111()2(2)2(0)(0) 11cossin1 2 (0)11cossin1.(3-13) 2z ud zESSS Y SS SS SS E SS In the relation, the term 2()(0)zE is the same as the 2 band edge shift of an in-plane (xand y-direction) biaxial stress and th e additional term arises from the anisotropic property of Youngs modulus of Si. Its minimu m and maximum values are approximately 130 GPa at = 0 and 169 Gpa at = 45, respectively, and repeated every 90 as shown in Fig. 3-3. Considering this anisotropic property, we need to take an average value of 2()()zE as a meaningful quantity: 22 21 ()()44 1244 1111 1 44 1244 2 0 1111 () 2 0()(0)11 cossin1 2 11cossin1 2 (0)zz zSS EE SS SS d SS E d (3-14) PAGE 43 43 Let us define the integral part in Eq. (3-14) as a conversion factor between the ring-type biaxial and in-plane biaxial stresses. is calculated to be 1.139. That means the ring-type biaxial stress causes a larger band edge shift by a factor of for the same strain. Since this difference comes from Youngs modulus anisotropy alone, we can simply include this factor into the stress-strain relation of Eq. (3-2): ()() .bi bi bi ringring ringinplaneY (3-15) The value of 1.139 is evaluated for a (001)-Si wafe r, which can be neglected roughly, but for a (110)-Si wafer is expected to be larger since we ha ve larger anisotropic values of Youngs modulus [37]. The value for a (110)-wafer is calculated in the appendix B. During this proposal, however, we will only deal with a (001) -Si wafer, and actually all measurements have been made on (001)-wafer Si MOSFETs. Strain Effect on Threshold Voltage It is shown in Fig. 2-9 that the energy bandgap of Si is changed with strain. This bandgap change (specifically, bandgap narrowing) is a critical factor to affect MOSFET operation properties. In thermal equilibrium, the new bandga p causes charge carriers in the energy bands to additionally increase, and accordingly Fermi ener gy level will be adjusted to meet the charge neutrality condition. Strain-Induced Fermi Energy Level Shift In a non-degenerate p-type s ubstrate, the mass action law (a product of the conduction and valence band charge carrier densities, n and p) under stress is stated, on the assumption that all the acceptor impurities are ionized, as 22() ()()()(0)expg iiE npnn kT (3-16) PAGE 44 44 2(0) (0) (0),i A An withpNandn N where in the parentheses represents zero stress, NA is the p-type impurity doping density, and ni is the intrinsic charge density. Based on Eq. (3-16), the additi onal charge Carriers, n() and p(), generated by the bandgap narrowing are expressed as 2() (0) ()()exp 1, ()0.g i A gE n np Nk T withE (3-17) Even at a small stress, the minority charge carrier density, n (), increases noticeably. For example, n () = ~ n (0) for an in-plane biaxial tension of 200 MPa. However, there is a negligible increase in the major ity charge carrier density, or p () NA. Note that the impurity doping density ( NA) is usually on the order of 1015 1018 cm-3, but the intrinsic charge density ( ni) is on the order of 1010 cm-3. Hence, the new Fermi energy level is determined so that the minority carrier density increases and the majority carrier density remains unchanged. Under the invariance condition of the majority carri er density with stress [38, 39], or ()() ()()exp ,FV VAEE p NN kT (3-18) we obtain the following expression for Fermi energy level shift: () ()()(0)()ln. (0)V FFFV VN EEEEkT N (3-19) Roughly speaking, if all the holes always remain at the valence band edge, the second term is not necessary, but in real case it is required to account for a carri er repopulation mechanism between the heavyand light-hole bands due to the stressinduced band splitting [40] as explained in the section 2.2.1. In Eq. (3-19), the vale nce band effective density of states (NV) is defined more PAGE 45 45 specifically in terms of density of state (DOS) heavyand light-hole effective masses (* H Hm and LHm ) [40]: 3/2 2/3 3/2 2 *3/2*3/222/VH H L HNkThmm (3-20) 2/3 **3/2*3/2,pHHLHwithaholeDOSeffectivemassmmm where all the DOS effective masses are 3-dimenti onal (3-D) ones and quite different from 2-D or 1-D effective masses dealt with in a quantized pote ntial well formed by a gate bias. Before stress is applied, the hole effective mass, *(0),pm is represented as a sum of heavyand light-hole effective masses, each with the same weighting factor. With increasing stress, *()pm gradually changes and finally will be one of the heavyand light-hole effective masses depending on the stress type. For example, the light hole band is in lower energy state for in-plane biaxial tension, and all the heavy holes will transfer to the light hole band at infinite stress. As a result, the total hole effective mass will be a light hole effective mass. If stated more concisely, 2/3 *3/2*3/2 **(0)(0) () ()().HH LH p HH LHmm atzerostress m mormat (3-21) Also, the ratio of the heavyand light-hole numbers are expres sed at a certain stress level based on a Maxwell-Boltzman distribution function for the non-degene rate valence energy band. ()() # exp () # ,Shear Shear HH LH holeEE ofHH H ofLH kT foralowerenergystateofthelightholeband (3-22) PAGE 46 46 where ()Shear HHE and ()Shear LHE are band splitting energies fo r heavyand light-hole bands. For an in-plane biaxial stress as an example, *()pm is written, using Eq. (3-21) and (3-22), as follows [28]: 2/3 3/2 ** 3 / 2 2/3 3/2 *3/2 *()(0)(0) () (0)()(0),holeHH LH p HH holeLHHmm fortension m mHm forcompression (3-23) where 0(0)0.49HHmm and 0(0)0.16.LHmm Using the same procedure, we also obtain the following expressions for a non-degenerate n-type Si substrate: (0) ()()ln, ()C FC CN EEkT N (3-24) 243/2 22()22/()(),C ltNk Thggmm (3-25) with 2 24 24 2 4 4(0)()(0) ()() ()(0)(0) .elec elecgHg f orvalleysinlowerenergystate gg Hgg f orvalleysinlowerenergystate (3-26) Here, 2(0) g and 4(0) g are defined as zero stress degeneracy factors and related to the six-fold degeneracy factor ( 6) as follows: 24 6 24(0)(0)60 () ()()2,4. ggat g ggorat (3-27) The repopulation factor for electrons, (),elecH is also represented similarly to that of holes in Eq. (3-22): PAGE 47 47 24() () __ 4 2 2()() # ()exp # .zz Spl Spl elecEE ofvalleyelectrons H kT ofvalleyelectrons foralowervalleyenergystate (3-28) In Fig. 3-5, Fermi energy level shif ts are plotted as a function of in -plane biaxial stress for both nand p-type substrates together with each term of band edge shifts and DOS changes. Figure 3-5. Fermi energy level shift vs. in-plane biaxial stress. (a ) For an n-type substrate, the lowest energy state is a LH-band fo r tension and a HH-band for compression, respectively. (b) For a p-type substrate, the lowest energy state is a 2 band for tension and a 4 band for compression, respectively. PAGE 48 48 The band edges (lowes t energy states) are 2 and LH bands, respectively, for an n-type and a ptype substrate under tensile stress, and 4 and HH bands under compressive stress. Strain-Induced Threshold Voltage Shift In this section, we briefly review the thres hold voltage expressions for both uniaxiallyand biaxially-strained n-MOSFETs. The formulas have been already obtained and published in [28, 41], but it would be worthwhile to comment on them because we have recently found some parts incorrect. As shown in [41], the threshold voltage ()thV shift is expressed in terms of the flatband voltage ()FBV shift and surface potential ()S change: ()()(),th FB SqVqVm (3-29) where m is the body effect coefficient and defined as 1/doxmCC [40]. The ratio of the depletion capacitance to oxide capacitance (/) D OXCC depends on both Si-channel doping density and oxide thickness, but in general m lies between 1.1 and 1.4. If we neglect the strain effects on the oxide charge, ()FBV is directly related to ()'FEs in Eq. (3-19) and (3-44) as 1(0) () ()()()ln()ln () (0) () () ()()ln / ,(3-30) (0)Gate Si CV FFC V CV FB Si V FV xx VNN EEEkTEkT NN qVforanitridecappinglayeredMOSFET N EEkTforaSiSiGeMOSFET N where the flatband voltage shift ( VFB) is only a function of the work function difference between the polygate and Si substrate, and the two different expressions are attributed to the different device structures, whether the gate is stresse d or not. It is also assumed that the strained poly-crystalline gate has the same (001) growth dir ection as that of the Si substrate, so applied stresses affect the same effects on the gate conduction band. The schematic energy band diagram for the n+-polygate and p-type substrate is shown in Fig. 3-6. As derived in [40, 41], in order to PAGE 49 49 get the expression for ()S in terms of band parameters, we use the relation between the quantized inversion charge de nsity and surface potential, 222 2 28( ) () () () expexp, ()D QM iS in ACqkTm E nq Q hNNkTkT (3-31) where 22 D m is the 2-D DOS effective mass and 2() E is the lowest energy state in the inversion potential well. The wrong part in [41] is our assumption that the ground state energy 2() E is not changed with stress, contrary to other literatures [13, 42]. Note in Eq. (3-31) that the total inversion charge density ()QM inQ has been approximated to that of the ground energy state since most electrons (particularly, in current short-ch annel devices) occupy the lo west energy state at a threshold voltage level and moreover, the ground and second lowest energy states are lowered and raised by a tensile stress, respectively. Fig. 3-7 illustrates the energy level shifts of the ground 2() E and second lowest energy state 4() E in the potential well for a tensile stress. In Eq. (3-31), taking the logarithm of both sides first and applyi ng the same inversion charge condition at threshold before and after stress leads to: 22 2()() ()/() ln 0. (0)/(0)S iC iCEq nN nNkT (3-32) Here we need to notice the carrier density product term, 2().in As already stated in Eq. (3-16), the charge carrier densities (n and p) in the energy bands increase or decrease only through the energy bandgap change ( Eg). The strain-induced band spli tting causes the carriers in each subband ( 2/ 4, or HH/LH subbands) to repopulate favorably in lower energy states, but does not change the carrier densities in real space. PAGE 50 50 Figure 3-6. Energy band diagram of the n+-polygate and p-type Si substrate. q Gate and qS are the work functions of the gate and substrate, respectively, and q S is the electron affinity. Figure 3-7. Lowest two energy levels of invers ion electrons at the th reshold voltage. Under uniaxial [110] or in-plane te nsile stress, the ground and second lowest energy levels are shifted oppositely along the out-of-plane di rection (field direc tion) as shown in the figure. In conjunction with a unit of carrier densities (# of carriers per unit area or unit volume) as well, the carrier densities are not cha nged spatially since repopulation is only a carrier redistribution PAGE 51 51 process occurred in the energy domain. Another mist ake has been made in this part [41]. As a result, 2()in is written as 22() ()(0)exp.g iiE nn kT (3-33) From Eq. (3-32) and (3-33), the surface potential change ()S is expressed as 2() ()()()ln. (0)C Sg CN qEEkT N (3-34) Finally, plugging Eq. (3-30) and (3-34) into Eq. (3-29) yields the follo wing threshold voltage shift expressions for uniaxial and biaxial strained n-MOSFETs, 1() () ()ln ()(1)()(1)ln (0) (0) () () () ()ln ()()ln (0) (0) /, ( 3 3 5 )V C VgC V C th VC Vg C VC xxNN EkTmEmEmkT NN foranitridecappinglayerednMOSFET qV NN EkTmEEkT NN foraSiSiGenMOSFET where 2() E has been replaced by ()CE since the ground energy leve l shift is equivalent to the conduction band edge shift for tensile st resses. Basically, these equations have been derived on the assumption that all the physical quantities vary linearly with stress, namely x ( ) = x ( ) x (0). Therefore, it is expected that they will be fitted better to a lower stress level. Each term in Eq. (3-35) reflects each physical phenomenon occurred in strained Si MOSFETs. The first two terms are introduced due to the Ferm i energy level shift and account for the valence band offset and repopulation between HH and LH subbands. Symmetrically, the conduction band offset (or electron affinity change) and repopulation (between 2 and 4 subbands) terms are included in the formulas. Lastly, the bandgap change term, which causes the conduction PAGE 52 52 band carriers to increase, is also a main com ponent. In Fig. 3-8, the physical phenomena occurred in strained MOSF ETs are illustrated. Figure 3-8. Illustration of each component constituting the threshold voltage shift formulas. The diagram is drawn based on an in-plane bi axial tension. Totally there are five components in the formulas, and each component corresponds to each physical phenomenon occurred in strained MOSFETs. Results and Discussion In order to check the validity of the newly corrected formulas of Eq. 3-35, let us first examine the electron occupancy in each energy st ate of the inversion la yer at the threshold voltage. In Table 3-1, each por tion of the electron populat ions in three lowest energy states is listed for different uniform substrate doping densities of NA. At lower doping densities ( NA PAGE 53 53 1017cm-3), electrons are more populated in the second lowest energy state ( 4 subband) than in the ground state (2 subband) due to the la rger degeneracy factor 24(2.4) gvsg and the 2D density of state effective mass 2422 00(0.190.0.417).DDmmvsmm At NA = 1018 cm-3, about 80 percent of the total inversion electrons occupy the ground energy st ate. Then, our formulas of Eq. 3-35 based on one band approximation should be best fit for NA = ~1018 cm-3 with a little substrate degeneracy. In addition, the directions in energy level sh ifts, as shown in Fig. 3-8, are a favorable factor for the formulas to become accurate. Table 3-1. Electron population in three lowest energy states at the threshold voltage for a uniform substrate doping density of NA. NA [cm-3] ES [V/cm] E0( 2) [%] E1( 4) [%] E2( 2) [%] Sum [%] 1015 1.503 104 6.5 21.7 4.8 33 5 1015 3.547 104 13.0 34.7 7.5 55.2 1016 5.125 104 17.2 40 8.5 65.7 5 1016 1.201 105 31.9 45.7 9.3 86.9 1017 1.730 105 41.1 43.3 8.5 92.9 5 1017 4.031 105 69.9 24.9 4.4 99.2 1018 5.797 105 82.6 14.8 2.4 99.8 5 1018 1.3448 106 98.2 1.6 0.2 100 *ES is the surface electric field right below the oxide layer. Lowest 60 energy levels have been involved in the calculation. In Fig. 3-9, the threshold voltage shifts are plotted as a func tion of stress for both uniaxiallyand biaxially-straine d n-MOSFETs. The uniaxial data have been obtained by the wafer-bending experiments in which both the gate and Si-channel are stressed as in the case of tensile-strained capping layered MOSFETs. Th e n-MOSFETs used in the experiments have [110]-channel directions, and mechanical stre sses are applied along th e channel direction (longitudinal stresses). The biaxia l data cited in the references are measured ones for tensilestrained Si/Si1-xGex MOSFETs. In these Si-Ge heterostructured MOSFETs, the magnitude of internal stresses applied to the channel de pends on Ge contents. For Ge-contents of x the applied stress can be calculated as follows: PAGE 54 54 (1) ,SiGeSi Si Ge Sixaxaa strain withlatticeconstantsofaanda a [100](1)(54.3)(56.6)54.3 129 54.3 x nmxnmnm stressYGPa nm 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -400 -350 -300 -250 -200 -150 -100 -50 0 Stress / GPaThreshold Voltage Shift / meV Uni-Data Bi-Data Uni-Model Bi-Model Uniaxial [110] Tension In-Plane Biaxial Tension [44] [45] [46] Figure 3-9. Threshold voltage shif ts vs. stress. The plot has been made for tensilestrained nMOSFETs at m = 1.3. The numbers in the sq uare brackets represent the experimental data obtained in the references. The three data used are for 20%and 30%-Ge rela xed layers of MOSFETs, and the applied stress to each device channel is calculated as 1.1 GPa and 1.65 GPa, respectively. In plotting the models for both uniaxial and biaxial stresses, a field-induced repopulat ion effect has been considered. At the threshold voltage, a ll the valley electrons already split into 2 and 4 subbands in the inversion potenti al well. As shown in Table 3-1, more than 70 % inversion electrons are in the ground energy state for NA > 5 1017cm-3 even without stress. In consideration of this effect, the conduction band DOS terms (the fifth term in Eq. 3-35) contribution is limited PAGE 55 55 to its 20%. Even if our one band threshold volta ge models seem rather rough, they agree well with the measured data as shown in Fig 3-9. More accurate and desirable models can be obtained simply by including a 24 coupling term. Now, let us rewrite the total inversion charge density expression of Eq. (3-31) including the next lowe st energy state [40]. 24 22 442 22 2() () () 8 () exp exp () () expQM D D i in AC SEE n qkT Qg mg m hNN kT kT q kT (3-36) A simple manipulation of Eq. (3-36) leads to 2 42 4244 2212 12 2 12 2() ()()()ln (0) ()() (0)(0)exp ln (0)(0) (0)(0) (0)exp (0).C Sg C D DN qEEkT N EE kT kT EE gm with and kT gm (3-37) Comparing this new coupling term with the already existing DOS change term, () ln, (0)C CN kT N we can find some similarity between them, that is 24 24 4422 24 442233 33 3 33 2() () lnln fromEq.(3-24)through(3-28) (0) 1()/ ln ,with 1/ foralowerenergystateofsubband,elec C C DD elec DD x yz DDgHg N Ngg Hgmgm mmmmm gmgm (3-38) PAGE 56 56 4212 21 12 21()() (0)(0)exp 1()(0)/(0) andln ln (0)(0) 1(0)/(0)elecEE kT H 42 4422 42 4422(0) (0) 22 (0) (0) 22 21() / ln 1/ foralowerenergystateof subband,EE DD elec EE DDHgmegme gmegme (3-39) where the same definition has been used for the electron repopulation factor,(),elecH as in Eq. 3-28. The difference between Eq. (3-38) and (3 -39) can be easily understood as two different repopulation mechanisms; one is a repopulation pr ocess occurred in the conduction band with no electric field (3-D repopulation), and the ot her is a repopulation pro cess occurred in the quantized potential well under the gate field (2-D repopulation). The exponential terms in Eq. (339) can be defined as preoccupation factors (or the initial conditi on of a 2-D repopulation process) since they represent the relative occupa tion probabilities (or the relative initial energy levels) between the 2and 4-subbands in the inversion layer before stress is applied. In a same sense, the preoccupation factors wi ll be unity for a 3-D repopulat ion process, which means there is no band splitting between the subbands before st ress. A strain gauge is a good example of the 3-D repopulation. Under the gate bias, two diffe rent repopulation processes (2-D and 3-D) are occurred simultaneously in strained MOSFETs, one in the conduction band and the other in the valence band. Fig. 3-10 shows the analogy betw een the 2-D and 3-D repopulation processes and their related parameters. Note that the effective masses of 2and 4-subbands are the same for a 3-D repopulation process, but di fferent for a 2-D one, that is 242 4223 3 3 000.19, 0.92, .DDD D x yy z x y zmmmmmmmmandmmmmm (3-40) PAGE 57 57 Now, we can obtain a more accurate two-band model from the one band model, simply by replacing the 3-D repopulation term (the fifth term in Eq. 3-35) with a 2-D one. This two-band model should be applied to MOSFETs with lower substrate doping densities such as NA < 1017 cm-3 beyond the coverage of the one-band model. A three-band model can be also easily made. This time, however, we have to add anothe r 2-D repopulation term to the existing five components of the two-band model instead of repl acing it. For example, a three-band model has the following form: 42 2 22 22412 12 '' 12 '' 12 '' 12 2()() (0)(0)exp ()()()ln (0)(0) ()() (0)(0)exp ln .(3-41) (0)(0) (0)(0) (0)exp (0)(0)SgEE kT qEEkT EE kT kT EE g with and kT 4 222 2.D Dm gm Interestingly, this new term is zero since the third lowest energy state is the same kind of 2 subband as the ground energy state, and thus has the same band splitting energy. For the same reason, we do not have five-, se ven-, and nine-band models and etc. Here, we notice that there is no repopulation among 2 subbands, or among 4 subbands. A four-band model has non-zero sixth term in addition to the five components of the two-band model. The sixth term is of the same form as that of the 2-D repopulation term of the two-band model w ith the following initial coefficients: 42 44 22'2 '' '' 12 2 2(0)(0) (0)exp (0)(0). D D EE gm and kT gm (3-42) PAGE 58 58 23D m2 g 43D m4 g 22D m2g (0)2E expkT42D m4 g (0)4E expkT Figure 3-10. Illustration of 2D and 3-D repopulation processes occurred in the conduction band edges under uniaxial [110] or in-plane tens ile stress. (a) In a 3-D process, 3-D parameters (degeneracy factor s and 3-D DOS effective masses) are related. (b) In a 2D process, which is occurred under gate bi as, in addition to the 2-D parameters (degeneracy factors and 2-D DOS effec tive masses) preoccupation factors are involved, and they determine their initial en ergy levels in the inversion layer before stress is applied. Similarly, the six-band model has non-zero seventh term in additi on to the six components of the four-band model. The seventh te rm is all the same as the sixth term of the four-band model except for 42'' ''' 1(0)(0) (0)exp EE kT (3-43) (a) (b) PAGE 59 59 The six-, eight-, ten-band models and etc. can be made repeatedly simply by adding one more 2D repopulation term at a time with a correspond ing initial energy leve l in the constant () 1(0).n The initial energy levels can be also determined quite accurately based on the energy eigenvalue formula obtained from the triangular potential well approximation [40]. 242/3 /3 3 ,0,1,2,..... 4 42s j zhqE Ejj m (3-44) Note that the energy eigenvalues of this formula are very accurate at the threshold voltage level unlike in the strong inversion region. The surface electric field () s E is also given by [40] 1 1 2 2 ,max2(/)ln(/)2 2ln.d SAS AA s SS SikTqNn i A SQ qN NkTN E n (3-45) Using Eq. (3-44) and (3-45), we can dete rmine the initial energy levels of all 2and 4subbands for each 2-D repopulation term, and t hus easily produce any order of n-MOSFET threshold voltage shift models no matter how large they are. A model containing more subbands will be better fitted, especially for lower s ubstrate doping MOSFETs since their subband energy levels are located more closely in the inversion potential well due to lower surface electric fields as shown in Table. 3-1. Summary The threshold voltage shift models a pplicable to both uniaxiallyand biaxially-strained n-MOSFETs have been reviewed and corrected based on the alrea dy published papers [28, 41]. Each model contains five components, each co mponent representing its corresponding physical phenomenon occurred in the strained MOSFETs. The existence of the two components (the valence band offset and subband repopulation terms) is attributed to the ma jority carrier charge neutrality condition. The ot her three components (the ener gy bandgap change, conduction band PAGE 60 60 offset and subband repopulation terms) are requir ed to account for the change in the inversion charge carriers. First, the energy bandgap change (bandgap narrowing) causes each subband carrier density to increase. Unlike in the valen ce band, the inversion carrier densities of the conduction subbands are not so hi gh at the threshold voltage le vel that even a small bandgap change brings about non-negligible increase in the carrier density of each subband, hence making the threshold voltage ( Vth) lower. The conduction band offset which is equivalent to the ground energy level shift in the inversion layer for tens ion, also affect the threshold voltage. For example, the downshift of the conduction band offset lowers Vth as in a tensile stress, while the upshift raises it as in a co mpressive stress. Lastly, the 2-D repopulation term causes Vth to be lowered or raised depending on the carrier transfer directio n. The concept of 2-D and 3-D repopulation processes has been introduced, based on which we can build up the multiband threshold voltage shift models easily in a repeti tion way. Especially, the multiband models are required to precisely describe the threshold voltage shift for p-MOSFETs. Fig. 3-11 shows two types of strained p-MOSFETs which are commonly adopted in the industry as a standard. Since HH and LH out-of-effective masses are similar as listed in Table 2-3, their energy levels are formed very closely in the inversion layer at th e threshold voltage level. For example, it is calculated using the out-of-plane effective masses in Table 2-3 that the two lowest energy level differences are only ~10meV and ~20meV for a uniaxial [110] and an in-plane biaxial stress, respectively, at a doping density of ~ 1018 cm-3 (compared with ~80meV for n-MOSFETs at the same condition). Note that hole effec tive masses are dependent upon a stress type. PAGE 61 61 Figure 3-11. Two types of stra ined p-MOSFETs commonly adopted in the industry as a standard. (a) In-plane tensile-strained and (b) uni axially compressive-strained p-MOSFETs are shown to have their lowest two subband en ergy levels closely located each other at the threshold voltage level due to simila r HH and LH out-of-plane effective masses. (a) (b) PAGE 62 62 CHAPTER 4 STRAIN EFFECT ON GATE TUNNELING CURRENT Strain effects on the energy bands in Si MO SFETs were shown to be classified into two main categories. One is the bandgap narrowing effect which causes th e minority carriers to increase in the energy subbands, thus affecting th e device operation properties such as threshold voltage shift. The others are th e band edge shifts and splitting. These effects make the charge carriers repopulate favorably in lower energy states In this chapter, we discuss how strain affects and alters the gate tunne ling current, especially in the region where the gate is biased above the threshold voltage (VG > Vth), and also introduce an experimental method of determining deformation potential constants for n-MOSFETs. Unlike in the subthreshold voltage region, the inversion charge carri ers start to ab ruptly increase beyond th e threshold voltage, so that we can neglect the charge carriers ge nerated by the bandgap narrowing in the strong inversion region. Instead, the subband ener gy level shift and the repopulation between the subbands are dominant mechanisms to account for th e change in the gate tunneling current with strain. Measurement of Direct Tunneling Current Measurements on the gate tunneling currents ha ve been made for both nand p-MOSFETs, with the drain, source, and body all tied to ground and the gate positively-biased using a Keithley 4200 DC characterization system, un der all types of mechanical st resses; uniaxial and biaxial, longitudinal and transverse, and compressive and te nsile. The stresses were applied to measure the industrial long channe l devices ranging from 1 m to 4 m, using the four-point bending jig for a uniaxial stress and the ring-type jig for a biaxial stress, as show n in Fig. 3-1. The n(or p-) MOSFET samples used consist of arsenic doped n+ (or p+) poly Si gate on top of 1.3nm nitrided SiO2 gate dielectric on ~1017 cm-3 boron doped p (or n) well. In Fig 3-2, the measured data at a gate voltage of 1.0 V are plotted as a function of the applied stress [20]. Under all types of tensile PAGE 63 63 stresses, the hole and electron ga te direct tunneling (DT) curre nt increases and decreases, respectively, while the trend is opposite under compressive stresses. Figure 4-1. Change in na nd p-MOSFET gate tunneling current ve rsus stress [20]. All types of tensile stresses increase the hole gate tunneling current, wh ile decrease the electron gate tunneling current. For compressi ve stresses, the trend is opposite. Direct Tunneling Current Model from Inversion Electron Before modeling the gate tunneling current, we explain qual itatively the experimentally observed trend in both nand p-MOSFETs. In the strong inversion region (e.g., VG = 1.0 V as in the plot), the charge carrier de nsity is high enough to neglect th e electron-hole pair carriers created by the strain-induced bandga p narrowing. Therefore, the stra in effects left to affect the inversion charge carriers are the subband energy le vel shift, repopulation a nd Fermi level shift as explained in Chapter 3. Explanation for Direct Tunneling Current Change with Stain To simplify our analysis, consider on ly the lowest two energy levels (E0, E1) in the inversion layer. In n-MOSFETs, the electron tunneling cu rrent decreases for tensile stress since the E0 state ( 2 subband) lowers as shown in Fig. 3-7, thus making 2 subband electrons bound in IG( ) / IG(0) [%] PAGE 64 64 a higher potential barrier as we ll as decreasing the population in E1 state (4 subband with higher tunneling probability) through repopul ation. Reversely for compressi ve stress, the electrons in E0 are in a lower potential ba rrier and the population in E1 increases, which causes the tunneling current to increase. It is also expected that Fermi level shift alters the tunneling current by affecting the inversion charge carriers. As stat ed in the mass action law of Eq. 3-16, the carrier density, n ( ), is not changed with Fermi level shift, but the carrier population in each subband will be altered. When Fermi level moves to ward the conduction band, the relative electron population (# in E1 / # in E0) increases. This effect increases the tunneling current slightly. In pMOSFETs, the tunneling current change is some what complicated to explain since the out-ofplane effective masses vary not only with the magn itude of applied stresses [4, 7], but with the stress type as shown in Table 23. According to the calculation results in Table 2-3, the out-ofplane effective masses are as follows: 00 000.28/0.20 / 0.21/0.26[110]. mmforaninplanebiaxialstress HHLH mmforauniaxialstress (4-1) Fig. 4-2 shows the ground and second lowest en ergy states in the i nversion potential well where the ground energy state is a heavy-hole (light-hole) band for an in-plane biaxial stress (a uniaxial [110] stress) sin ce the out-of-plane effective mass of the HH (LH) is heavier. When compressive stress is applied, HH ( ) and LH ( ) for a biaxial (uniaxial) stress as shown in the figure. These subband energy shifts cause the tunneling current in p-MOSFETs to decrease for both uniaxial and biaxial stresses. On the other hand, HH ( ) and LH ( ) for a biaxial (uniaxial) tension, which in creases the tunneling current. Our tunneling current model for pMOSFETs explains this trend well [45]. PAGE 65 65 Figure 4-2. Ground and second lowe st energy states of p-MOSFETs in the inversion potential well. A heavy-hole (light-hole) band is a ground energy state for an in-plane biaxial (a [110] uniaxial) stress. The arrow direc tions represent the s ubband energy level shift under compressive stress. Physical Model for Direct Tunneling Current in n-MOSFET It has been previously reporte d that stress alters the gate tunneling currents on both (001) nand p-MOSFETs, with tensile stress typically used for n-MOSFE Ts [46]. For the purpose of deformation potential measurements, however, we use compressive stress since electrons primarily only populate the lowest two energy levels under this c ondition. In other words, the electron population of the 2 valley next lowest subband, 2'(), E is negligible under compressive stress, which simplifies the interpretation of the tunneling current data. A schematic drawing of the direct tunneling process from the 2 and 4 subbands and the effect of stress on the subband energies are shown in Fig. 4-3. The direct tunneling electron current density JG can be expressed in terms of the charge density ( Nij) and lifetime (ij) of each energy subband in the inversion layer [47, 48], which are functions of stress, ji ij ij GqN J,, )( )( )( (4-2) PAGE 66 66 where the subscript i denotes 2 (or 4) valley and j each subband belonging to one of these two valleys, respectively. When stress is applied, the change of tunneling current density JG() is written as, ji ij ij G ij ij G GJ N N J J,)( )( )( (4-3) Above the threshold voltage, most electrons (e.g., ~90% at VG = Vth and NA = ~1017 cm-3 as shown in Table 3-1) occupy the lowest tw o energy states, or each ground state for 2 and 4 valleys as shown in Fig. 4-3. Figure 4-3. (a) Schematic band diagram for the gate direct tunnel current in an n-MOSFET on a (100)-wafer. (b) 2 and 4 energy level shifts under compressive stress and MOSFET inversion layer confinement. Under compressive stress, 2 energy levels are raised while 4 energy level (dotted line) is lowered. From Eq. 4-2, the relative change of tunneling current IG()/IG(0), referenced at zero stress, is calculated as follows, to first order in ()/(0), )0( )( )0( )0( )( )0( )0( )( )0( )0( )(4 4 2 2 4 4 C B N N A I IG G (4-4) (a) (b) PAGE 67 67 where )0(/)0()0(/)0( )0(/)0(1 )0(4 2 4 2 4 2 NN A )0(/)0()0(/)0( )0(/)0( )0(4 2 4 2 4 2 NN NN B and )0(/)0()0(/)0( )0(/)0( )0(2 4 2 4 2 4 NN NN C. Note in the approximation we use N2() + N4() 0 (neglecting higher subbands), and assume linear relationships for parameters, and N are valid since the applied stresses are small (<300 MPa). The approximation has been checked numerically and introdu ced little error. The tunneling lifetimes of the electrons in each 2 and 4 ground state under compressive [110] stress are given by [47]: )()( 1 )(4/2 4/2 4/2 ET (4-5) and quantized energy levels, 2() E and 4(), E are expressed, using Table 2-2 and Eq. (3-44), as 22/3 ,2 1112 1211 29 () ()(2)()(), 33 162eff uu dhqF ES S S S m (4-6) 42/3 ,4 1112 1211 49 () ()(2)()(), 36 162eff uu dhqF ES S S S m (4-7) where T2/4() is the transmission probability of a modified WKB method [47, 48], and the quantization effective masses, m*2 = 0.92 m0 and m*4 = 0.19 m0 and the elastic compliance constants, S11 = 7.6810-12 m2/N and S12 = 2.1410-12 m2/N. In the above expressions, the 2nd and 3rd terms are hydrostatic and shear strain components, respectively, and the stress, has negative values for compression, but the plots (Fig. 4-4 and 4-6) are made on a positive scale for PAGE 68 68 convenience. The effective electric fields ( Feff, 2 and Feff, 4) are introduced into our model to compensate for triangular potential approximation errors in the inversion condition, Si inv d effQ Q F 4/2 max, 4/2, (4-8) where Qd,max is the maximum depletion sheet charge density, Qinv is the inversion sheet charge density, and Si is the Si dielectric constant. The co rrection factors used in our model are 2= 0.75 [48], and 4= 1 [47]. Figures 4-4 (b)-(d) show whic h terms in Eq. (4-4) contribute to the change in the gate leakage at low and high gate voltage. At low gate biases, the stress-induced repopulation term, the 1st term in Eq. (4-4), contributes greatly to tunnel current as shown in Fig. 4-4 (b), while its effect reduces gradually as the gate bias increases as shown in Fig. 4-4 (c) and (d). Extraction of Conduction Band De formation Potential Constant Stress alters the tunneling current in two ways: (1) Stress-induced energy level splitting causing a repopulation between 2 and 4 subbands as shown in Fig 4-5. The life time of 2 subband is significantly longer due to the high out-of-plane mass (0.92m0 vs. 0.19m0). (2) The shift in the energy levels alters the SiO2/Si barrier height. The change in the gate current versus applied compressive stress for diffe rent gate voltages is shown in Fig. 4-4 (a). We observe the change, IG()/ IG(0), is positive (increases) and is larg er at low gate voltage. At high gate voltage, the change is a weak function of voltage Both of these trends can be understood from how the vertical electric field and compressive stress shift and split the energy levels. At high gate voltage (high vertical field), 2 is many kT below 4, electron primarily populating 2 subband. Hence at high gate voltage, a smaller change in the tunneling current results since compressive stress only alters the gate current by lowering the SiO2/Si barrier height (shifting 2 to higher energy as seen in Fig. 4-3). PAGE 69 69 Figure 4-4. (a) Relative direct tunnel current change [IG()/IG(0)] versus applied compressive stress at different gate voltages. Data (squares) were measured on industrial MOSFETs. The solid lines are our model. (b)-(d) Breakout of the various contribution to IG()/ IG(0) at different gate voltage s: [electron repopulation from 2 to 4 (dominates at low VG), change in lifetime of 2 or 4 electrons due to change in barrier height]. For low gate voltage, electrons populate both 2 and 4 subbands, so in addition to the stress-induced change in barrier height, stress also increases the population of electrons in 4 which further increases the tunneling current (due to the shorter lifetime in 4 than 2). Capturing the change in tunneling cu rrent with stress (or slopes of curves in Fig. 4-4) for the full PAGE 70 70 range of gate voltage, Fig. 4-6 shows the change in d[IG()/IG(0)]/ d versus applied gate voltages. Figure 4-5. Stress-induc ed repopulation between 2 and 4 ground state electrons. Negative signs in stress mean compression, and the electron population in 2 and 4 subbands decreases and increases, respectively, with increasing compressive stress. The slope was extracted from the raw data of IG() with the method of least squares. In Fig 4-6, two deformation potential constants (d and u) are used to match the measured data. The model fits well with the data over a gate bi as range of 0.4V to 1.6V, which encompasses the entire direct current re gion [48]. The best fit [2() E 1.7051011 [eV] and 4() E 2.9951011 [eV] with in units of Pa] results in d 1.0 0.1 eV and u 9.6 1.0 eV. The obtained values of deformation potential constants are very close to theoretical values for bulk Si by Fischetti and Laux (d = 1.1 eV, and u= 10.5 eV) [12]. PAGE 71 71 Figure 4-6. Change in slopes ( d [IG()/ IG(0)]/ d) versus gate voltage with 95% confidence error bars. Best fit for the entire data set occurs for d and u of 1.0 eV and 9.6 eV. Deviations from the best fitting values are shown by changing the deformation potentials by ~10%. The insets represent schematic band diagrams at low and high gate biases. Higher slopes at low VG are due to a larger st ress-induced repopulation between 2 and 4 subbands since their energy levels are closer to each other. The sensitivity to different values of deformatio n potentials is also s hown in Fig. 4-6. The low gate bias slopes are set by the stress-induced band splitting energy, 4(). E The entire curve is adjusted up or down by the magnitude of the SiO2/Si barrier height [2() E ] since the change in 2() E results in a parallel shift without a ny change in the already determined low gate bias slopes. To illustrate the goodness of the model fit, 10% deviations in both 2() E and 4() E are plotted, in which d ranges from 0.873 to 1.141 eV, and u ranges from 8.61 to PAGE 72 72 10.5 eV. These fits are shown as the dashed lines in Fig. 4-6 and si gnificant deviation is observed for 5% differences in the deformation poten tial constants. Stress effects on the oxide thickness are less than 0.05% at = 300 MPa assuming SiO2 contracts the same amount as Si along the stress direction, and has negligible effects on the model prediction and least-squares fit of deformation potential constants. Summary The gate tunneling current has been shown to increase or decrease depending on the stress type, and exhibit an opposite trend for nand p-MOSFETs. This tunneling current change is explained well by the strain-induced energy level shift and repopulation. The gate bias dependence on IG()/ IG(0) as a function of mechanical stre ss for n-MOSFETS has been used to extract the conduction band deformation potent ial constants. The hydrostatic deformation potential constant, which is traditionally hard to measure, is extracted from the tunneling current and shows excellent agreement with recent numerical calculations [12, 29, 30]. These values of deformation potential constants suggest th e straininduced co nduction band shift is approximately EC() = 1.711011 eV for uniaxial [110] strain and EC() = 3.651011 [or, EC( x ) = 0.23x in terms of Ge content x ] eV for in-plane biaxial strain. The measured conduction band shift in real MOSFET samples is sm aller than that typically assumed for biaxial strain, EC() = 9.041011 [or, EC( x ) = 0.57x ] eV [41, 49]. However, this smaller value of conduction band shift is consistent with both the theoretical cal culations and magnitude of the strain-induced MOSFET threshold voltage shift as shown in Fig. 3-9. PAGE 73 73 CHAPTER 5 STRAIN EFFECT ON LOW-FREQUENCY 1/F NOISE C HARACTERISTICS Introduction Low-frequency 1/f (or flicker) noise in strained-silic on MOSFETs is certainly an important and interesting research subject since it is often a limiting factor in the device design. It is known that 1/f noise is up-converted to produce phase nois es or degrade SNR (Signal to Noise Ratio) in RF and mixed-mode circuits. In the applicati on of strained MOSFETs to high-performance and high-speed CMOS or RF circuits, this low-freq uency noise specifically deserves attention and research. The 1/f noise in MOSFETs is believed to occur du e to carrier trapping and detrapping at the oxide-Si interface and severa l models for this surface effect have been proposed [50-53]. Ralls and etal. supported these models by ascribing their random telegraph signal observation in the channel conductance to a phenomenon of the electron capture and emission by the interface trap states [54]. In addition, Welland and Koch even obtained precise trap profile images on the silicon surface of the MOS structures throu gh the scanning tunneling microscopy [55]. To the contrary, bulk mobility fluctuations have been al so proposed by Hooge [56, 57] as a main source of 1/f noise generation in MOSFETs. He concrete ly pointed out an inversely proportional relationship between the magnitude in 1/f noise spectrum and total number of charge carriers based on his huge data. These two different viewpoints of a 1/f noise generation mechanism caused a long-term debate over bulk mobility fluctu ations versus carrier number fluctuations by a surface effect.[58] Currently, a combined mo del (the number and its correlated mobility fluctuations) is generally accepted as a main source of 1/f noise generation in MOSFETs [59-65]. Fig. 5-1 shows a 1/f noise generation mechanism at the oxide-silicon interface. During the current flow, channel charge carriers interact w ith interface traps (empty or filled) in the gate PAGE 74 74 oxide. As a result, the charge carriers can be captured, or emitted, or altered by the interface traps, which causes both carrier number and correlated mobility fluctuations. Figure 5-1. Illustration of 1/f noise generation at the SiO2/Si interface, adapted from [64]. Conventional Charge Trapping Model In MOSFETs, the most common 1/f noise model is a charge trapping model in which the carrier number and its correlated mobility fluctua tions are described by random telegraph signals (RTSs) in time domain. It is known that each trapping and the subsequent detrapping produce a RTS, and the observed 1/f power spectrum is a result of the superposition of each RTS. Actually in the submicron-size MOSFETs (channel area A<1m2), only a single trap can be activated near the quasi-Fermi level over the entire channel. Th e trapping and detrapping of a channel charge carrier by this trap result in discrete channel current resembling RTSs. Fi g. 5-2 shows a typical time-domain waveform of the drain cu rrent, in which the average capture ( c ) and emission time ( e ) and the average drain current RTS amplitude ( di ) are specified. Each single RTS contributes to the resultant 1/f noise power spectrum of drain current in larger channel-area MOSFETs. The drain current spectrum of a single RTS has a Lorentzian shape, PAGE 75 75 0 0.10.20.30.4 5 10 15 t [s] c e Figure 5-2. Discrete modulation of the cha nnel charge current due to a single trap. 2 22() () 1()dd iz Si z (5-1) where 1()1/()1/()cezzz is the effective time constant of a trap located at z from the oxide-channel interface. Based on the conventional number fluctuation model, we can obtain an expression for the drain current noise power spec tral density (PSD) due to total RTSs. According to the number fluctuation theory, the PSD of th e mean square fluctuation in the number of trapped carriers in the volume element V and energy tE and ttEE is given by 22(,) ()4(,)(1) 1(,)tNtttttEz SnEzffEV Ez (5-2) Since the fluctuation in the trapped carriers Nt is equal to the fluctuati on in the channel charge carriers N in strong inversion, or ()(),tNNSS we can write down the total drain current noise PSD()IDS using the general relation 22()/()/:DIDNSISN 2 22 000 2 22 0(,) () 4(,)(1) 1(,) (,) 4(,)(1) 1(,)(A) (B)ox D oxt LW D Itt t t t t D tttttIE z Sd E d x d y n E z ffd z NE z I Ez LWnEzffdE dz NE z (5-3) PAGE 76 76 where x-axis along the channel length, y-axis along the channel width, z-axis into the oxide from the interface 31() []N D tSNoisePSDofthemeansquarefluctuationinthenumberofoccupiedtraps IAveragechannelcurrentundertotalRTSs NTotalchannelchargecarriernumber nTrapdensitywithaunitofcmeV LChannellength WChannelwi dth 11exptf n tfn tTrapoccupationfunctionwithtrapenergyEandquasiFermienergylevelE EE f kT In Eq. (5-3), it is assumed that the oxide trap distribution is neg ligibly affected along the channel direction for a low drain bias (e.g., VD=0.1V for our measurements). The integrals (A) and (B) are commonly dealt with in most literatures under two assumptions: Uniform spatial distribution of oxide traps, (,)() ttttnEznE Trapping time constant, 04 (,)()exp() 2.oxBEzEzwithm h From the assumption the integral (A) in Eq. (5-3) is calculated as, (A)4()(1)tfntttnEffdE assuming tn is not changed much within a few kT around f nE 221exp exp 4() 4() 1exp 1exptfn tfn tfn ttfn t tfn tfnEE EE kTd kT kT nE dEnE dE EE EE kT kT 14()4()1exp.tfn tfntfnEE kT nE kTnEwithX Xk T (5-4) PAGE 77 77 0 0.5 1 1.5 0 0.05 0.1 0.15 0.2 0.25 Figure 5-3. Contribution to noise PSD by oxide traps only within a few kT around f n E As shown in Fig. 5-3, since the produ ct of trap occupancy functions (1)tt f f gives its sharp peak around f nE in the trap energy distribution oxide traps only within a few kT around f n E mainly contribute to 1/f noise power spectrum. The assumption has been made based on the WKB theory for the gate tunneling of channel carriers. 0() E is the time constant at the interface and is the attenuation coefficient of the carrier wave function in the oxide. Also, oxm is the effective mass of th e carrier in the oxide and B is the tunneling barrier height seen by the carriers at the inte rface. From the relation 0(,)()exp(), EzEz we obtain ddz Then, 1 001 221 (B) (0)() 1oxd withzandzt 1 0221 1 d (1).[]ttt f fvsEeV 0.9fn E eV a few kT ft(1ft ) Et [eV] PAGE 78 78 1 011 1 1011 tan() tan()tan() 2 f 1 4 f (5-5) since 11tan() 1tan()01. 2 forand for The lower and upper values in the integral, 01, and can be roughly estimated using the relation 0(,)()exp(), EzEz where0() E and have typical values of ~10-10 s and ~108 cm1, respectively. For a MOSFET with tox=5nm, 10 11 01~10 10. sands Thus in the frequency range of 8910210, HzfHz we can observe 1/f noise power spectrum unless it is buried by thermal noise at high frequencies. For a thinner oxide device (e.g., tox=1.3nm for our samples), the low frequency region of the 1/f noise spectrum is severely limited by the time constant1 (response of the slow surface st ates). Plugging (5-4) and (5 -5) into (5-3) leads to 2 2() () .Dtfn D IkTLWnE I Sf Nf (5-6) If we check the unit of the right-hand side, has a unit of 1[] cm and tn a unit of 31[] cmeV, which gives a unit of 2[/] AHz totally. It should be noted that the attenuation coefficient is obtained for the rectangular potentia l barrier (approximation of a trapezoidal potential barrier) at the oxide-Si channel interface. In the expression of a trapping time constant,0(,)()exp(), EzEz the quantity z is given for a trapezoidal barrier as follows [72], 3/23/2 *() 8 2, 3BoxB ox oxqz zm hq (5-7) PAGE 79 79 where ox is the electric field in the oxide. Expandi ng the nominator in Eq. (5-7) in terms of z, we obtain 2 1/2 2 2 3/23/2 1/2 1/233 ( ) ........ 28 3 1. 2oxB ox BoxB B oxB ox Bqzqz qz qzqz for (5-8) For a negligible oxide band bending as in operations of old devices (tox>5nm), we can approximate z as a rectangular barrier case with little errors, that is *4 2oxBzmz h (5-9) However, this rectangular barrier approximation l eads to a considerable error for modern thinoxide MOSFETs (tox<1.5nm) especially when a high gate bias is applied. In order to reduce this error, it is possible to include the 2nd term in Eq. (5-8), that is '*4 21, 4oxox oxB Bqt zm z h (5-10) where we replaced 2z with oxtz to keep a form of '.z The time constant then changes to '' 0(,)()exp() 1 4oxox Bqt EzEzwith (5-11) Also, it should be desirable to use a modifi ed WKB method for a thin oxide MOSFET. In gate tunneling models based on a modifi ed WKB method, a compensation factor (TR) is introduced to account for the disc ontinuity at the oxide-Si channe l interface in calculating the transmission probability of channel carriers [47,48]. If we use this modified WKB approximation method, the time constant (,) E z is further changed to PAGE 80 80 0() (,)exp(),RE Ez z T (5-12) since the time constant is inversely proportional to the transmission probability. Therefore, the new time constant, (E, z), is a function of gate bias. Return ing to Eq. (5-6), the drain current noise PSD can be expressed as, when we include the correlated mobility fluctuations [59, 65, 72], 2 2 2() () 1.,Dtfn D IkTLWnE I Sf SN Nf (5-13) where S is the scattering coefficient and is the correlated mobility fluctuation. In general, the scattering coefficient S is a function of both the channel ch arge density associated with a screening effect and the trap distance from channel charge carriers. The conventional charge trapping model has be en briefly reviewed so far. The exact /1f noise spectrum results from the assumption that th e oxide trap distributio n is spatially uniform for the calculation convenience in Eq. (5-3). No w, we examine a special case of the trap distribution to obtain a more general form of the /1f spectrum. In general, a trap distribution is functions of energy and space, and often treated as exponential functions in literature. We choose then a functional form of the trap density as, 0(,)exp( ),tto x oxz nEznqVz t (5-14) where and are trap distribution coefficients ov er energy and space with units of meV-1 and nm-1, respectively, and the two terms in the expone ntial argument are due to oxide band bending and nonuniform spatial distribution in to the oxide depth. Fig. 5-4 shows the coordinate system in which the traps are distributed along the z-axis an d the field direction. When a gate voltage is applied, traps are shifted from the original place by an amount of oxide band bending. PAGE 81 81 The integral part (B) in Eq. (5-3) is written using the trap distribution (,)tnEzin Eq. (514) as, 22 0exp( ) (B) 1oxt ox oxz qVz t dz (5-15) Figure 5-4. Shifts in trap lo cation due to oxide band bending. Similarly as carried out in Eq. (5-5), we can calculate the integral (B) in Eq. (5-15). 1 0 1 022 (/)/ 22 (/)/1(/)/ 0exp(/) 1 (B) 1 11( ) () 1 ()Coxox oxoxoxoxoxox qVt qVtqVtqVtz d d (5-16) The integral (C) in Eq. (5-16) can be easily calculated if we use the following relation, 1 0()(1) 01. 1s i npx dxppwherep xp (5-17) PAGE 82 82 We change the integral (C) into the form of Eq. (5-17). 1 01/ 2 2 1/ 1 2 2 222 01 2 01() (C) () 21() 1[()] ()for()1and()1 21() (5-18) 1/ 2sin 2 d d Using Eq. (5-3), (5-13), (5-16), and (5-18) we obtain for the drain current noise PSD, 1 2 2 0 2() 2() 1 () 1 sin(/2)(2)Dtfn D IkTLWnE I Sf SN Nf f (5-19) where / 1.oxoxqVt The exponent in 1/ f noise spectrum is expressed with two terms; one is due to energy distribution of traps, which is dependent of gate bias, and the other is due to spatial distribution of traps, which is independent of gate bias. Typical values of and are ~0.02/mev and ~3/nm. Wafer Bending Experiment on 1/f Noise Measurements of dc Currents and Drain Current Noise P SD A schematic block diagram for the measurements of 1/f noise and dc currents is shown in Fig. 5-5. The dc currents (drain and gate currents) a nd drain current noise PSD were measured for each applied mechanical stress at the same bias condition. A Keithley 4200 dc characterization system was used to measure dc currents at the drain and gate terminals, and a Stanford Research (SR) 785 spectrum analyzer wa s used to measure the drain current noise PSD, respectively. One battery power ed SR 570 current amplifier was used to amplify the drain current noise (SID) under an applied drain bias (VD). A second battery powered LNA was employed to apply a variable gate bias (VG). In order to minimize external electromagnetic PAGE 83 83 interference, all equipment and cables were placed inside a shielded pr obe station and battery powered except for the spectrum analyzer a nd semiconductor parameter analyzer (Keithley 4200). The noise PSD data were obtained usi ng a SR 785 spectrum analyzer up to 12.8kHz by measuring five frequency spans (100Hz, 400Hz, 1.6kHz, 6.4kHz, and 12.8kHz) using a Hanning window. Each frequency span contained 800 data points (800 FFT lines), and each data record was typically averaged from 100 to 1000 times by the SR 785 spectrum analyzer. Extraction of the current noise PSD of an nchannel MOSFET device under test ( DUT) is illustrated in Fig. 56. In order to measure the curren t noise PSD of the setup, the i nput of the amplifier (SR 570) is open circuited. The plot of the current noise PSDs of the setup and a DUT with and without the subtraction of the setup noise PSD is shown in the figure. The noise PSD of the DUT is extracted by subtracting the noise PSD contribution of the setup fr om the measured total current PSD. All the measurements on our MOSFET sample s were made using the same sensitivity of 20 A/V for the SR 570. Figure 5-5. Schematic block diagram for 1/f noise and dc current measurements. 1/f noise and dc currents (both drain and gate currents) were measured for each applied mechanical stress using a spectrum analyzer, two L NAs, and a Keithley 4200 system. The SR 570 current amplifier has a current input (SID) and a dc voltage output. PAGE 84 84 Figure 5-6. Noise power spectrums for an n-channel MOSFET with and without the setup noise. Measurements were made on a sample with a channel length L=2 m, a width W=50 m, and a threshold voltage Vt=0.36V at dc biases VG=0.6V and VD=0.1V. The DUT noise power spectrum is obtained by subtracting the setup noise from the measured total noise. Measurement Results n-MOSFET under tensile stress The drain current 1/f noise was measured on an n-channel MOSFET with a channel length L=2 m, a width W=50 m, and a threshold voltage Vth=0.36V under tensile stress. The MOSFET was biased in the linear region at VG=0.6V and VD=0.1V, and six uniaxial tensile stresses were applied up to 225MPa. The measured dc currents and 1/f noise power spectrums are shown in Fig. 5-7. The drain (ID) and gate currents (IG) are observed to consisten tly increase and decrease, respectively, with increasing tensile stress as e xpected from previous st udied [46, 73]. Under a stress of 200MPa, the drain current increases by about 7%, and the gate current decreases by about 2.4%. The drain current noise PSD is observe d to increase for tensile stress as shown in Fig. 5-7 (d), although the trend is not clear at some frequencies without further averaging as discussed in section 5.3.2.4. PAGE 85 85 Figure 5-7. Measurements of an n-MOSFET unde r tensile stress. Measurements were made on a sample with a channel length L=2 m, a width W=50 m, and a threshold voltage Vt=0.36Vat dc biases VG=0.6V and VD=0.1V. (a) Relative changes in drain and gate tunneling currents. (b)-(c) Obse rved noise power spectrums for different stresses. (d) Comparison of drain current noise PSD for different stresses. n-MOSFET under compressive stress The measured n-channel sample has a channel length L=2 m, a width W=50 m, and a threshold voltage Vth=0.28V. The sample was biased in the linear region at VG=0.6V and VD=0.07V, and six uniaxial compressive stresse s were applied up to 189MPa. The measured dc currents and 1/f noise power spectrums are shown in Fig. 5-8. As opposed to the tensile stress case, with increasing compressive stress, the drain current (ID) decreases and the gate current (IG) increases as also shown previously [13, 74]. Th e drain current decreases by about 6%, and the gate tunneling current increases by about 2.5 % at a compressi ve stress level of 200MPa. The drain current noise PSD is shown to decrease in Fig. 5-8 (d). PAGE 86 86 Figure 5-8. Measurements of an n-MOSFET u nder compressive stress. Measurements were made on a sample with a channel length L=2 m, a width W=50 m, and a threshold voltage Vt=0.28Vat dc biases VG=0.6V and VD=0.07V. (a) Relative changes in drain and gate tunneling currents. (b)-(c) Observed noise power spectrums for different stresses. (d) Comparison of drain current noise PSD for different stresses. p-MOSFET under compressive stress The measured dc currents and 1/f noise power spectrums are also plotted for a p-MOSFET in Fig. 5-9. Measurements were made under co mpressive stress for two different gate biases, VG= -0.6V and -0.8V, at the same drain bias, VD=-0.1V. Compressive stresses were applied up to 189MPa. For these two measurements, the drain (ID) and gate (IG) currents are commonly observed to increase and decrease, respectively, with increasing compressive stress [45, 74]. The change is about 10% for drain cu rrents and -2% for gate currents at a compressive stress level of 200MPa. The noise PSDs increase for both of the tw o measurements as shown in Fig. 5-9 (d). PAGE 87 87 Figure 5-9. Measurements of a p-MOSFET under compressive stress. Measurements were made on a sample with a channel length L=1 m, a width W=50 m, and a threshold voltage Vt=-0.36Vat dc biases VD=-0.1V and VG=-0.6V, and VD=-0.1V and VG=-0.8V. (a) Relative changes in drain and gate tunneli ng currents. (b)-(c) Observed noise power spectrums for different stresses. (d) Co mparison of drain current noise PSD for different stresses. Data analysis In this subsection, we first examine the av eraging of the raw drai n current noise data obtained by the spectrum analyzer. Unlike drai n or gate tunneling current measurements, the noise PSD values are measured with large uncertainties. More accurate PSD data can be obtained through a higher number of averages, which are generally limited by the measurement time. As shown in Fig. 5-7 (d) through 5-9 (d), the magnitude of the fluctuations is almost comparable to the maximum noise PSD change fo r our stress level, although the noise data have been already averaged from 100 to 1000 times by the spectrum analyzer. More av eraging is then required to differentiate even a few percent change in the noise PSD which may be accomplished using linear regression analysis either globally or loca lly by noting the 1/f frequency dependence of the PAGE 88 88 noise spectra. Since the drain current 1/f noise PSD generally follows the following frequency dependence, (1) (),ID IDSHz Sf f (5-20) we can extract the noise magnitude and exponent by a least squares f it (LSF) of the noise spectrum on a log-log plot. Over the frequency ra nge that Eq. (5-20) holds, the PSD can be expressed as a linear function of frequency on a log-log plot, log[()]log[(1)]log,ID IDSfSHzf (5-21) as illustrated in Fig 5-11 (a). If the PSD data follows a 1/ f dependence over a wide frequency range, then a frequency-i ndependent noise exponent, and magnitude, SID(1 Hz ), may be extracted via LSF. This is the case for the meas ured p-channel MOSFET PSD data shown in Fig. 5-11 (b)-(c). However, in some devices, the lo w frequency PSD data exhibits some frequency structure associated with a trap-related Lorentzi an such as in the measured n-channel MOSFET PSD data seen in Fig. 5-8 (b)-(c). When Eq. (5 -20) applies only locally for a smaller range of frequencies, the PSD at each frequency is firs t averaged using a moving average, and then the best fit slope is computed. Fig. 5-10 illustrates the method used to extr act the average values locally from the measured raw data. The moving av erage is obtained as follows. Consider an average value of noise PSD, (),IDMSf at a specific frequency of fM. The neighboring data point pairs are chosen such that each pair satisfies the condition loglogloglog,iiMLHM f fff where logiL f and logi H f are each frequency pair centered about a target frequency, log fM. N pairs are averaged to estimate an aver age value of the noise PSD, (),IDMSf at a specific frequency of fM. On a linear scale, (),IDMSf at M f is obtained as follows, PAGE 89 89 1/2 1 0()()()iiN N IDM IDLIDH iSfSfSf (5-22) where 00()and()IDL IDHSfSf are defined as 00()()(),IDLIDHIDMSfSfSf and N is the number of chosen neighboring data pairs. Figure 5-10. Schematic illustration for extracting a local average value of noise PSD at a specific frequency of interest from the measured raw data. Raw noise data fluctuate against a piecewise linear line on a log-l og plot. To reduce the deviation in PSD values due to these fluctuations, a local averaging method can be used. The local average value of the PSD, log(),IDMSf at a frequency of M f is obtained through averaging of chosen neighboring data pairs of PSD values,log()iIDLSf and log(),iIDHSf which are chosen such that loglogloglogii M LHM f fff The extraction of the noise exponent, and magnitude, SID(1 Hz ), is illustrated for the measured p-channel MOSFET PSD which follows a 1/ f dependence over a wide frequency range as shown in Fig. 5-11 (a). For each applied uniaxial longitudinal compressive stress, and SID(1 Hz ) are extracted via LSF of the PSD over a frequency range of 30Hz to 1 kHz and plotted in Figs. 5-11 (b)-(c) as a function of stress for a gate bias of -0.6V. The normalized change in SID(1 Hz ) relative to the unstressed case, SID(1 Hz ; )/SID(1 Hz ;0), is plotted as a function of stress in Fig. 5-11 (d). When the PSD follows a 1/ f dependence over a wide frequency range, the global and local LSF results show good agreement as seen in Fig. 5-11 (d) for two gate biases. PAGE 90 90 Figure 5-11. Analysis of p-channel MOSFET data under compressive stress. (a) Extraction of the noise magnitude and exponent on a log-log plot. (b) Extracted exponent value vs. applied stress (c) Extracted noise magnitude vs. applied stress (d) Relative changes in noise PSD vs. frequency at di fferent gate voltages. The lines are plotted based on the extracted noise magnitude and exponent valu es, and the symbols are averaged values obtained using 50 pairs of ne ighboring noise PSD data. From the measured n-channel MOSFET PSD for different applied mechanical stresses the local averaging technique is used with 50 pairs of data points to extract the average drain current noise PSD, ;,IDSf as a function of stress at specific frequencies. Figs. 5-12 and 5-13 plot the extracted average PSD values for n-ch annel MOSFETs as a function of applied tensile and compressive stress for specific freque ncies and the normalized change in PSD, ;/;0,ID IDSfSf as a function of frequency for a specif ic stress. Both the locally averaged PSD and the raw PSD are shown. The PSD values corrupted by external noise sources such as 60Hz and its harmonics were exclud ed in the average calculations. PAGE 91 91 Figure 5-12. Analysis of n-ch annel MOSFET data under tensile st ress(a)-(c) Changes in noise PSD vs. applied stress at different freque ncies. The solid squares are extracted average values at each stress level usi ng 50 pairs of neighboring PSD data, and the empty triangles are raw values directly read from the SR 785. Th e lines are fitted to the extracted average values. (d) Relative change in noise PSD vs. frequency at a stress of 200MPa. To summarize our measurements, it is observed that: (1) 1/f noise drain current PSD increases and decreases for n-channel MOSFETs w ith increasing tensile and compressive stress, respectively, while it increases for p-channel MOSFETs with increasing compressive stress. (2) The relative change in PSDs shows a frequenc y dependence (i.e., larger changes for lower frequencies) for both nand p-MOSFETs. (3) With applied mechanical stress, the changes in drain current have a similar trend to the cha nge in the drain current noise PSDs while the changes in gate currents have a tr end opposite to that of the change in drain current noise PSDs. PAGE 92 92 Figure 5-13. Analysis of n-channel MOSFET data under compressive stress. (a)-(c) Changes in noise PSD vs. applied stress at different frequencies. The solid squares are extracted average values at each stress level usi ng 50 pairs of neighboring PSD data, and the empty triangles are raw values directly read from the SR 785. Th e lines are fitted to the extracted average values. (d) Relative change in noise PSD vs. frequency at a stress of -200MPa. In the noise measurements, it has been observed that both 1/f noise magnitude and exponent are functions of applie d mechanical stress. Therefor e, the strain-induced relative change in 1/f noise PSD, referenced at zero stress, can also be expressed from Eq. (5-20) as (;) (1;) ln1 ln1 ()ln, (;0) (1;0)ID ID ID IDSf SHz f Sf SHz (5-23) where (;)(;)(;0)ID ID IDSfSfSf and ()()(0). We assume that linear relationships for the parameters, (;)IDSf and (), are valid since the appl ied stresses are small (< 250MPa). Table 5-1 lists the stress-dependent dr ain current PSD parameters obtained from our measurements. In this study of strain effects on noise PSD, (1;)IDSHz and () are the PAGE 93 93 focus of the noise model since they vary indepe ndently with applied st ress. The next section develops a theoretical model for the stress dependence of drain current noise PSDs. Table 5-1. Relative change in noise PSD pa rameters for an applied stress of 100MPa. Stress/Device ID( )/ID(0) IG( )/IG(0) ( ) SID(1 Hz ; ) /SID(1 Hz ;0) Tensile n-channel MOSFET 3.78 % -1.20 % 0.66 % 11.9 % Compressive n-channel MOSFET -2.97 % 1.24 % -0.75 % -16.2 % -0.6V 5.07 % -0.957 % 0.82 % 19.2 % Compressive p-channel MOSFET -0.8V 4.80 % -0.769 % 1.03 % 22.9 % *The relative changes in the noise magnitude and exponent ar e extracted using Eq. (5-23) for nchannel MOSFETs and are extracted using Eq. (5-21) on the log-log plot for p-channel MOSFETs. Charge Trapping Model under Strain Mechanism for Change in Noise PSD under Strain A charge trapping model is considered to explain the strain dependence of 1/f noise in MOSFETs. The charge trapping model ascribes the origin of 1/f noise to charge trapping and detrapping of channel charge carriers by oxide trap s [59, 65, 72, 75-78]. In ultrathin gate oxide MOSFETs, contrary to the conventional treatment of 1/f noise, even a relatively low gate bias can cause significant band bending in the Si-channel, thus causing the Fermi level to lie above the conduction band edge or below the valence band edge [47-48]. Under this condition, the contribution to 1/f noise PSD mostly results from trapping/ detrapping of channel carriers at oxide traps existing above the Si conduction band edge or below the Si valence band edge. Thus, trapping at bandgap traps via two-step or mu lti-phonon processes can be neglected compared with trapping via elastic dire ct tunneling [65, 72, 75]. Fig. 5-14 (a) shows a schematic band diagram of n-channel MOSFETs under mechanical stress. Applying uniaxial tensile stress shifts the ground energy level (E0) lower in the inversion layer [13, 46]. As a result, the tunneling probability of channel electrons at E0 decreases because of the highe r potential barrier while the PAGE 94 94 trapping probability of tunneling electrons by oxide traps increases since th eir energy level shifts closer to the quasi-Fermi level (EFN) as shown in Fig. 5-14 (b). These two effects oppositely affect the change in noise PSD; the former decr eases the noise PSD, but the latter increases it. However, as indicated in Table 5-1, th e decreasing effect (the reduction in IG) is not dominant in determining the overall change in noise PSD. In a ddition, the oxide trap distribution is another important factor in determining the noise PSD change. Tunneling electrons encounter less or more traps depending on the oxide trap distri bution in energy space. Trap distribution is represented over space and energy as dots in Fig 5-14 (a). Strain also affects the correlated mobility fluctuations in the charge trapping mo del through both alteration of channel carrier mobility and repopulation among inversion subband carriers. In the following subsection, we discuss these strain effects on 1/f noise PSD in more detail with some numerical examples. Charge Trapping Model under Strain In the conventional charge trapping m odel, the drain current noise PSD,(),IDSf is given by [59, 65, 72, 78] 22 FN 2() () ,Dt IDInE SfAkT Nf (5-24) where A is the gate area, kT is the thermal energy, N is the total number of channel carriers per unit area, is the attenuation coefficient in the WKB approximation, is the parameter combining carrier number and correlated mobility fluctuations ( =1 for the number fluctuation model and >1 for the unified model), and nt( EFN) is the trap density at the quasi-Fermi level with a unit of cm-3eV-1. Rewriting this equation for applied stress in terms of the measured quantities in Table 5-1, we obtain PAGE 95 95 FN FN(;) (;)2() 2() ln1 ln1 ln1 ln1 (;0) (0) () (;0) ()2() ln1ln1 ()ln. (0) (0)t ID D ID D tnE SfI Sf I nE N f N (5-25) The above expression is further si mplified since the fourth and fifth terms can be neglected. The attenuation coefficient is defined by B2 2,oxm (5-26) where oxm is the effective mass of cha nnel carriers in the oxide, and B is the oxide barrier height seen by channel carriers at the interface. Here B is a function of stress a nd the strain-induced change B( ) is only a few meV for our stress level of 200MPa compared with B(0)=3.15eV for conduction band electrons and 4.5eV for valence band holes [13, 47-48]. Thus, ( )/(0) 1. The total number change in channel carriers due to stress, N ( )/ N (0), can be written in terms of the drain ( ID) and gate tunneling ( IG) currents in steady-state condition, (0)() () (0)(0)(0)(0)GG DGGII N NIII (5-27) This quantity is also very small. Then, fr om Eq. (5-23) and (5-25) through (5-27), PAGE 96 96 ECE0( ) qVOXEFN z z=0 z=toxOxideSi-channel Gate q B( ) EV EFG(a) EFNft (1-ft)0 EC(z=0) Et E0( ) (b) Figure 5-14. Schematic band diagram of an nchannel MOSFET under mechanical stress. Dots are symbolized as oxide traps. Traps are shifted by the amount of oxide band bending (qVox) when a gate bias is applied. Applyi ng stress alters inversion subband energy levels at which tunneling electrons enc ounter less or more traps depending on the energy distribution of traps. (a) Trapping of channel charge carriers through an elastic direct tunneling mechanism. With increas ing tensile stress, the ground energy level (E0) of the inversion channel electrons continues to lower and get closer to the electron quasi-Fermi level (EFN). The noise PSD increases under tensile stress since trapping probability increases as the energy level of tunneling electrons move closer to EFN. (b) Trap occupation function product ft(1-ft) versus trap energy Et. Trapping probability is proportional to ft(1-ft). ,effFN ,effFN(;) (;)2() 2() ln1 ln1 ln1 ln1 (;0) (0) () (;0) ()ln,t ID D ID D tnE Sf I Sf I nE f (5-28) ,effFN ,effFN(1;)2() 2() andln1 ln1 ln1 (1;0) (0) () (;) ln1 (;0)ID D ID D t tSHz I SHzI nE nE (5-29) The magnitude change in noise PSD due to strain has been expressed with three terms in Eq. (529). Roughly estimated using our measured data in Table 5-1, half of the total magnitude change in PSDs comes from the last two terms in Eq. (529). It is mentioned by some literature that the oxide trap density nt(EFN) should change with strain, but in our experiments, the 3rd term is not necessarily a result of the trap density change. As mentioned earlier, it is due to strain-induced PAGE 97 97 trapping position change in energy space, and is also related to spatial trap distribution. In this sense, we use the effective change in trap density ( nt ,eff/nt ,eff) instead of the direct change ( nt/nt) as in Eqs. (5-28) and (5-29) Now, we examine the last two terms in Eq. (5-29) in more detail. In order to obtain a general form of the/1f spectrum since our measurements extract the stress dependence of the exponent we assume that the oxide tr aps are distributed exponentially over energy (E) and space (z) as assumed elsewhere [76]. It is also assumed that the traps are distributed continuously along the oxide depth di rection for our ultrathin (1.3nm) gate oxide samples since the gate areas (50 m2 and 100 m2) are very large. Thus, the trap density is represented as 0(,)exp,ttnEznEz (5-30) where 0 tn is the trap density at the interf ace (z=0) with the Si band edge (EC or EV) at z=0, and and are the trap distribution coefficients ov er energy and space with units of meV-1 and nm-1, respectively. Referring to Fig. 5-14 (a) for the coordinate system, under mechanical stress and gate bias, the trap distribution seen by tunneli ng channel carriers at the ground energy level (E0) is given by 00 B(,;)(exp (),tt o x oxz nEzn qV z t (5-31) where the terms in the exponential argument are due to oxide band bending, applied stress, and spatial trap distribution, respectiv ely. In general, the parameters, 0,tn and are functions of mechanical stress since applied stress can alter the trap distribution by affecting both trap energy and existing interface strain between the Si-channel and the oxide. The signs for and are positive (negative) for the exponential increase (decrease) for increasing distance from the PAGE 98 98 interface and increasing energy above the Si band e dge. For clarification, we also state the signs of B( ), that is B-0fornchannelMOSFETundertensilestressand pchannelMOSFETundercompressivestress () 0fornchannelMOSFETundercompressivestressand pchannelMOSFETundertensilestress. (5-32) These signs of B( ) reflect the ground energy level shifts in the inve rsion layer for applied different types of stresses. Trapping by channel carri ers in higher energy levels is neglected since the contribution to noise PSD is much smaller. The integral form of the drain current 1/f noise PSD in the charge trapping mode l is written as [59, 75-78] Cox ox Vox2 Et 2 22 E0(,) () 4(,)(1) 1(,)D IDtttI Ez SfA nEzffdE dz NE z (5-33) where ft is the trap occupation function, is the trap time constant, and ECox and EVox are the oxide conduction and valence band edges, respectively. The expre ssion is valid for a low drain bias [59, 75-76]. Eq. (5-53) can be rewritten for applied mechanical stress, using Eq. (5-31), as Cox Vox ox2 E 2 B0 E t 22 0() (;) ()exp()()4()()1() ()exp()/() (5-54) 1()D ID ttt oxoxI SfA qnffdE N qVtz dz 1 FNB 0() where()1exp and()=exp().tEEq f z kT The stress-dependent tr ap occupation function, ft( ), is introduced to describe the stress dependence of the trapping probability of tunneling channel carriers by oxide traps. Strain effect on the exponent in 1 / f noise power spectrum From the second integral in Eq. (5-5 4), we obtain a general form of the ( )/ 1fspectrum, where is defined by PAGE 99 99 ()/() ()1. ()oxoxqVt (5-55) The signs of and are >0 and <0 in the literature [65, 72, 7677, 79]. We also confirmed that >0 with our gate bias dependen ce measurements of the exponent for both nand p-channel samples. Typical values of and are cited to be ~0.02/meV, ~ -3/nm and ~10/nm, respectively [72, 75-77, 79]. Referenced to the measured values of ( ) in Table 5-1 and using Eq. (5-55), ( ) is estimated approximately as G()/() () (0) ~0.008for100MPa at V0.6V,oxoxqVt (5-56) where ( ) and ( ) are changes due to trap redistribution over energy and space under stress. Physically, both changes are possi ble since stress alters both trap energy states by affecting the bonding energy of SiO2 and the existing interface strain betw een the Si-channel and oxide. It is roughly estimated that ()= ~5 10-4 /meV and ()~0.05/ nm at a stress of 100MPa. The sign of ( ) is positive (negative) for n-channel MOSFETs under tensile (compressive) stress. In order to account for our measured data of ( ), traps must redistribute over energy and space for applied stress. Fig. 5-15 illustrates th e stress dependence of the trap distributions in energy and space. As shown in the figure, applying tensile (compressive) stress to an n-channel MOSFET causes the oxide traps to move toward higher (lower) energy and spatially deeper (shallower) oxide regions. The exponent ( ) then increases (decreases) for tensile (compressive) stress since trapping occurs more at spatially deeper (shallower) regions into the oxide. Note that trap redist ribution to a higher (lower) en ergy region under oxide band bending yields the same effect on the exponent change ( ) as trap redistribution to a spatially deeper PAGE 100 100 (shallower) oxide region as implied in Eq. (531). Under an assumption of sorely pure trap redistribution due to applied stress, there is no net increas e or decrease in number of traps, that is ,,(,;)(,;).tt Ez EznEzdzdEnEzdzdE (5-57) This assumption is likely to be valid for our maximum applied mechanical stress level of 200MPa since the strained energy is much less than the chemical bond energy of a Si-O bond in SiO2. Figure 5-15. Trap redistribution for an nchannel MOSFET under mechan ical stress. In our wafer bending experiments, the oxide layer is also stressed as well as the Si-channel. Applied mechanical stress may cause oxide traps to redistribute over energy and space. (a) Energy distribution of traps from the conduction band edge. (b) Spatial distribution of traps from the oxide-channel interface. Strain effect on the noise PSD magnitude Change in trapping probability of tunneling carriers by oxide traps Assuming that the total number of oxide traps remains constant during redistribution as defined in Eq. (5-57), the trap redistribution effect can be negl ected compared with the other strain effects although it can affect the magnitude change in noise PSD In the first integral of Eq. (5-54), the maximum relative change in tr apping probability under strain is given by B()1()(0)1(0) ()/. (0)1(0)tttt ttffff qkT ff (5-58) PAGE 101 101 This relation is obtained for trapping occurring far aw ay from the Fermi level (E0-EF 3 kT ). At a stress of 100MPa, the maximum values of th e relative change in trapping probability are calculated to be 0.066 for an n-channel MOSFET and 0.104 for a p-channel MOSFET. Fig. 5-16 shows the relative change in trapping probability as a function of energy of tunneling carriers at an applied stress of 100MPa. Figure 5-16. Relative change in trapping probab ility vs. tunneling channel carrier energy for an applied stress of 100MPa. The plots ar e made with an energy level shift q B( )=1.7meV for an n-cha nnel MOSFET and 2.7meV for a p-channel MOSFET at 100MPa. The energy level shift of 2.7meV for valence bands has been calculated based on the 4-band strain Hamiltonian [8]. At low gate biases the ground energy level E0 of the inversion channel carriers are higher than the Fermi energy, EF. The trapping probability for these tunneling carriers increases for nchannel (p-channel) MOSFETs with increasing tensile (compressive) stress, whereas its relative change decreases. At high gate biases where E0 lies below EF, the trapping probability decreases and the magnitude of its relative change increases. This behavior is desirable from the viewpoint PAGE 102 102 of device applications si nce both gate leakage and 1/f noise can be reduced while drain current is enhanced. Change in correlated mobility fluctuations of channel charge carriers The factor in Eq. 5-54 is given by [59, 78] 1,effSN (5-59) where S is the scattering parameter, eff is the effective mobility limited by all the scattering mechanisms except for Coulombic scattering by oxide charges, and N is the total number of channel carriers per unit area. The strain induced change ( )/(0) is then (0)()(0)() () ()1(0)(0)eff eff effNSSN SN (5-60) where the stress dependence of S is caused by the fact that stre ss alters the average distance of the channel carriers in the ground energy level fr om the oxide/channel interface through carrier repopulation among inversion subbands [78, 80]. The scattering parameter is approximately dependent upon an average distance ( dave) between the scattering charge and channel carriers [65, 72, 78, 81], that is ave0 maxave max()2ln1(/)with()/().oxSioxSiSdSrdr CC (5-61) For our ultrathin gate oxide samples, rmax is calculated to be ~5nm. The value of S is typically taken ~2 10-15Vs [59, 65, 72]. With this value of S and a typical mobility value of S and a typical mobility value of 500 cm2/Vs, we can rewrite Eq. (5-60) as () ()4() ()5(0)(0)D DI S SI (5-62) It should be noted that eff( )/eff(0) ID( )/ ID(0) for long channel devices at low gate and drain biases [73], and that N is calculated to be ~41012/cm2 for our MOSFET samples at the measurement bias condi tion using the relation N =Cox (VG-Vt)/q PAGE 103 103 Estimation of total magnitude change in noise PSD Returning to Eq. (5-29) compared with Eq. (5-54) and using Eqs (5 -58) and (5-62), we finally obtain an expression for the magnit ude change in noise PSD as follows, B B() (1;)2() () 8() ln1 ln1 ln1 ln1 (1;0) (0)5(0)(0) (0) () ln1 ()(), (5-63)G IDDD IDDDGI SHzII S SHzI SI I q q kT where is a gate bias dependent para meter with its value ranging from -1 to 1. The third term in the equation is included to account for the stra in-induced change in tunneling probability of channel carriers and the last term arises from the nonuniform energy distribution of traps which describes whether the tunneling ch annel carriers encounter less or more traps. In estimating the noise PSD magnitude for our measurements, we us e the following values for nand p-channel MOSFETs for an applied stress of 100MPa: q B( )=1.7meV (n-channel) and 2.7meV (pchannel), ( )=0.02/meV (both nand p-channel), =0.6 (n-channel) and 0.8 (p-channel), and S ( )/S (0) = 0.01 (both nand p-channel). Here has been determined on the basis of the triangular well approximation of i nversion layers. Note that the quantization effective mass of the carriers occupying the ground energy level is about three times larger for electrons than for holes (0.92 m0 vs. 0.29m0). Based on Eq. (5-61), S ( )/S (0) can be also estimated using the triangular well approximation a nd charge density expression for inversion subbands. The negative value of S ( )/ S (0) is applied to an n-channel MOSFET under compressive stress. The calculated values for SID(1 Hz ; )/ SID(1 Hz ;0) are listed in comparison with the measured ones in Table 5-2, where the measured valu es of drain and gate currents [ID( )/ID(0) and IG( )/IG(0)] are partly used for the calculations. In Eq. (5-63), the fourth and fifth terms have opposite signs for low gate and drain biases Gt D(..,VV~0.3V,V~0.1V). eg As a result, the actual sum of PAGE 104 104 the last three terms which is equivalent to the effective oxide trap change, ,effF,effFln1(;)/(;0),ttnEnE in Eq. (5-29) is much smaller than the first two terms. In these bias ranges, the stress altered channel mobility eff( ) is primarily responsible for the total change in drain current PSD under strain since both the change in drain current, ID( )/ ID(0), and correlated mobility fluctuations, ( )/(0), mainly results from the change in channel mobility, eff( )/eff(0). Thus, Eq. (5-63) can be simply approximated as, (1;)2() () 8() ln1 ln1 ln1 (1;0) (0)5(0)(0) 4() ln1. (0)ID D D ID D D D DSHz I I S SHzI SI I I (5-64) Eq (5-64) seems to be very useful from the practical point of view because we can easily estimate the noise magnitude change, SID(1 Hz ; )/SID(1 Hz ;0), after simply measuring or calculating only the drain current change with strain, ID( )/ ID(0). As shown in Table 5-2, the calculated values of SID(1 Hz ; )/ SID(1 Hz ;0) are not so different for Eqs. (5-63) and (5-64). In this section, we have discussed four causes of the magnitude change in 1 / f noise PSD under strain; changes in (1) trapping probability of tunneling carrie rs by oxide traps (2) tunneling probability of channel carriers (3) correlated mobility fluctuations of channel charge carriers, and (4) available traps encountered by tunneling channe l carriers due to nonuniform trap distribution in energy. These effects result directly from the modification of Si inversion subband energy levels by strain. In addition, strain effects on oxide traps were invest igated. To explain our experimental observation that the exponent in 1 / f noise spectrum is varied with applied stress, oxide traps must be redistributed over energy, or space, or both. The strain-induced change in the exponent, is predicted to be a functi on of gate voltage as indicat ed in Eq. (5-56). A larger PAGE 105 105 value of should be measured for a higher gate vol tage, which is consistent with our measurements in Table 5-1. Table 5-2. Comparison of the measured and ca lculated relative magnitude changes in drain current noise PSD, SID(1 Hz ; )/ SID(1 Hz ;0), at a stress of 100MPa. Device/Stress Measurement Calculation n-channel MOSFET under tens ion 11.9 % 14.9 (15.1) % n-channel MOSFET under compression -16.2 % -11.4 (-11.9) % VG= -0.6V 19.2 % 22.0 (20.28) % p-channel MOSFET under compression VG= -0.8V 22.9 % 19.9 (19.2) % *The relative magnitude changes in noise PSD, SID(1Hz; )/SID(1Hz;0), were estimated with some measured values [ID()/ID(0) a nd IG()/IG(0)] and calculated values ( B(), S) using Eqs. (5-63) and (5-64). The values in the parenthesis were calculated using Eq. (5-64) Summary We examined detailed mechanisms of strain effects on noise PSD based on our measured data. It was shown that the appl ied mechanical stress altered bot h the magnitude and exponent in the 1/f noise spectrum, resulting in larger change s in noise PSD at lower frequencies. The magnitude in 1/f noise drain current PSD was measured to increase and decrease for n-channel MOSFETs with increasing tensile and compressive stress, respectiv ely, while it increased for pchannel MOSFETs with increasing compressive stress. The dominant factors affecting 1/f noise magnitude were identified and investigated. One of the main factors is a trapping position change in energy space. Since its contribution to noise PSD is solely determined by the relative distance from the quasi-Fermi level, there is some possibi lity we can suppress the noise arising from this factor through proper ch oice of gate bias or strain e ngineering. More specifically, the quantization effective mass which determines the lowest energy level in the inversion layer is approximately three times larger for electrons th an for holes. At a relatively lower gate bias compared to p-channel MOSFETs, n-channel MO SFETs can be biased such that the ground energy levels are located below the Fermi level, and thus the 1/f noise PSD can be reduced by PAGE 106 106 applied stress. The energy distribution of the oxide traps was shown to reduce the 1/f noise PSD magnitude for both n-channel MOSFETs under te nsile strain and p-channel MOSFETs under compressive strain. The stress altered channel mobility, eff( ), was also shown to be a key contributor to the noise PSD change especially at low gate and drain biases. In long channel devices, the noise PSD magnitude change is approxi mately related to the drain current change as, (1;)/(1;0)ID IDSHzSHz 4()/(0).DDII PAGE 107 107 CHAPTER 6 SUMMARY AND RECOMMENDATIONS FOR FUTURE WORK Summary In this dissertation, the effects of strain on the CMOS transistor operation such as threshold voltage, gate tunneling current, and 1/f noise characteristics have been investigated. Strain effects on both conduction and valence ener gy bands are first presented in Chap 2. Using the elastic compliance constants (S -matrix elements), the deformation potentials of the conduction and valence bands are calculated, and the band edge shif t and splitting are discussed as strain effects. Each component of the model is also analyzed thoroughly in c onjunction with its underlying physical mechanism. In Chap 3, stress-applying apparatuses of uni axial and biaxial jigs ar e introduced. Based on the deformation potential calculation method in Chapter 2, key ba nd parameters to affect the threshold voltage are obtained as a function of strain. Large differences in the strain-induced threshold voltage shifts for uniaxial and biaxial tensile strained Si n-channel MOSFETs are explained and quantified. The calcu lated threshold voltage shift is significantly larger for biaxial than uniaxial strained MOSFETs and is in ag reement with uniaxial wafer bending and published biaxial strained Si on relaxed Si1-xGex experimental data. The larg e threshold shift for biaxial strain results from the strain-induced change in the Si channel electron affinity (or conduction band offset) and bandgap. The small threshold volt age shift for uniaxial pr ocess tensile strain results since the n+ polySi gate in addition to the Si channel is strained and significantly less bandgap narrowing occurs for uniaxial than biaxial tensile strain. In Chap. 4, strain effects on gate tunneli ng current are discussed. The detailed gate tunneling model for both nand p-MOSFETs is th en constructed. An experimental method to determine both the hydrostatic and shear deform ation potential constants is introduced. The technique is based on the change in the gate tunneling currents of Si n-MOSFETs under PAGE 108 108 externally applied mechanical stress and has been applied to industrial long channel (2 m and 4 m) MOSFETs. The conduction band hydrostatic and shear deformation potential constants (d and u) are extracted to be 1.0 0.1 eV and 9.6 1.0 eV, respectively, which is consistent with recent theoretical works. In Chap. 5, 1/f noise PSD (Power Spectral Density) fo r both nand p-MOSFETs has been measured and analyzed under externally app lied mechanical stress in conjunction with applications of strained devices to high performance RF or hi gh speed CMOS circuits. It is observed that (1) 1/f noise PSD increases and decreases for n-channel MOSFETs with increasing tensile and compressive stress, respectively, while it increases for p-channel MOSFETs with increasing compressive stress. (2) The relative change in PSDs shows a frequency dependence (i.e., larger changes for lower frequencies) since strain alters both the noise magnitude and exponent. The observed trends have been expl ained by a strain-induced energy level shift mechanism in the inversion layer. More specifi cally, four causes of the magnitude change in 1/f noise PSD were discussed: change s in (1) trapping probability of channel carriers by oxide traps (2) tunneling probability of cha nnel carriers (3) correlated mob ility fluctuations, and (4) available traps encountered by t unneling channel carriers due to nonuniform trap distribution in energy. These effects result directly from the mo dification of Si invers ion subband energy levels by strain. Strain effects on oxide traps were also investigated. To e xplain our experimental observation that the exponent in 1 / f noise spectrum is varied ith applied stress, oxide traps must be redistributed over energy or space or both. Recommendations for Future Work The demand for new material and technologies becomes in creasing in the nanometer CMOS regime. The downscaling of device dime nsions causes the gate leakage current to increase exponentially, thus increasing the 1/f noise significantly. Especially in conjunction with PAGE 109 109 high-k gate dielectric MOSFETs, low-frequency 1/f noise measurements can be used as a valuable tool for quality and reliability evalua tions of these devices since the low-frequency noise in a device is sensitive to the device technology such as the presence of traps, defects and crystal damage [82]. In the noise measurements of our ultrat hin (1.3nm) gate ox ide MOSFET samples, 1/f noise power spectrums have been observed in low fr equency regions (< ~10Hz), contrary to the conventional charge trapping model. It is expect ed in the conventional model that the tunneling time is very fast for this thin gate oxide. As a result, no 1/ f noise spectrums would be observed at such low frequencies as f < ~10Hz. A roll-off in the spectrum would be instead expected below a frequency corresponding to th e tunneling time to the fart hest situated traps at tox=1.3nm. This observation of 1/f noise spectrums in low frequency regions is important since it can be a new research topic for highly scaled CMOS devices. PAGE 110 110 APPENDIX A CONDUCTION BAND DEFORMATI ON POT ENTIALS FOR GE In Chapter 2, we reviewed the conduction band deformation potentials for Si. It was shown that the behavior of -valleys was critical to determine the physical properties of Si MOSFETs since the -valleys form the lowest conduction en ergy bands. In Ge, however, the conduction band minima are -valleys, which are located along the eigh t equivalent directions of [111]. In this case, Eq. (2-11) is no longer valid to us e. A more general expression can be found in the original notation of Herring and Vogt [2, 8, 29, 30]: 1:,i CduiiEa a (A-1) where and 1 are the 2nd order strain and unit tensors, respectively, ia is a unit vector parallel to the K vector of valley i and iiaadenotes a dyadic product. In Eq. (A-1), the hydrostatic and shear strain components are written as 1:, 3 1: 3Hydro u Cd Shear u CuiiE andEaa (A-2) As mentioned in Section 2.1, a dyad notation is equivalent to a second order tensor representation, and the double dot product ( : ) is defined as [66] :. ABCDACBDforanytwodyadsABandCD (A-3) The result of the double dot produc t produces a scalar quantity, and Eq. (A-3) can be written in a more convenient notation: : A BCDCABDorACDB (A-4) PAGE 111 111 The strain and unit tensors can be also expressed in th e dyad notations as follows: .........., 1.xxxy zziiij kk and iijjkk (A-5) Using Eq. (A-3) and (A-4), we can rewrite the en ergy shift expression of Eq. (A-1) in a simpler form: 1:[()][],i T CduiidijuEa aT rKK (A-6) where K and TK are row and column unit vectors, respectively, along the direction of the K vector of valley i As an example, let us calculate the deformation potentials of the Ge conduction band for a uniaxial [110] stress. Fig. A-1 shows the location of eight ellipsoids of valleys. Four ellipsoids lie on a (110)-plane and another four ellipsoids on a (110)-plane. When a uniaxial [110] stress is applied, these eight valleys can be divided into two groups according to their condition under stress. The four (110)-plane valleys (blue) form one group which is under the same stress condition, and the other four (110)-plane valleys (grey) another group. Since these two groups each are on the same condition under [110] stress, we choose 111and 111 valleys, each one from each group for calculation. Using Table 2-1 and Eq. A6, we can obtain the deformation potentials due to hydrostatic and shear strain components as follows: () () [111] [111] () () 1112[()] 3 2 3Ge HydroHydro Ge u di j G e Ge Ge GeGe u dG eEE Tr SS (A-7) () 44 () () 44(/6)[111] [()] 3 (/6)[111]GeGe uG e ijGe ShearGe T Cu GeGe uG eSforvalleys Tr EKK Sforvalleys (A-8) PAGE 112 112 [111] [110] [111]0 11 (1,1,1),(1,1,1),0. 33 00xxxy yxyy zzwithK K and Note that off-diagonal strain components, x yy xand are used in the calculation and the Herring and Vogts formula is cons istent with the conventional notation of stress-strain relation. The calculation result shows that the band splitting occurs in a way: For a tensile [110] stress, the energy of each four valley on a (110)and a (110)-plane are up-shifted and down-shifted along each valley direction, respectively, by the same magnitude. Typically MOSFETs are fabricated on a (001)-wa fer and the channel is located along the [110] direction. In this structur e, the gate bias is applied along the [001] direction. Therefore, we need to obtain the effective band splitting in the field di rection to correctly evaluate stress effects on the device properties. From Eq. (A-8), the effective band splittin g along the [001] direction is obtained by projecting each valley direction onto the [001] direction. () 444 () 444(/6) 3 (/6) 3Ge Ge u Ge Shear C Ge Ge u GeSforvalleys E Sforanothervalleys (A-9) The band splitting in Ge for other directions is listed in [8]. PAGE 113 113 [100] [010] [001][111] [111] [111] [111] Figure A-1. Eight-fold degenerate -valleys in the Ge conducti on band. Under uniaxial [110] stress, the four valleys (blue) on a (110)-plane are on the same stress condition and the other four valleys (grey) on a (110)plane are on another same stress condition. PAGE 114 114 APPENDIX B YOUNGS MODULUS IN A (110)-SI W AFER Youngs modulus is plotted as a function of angle in a Si (110)-wafer in Fig. B-1, and the analytical expression is given by [37] 1 (110) 4 4 11111244()(/2)(sincos/21) YSSSS (B-1) Similarly as in Section 3.1, we can calculate the conversion factor For the purpose of comparison with an in-plane (xand y-direction) biaxial stress, we take Youngs modulus in the [001] direction to be a reference value. 1 44 1244 1111 1 44 1244 0 1111 0()(/2) (/2)11(sincos/21) 2 11(sincos/21) 2 (/2) 1.256(/2) YY SS Y SS SS d SS Y d Y (B-2) The interval of the integral is chosen from 0 to 180 since Youngs modulus has a two-fold rotation symmetry about the [001] axis. We can also calculate the conversion factor in Ge simply by replacing the elastic compliance consta nts in Eq. (131) with Ge elastic compliance values. PAGE 115 115 50 100 150 200GPa 30 210 60 240 90 270 120 300 150 330 180 0 [110] [001] Figure B-1. Youngs modulus of Si as a function of direction in the (110) plane. PAGE 116 116 APPENDIX C STRAINED-SI MOFETS ON A (110) WAFER It is contended in [67, 68] that biaxiall y-strained S i MOSFETs on a (110)-wafer are advantageous over the conventional counterpart s on a (001)-wafer in respect of mobility enhancement. However, in order to accurately compare these two types of (001)and (110)wafer MOSFETs in terms of the stress effects on mobility enhancement, we need to first quantify an equivalent stress to the fiel d direction for (110) MOSFETs. In the appendix, we investigate which direction of the strain and channel is de sirable for uniaxially-strained (110) MOSFETs to obtain the highest mobility on the bases of deform ation potentials and carrier effective masses. Let us first obtain the stress tensor components fo r an arbitrary stress direction as shown in Fig. C-1. A simple calculation using Eq. (2-7) leads to 222 22 2 2sincossinsincossincoscos (,)(,)sinsincossinsinsincossin. sincoscossincossincosij (C-1) If we set 3 4 and 2 we obtain a stress tensor expr ession for an arbitrary stress direction in a (110)-plane, and then using Eq. (2-3) through (2-5) we can calculate the strain components as follows: (110) (110) 2 2 1112 12 (110) (110) 2 2 1112 12 (110) (110) (110) 2 2 (110) 11 12 44 (1 (110)()()cossin, 2 ()()cossin, 2 () ()()sincos,() sincos, 22 ()xx yy zz yz xzSS S SS S SS S 10) (110) (110) 2 44 44() () sincos,() cos,(C-2) 4 22xySS PAGE 117 117 [001] [110] 0 (110)-wafer [110] Figure C-1. (a) Representation of an arbitrary stress in a spherical coordinate. (b) New direction specified in a (110)-wafer. (a) (b) PAGE 118 118 where the superscript (110) denotes a (110)-wafer and is a newly specified direction in a (110)-wafer. In order to obtain the effective ba nd splitting along the [ 110] (or a gate field) direction, we need to first calculate each band splitting for the [100]and [010]-valleys. From Eq. (2-12), the average energy shift and band sp litting for the [100]-valleys are expressed as 2(110) (110) 1112 [100](110) 2 (110) 1112() [()] 2(), 33 [()] ()() 3 cos1 (). 23Hydro uu Cdijd ij uxx uET rS S Tr E SS (C-3) Note that the magnitude of this 2 band splitting is two thirds of the total splitting, as shown in Eq. (2-15). For [010]-valleys as we ll, we have the same magnitude of the band splitting since in-plane uni axial stresses in a (110 )-wafer are applied symm etrically to the [100]and [010]-valleys. Fig. C-2 (a) shows the locati on of six valleys in a (110)-wafer and a band splitting diagram. It is illustrated in the di agram how to calculate the effective band splitting along the [110]-direction. Along both [100]and [010]-directions, each 2 valley is shifted by two thirds of the splitting as denoted by symbols and in the figure. More specifically, we first consider the band sp litting between the [100] 2-valleys and [010]-[001] 4-valleys along the [100] direction. The two groups split oppositely ( and ); the [100] 2-valleys are upshifted by two thirds of th e splitting and the rest 4-valleys are down-shifted by one third. Again, along the [010] dir ection the [010] 2-valleys are up-shifted by two thirds of the splitting () and the [100] 2-valleys and [001] 2-valleys are down-shifted by one third ( and ). As a result, each pair of valleys of th e three directions reaches its final position which is a successive vector sum, that is, + for [100] 2-valleys, + for [010] 2-valleys, and + for PAGE 119 119 [001] 2-valleys. Therefore, the effective band split ting along the [110] di rection is expressed, using Eq. (A-6), as 2 (110) 11124 [110] 2 (110) 11122cos1 ()cos45 223 () cos1 ()cos45. 23u CSpl uSS forvalleys E SS forvalleys (C-4) Note that the average band energy level is not changed with shear stra in in all directions. Fig. C-2 shows the effective band splitting vs. st ress direction, and the conductivity effective mass of 4 valley electrons vs. channel direction. U nder a gate bias, th e ground energy state becomes a 4 subband since the out-of-effectiv e mass of the electrons in 4 valleys is larger, namely, 42** *[110]*[110]* 00 **2 0.315. 0.19.tl t tlmm mm v s m m m mm Consistent with these quantized energy levels, the strain-induced splitting directi on and magnitude must be determined for the carrier mobility to be maximized. In the figur e, the splitting direction is changed at 11 sin35.3 3 which corresponds to the [111] st ress as listed in Table 2-2. At = 90 ([001] direction) for a tensile stress, the splitting magnitude is maximum, and the direction is consistent with the quantized energy levels; the ground ( 2 subband) and 2nd lowest ( 4 subband) energy levels are lowered and raised, respectively. It is also shown that the selection of the channel direction for a (110)-w afer MOSFET has a strong influe nce on carrier mobility because of the anisotropic conduc tivity effective mass of 4 valleys [67, 68]. An actual calculation yields the following dependence of the eff ective mass on the channel direction : ** (110) *2*22 () cos(1sin)tl tlmm m mm (C-5) PAGE 120 120 Therefore, both the best stress and channel dire ctions are the [001] dir ection for a (110)-wafer nMOSFET as indicated in the plots. [001] 0 [110] [100] [010] [110]( 1 1 0 ) p l a n e ( 0 0 1 ) p l a n e Figure C-2. (a) Si x ellipsoids of -valleys, each lying along the six equivalent axes of [100]. 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PAGE 128 128 BIOGRAPHICAL SKETCH Ji-Song Lim was born in Korea. He received his Master of Scienc e degree in electrical and computer engineering from the University of Fl orida in 2002, where he is currently pursuing a Ph.D. degree focusing his resear ch on strain effects on silicon CMOS transistors such as threshold voltage, gate tunneling current, and 1/f noise characteristics. |