1 BULK Si (100) VALENCE BAND STRUCTURE UNDER STRAIN Sagar Suthram Computational Nanoelectronics Class Project

2 Outline  Brief history of MOSFET scaling and need for Strained Silicon.  Understanding Strain.  Si valence band structure calculation using k.p method.  Si valence band structure calculation using tight binding method.  Strain effects on Si valence and conduction band – qualitative picture.  Summary

3 MOSFET Scaling History  Si MOSFET first demonstrated in SSDRC in  Improved dramatically due to gate length scaling driven by  Increased density and speed  Lower costs  Power improvements  Semiconductor industry scaled the MOSFET channels based on Moore’s law (1965). Simple geometric scaling followed.  Constant field scaling introduced by Dennard et. al. (1974).

4 MOSFET Scaling History  Constant field scaling too restrictive  Subthreshold nonscaling  Power-supply voltage not scaled proportional to channel length  Generalized scaling is preferable which allows oxide field to increase  Shape of 2-D electric field pattern preserved (channel doping engineering)  Short channel effects do not become worse

5 MOSFET Scaling Limits  But conventional planar bulk MOSFET channel length scaling is slowing  Increased off-state leakage  Increased off-state power consumption  Degraded carrier mobility due to very high vertical fields (thin oxides <2nm)  Lithographic limitations  Little improvement in switching performance  Inability to scale supply voltage and oxide thickness

6 Continued Transistor Scaling  “No exponential is forever” – Gordon Moore  But present scaling limits for Si MOSFET are caused by materials and device structure and are not hard quantum limits  Continued scaling requires new materials and device structures  High –K dielectrics  Strained Si  Novel channel materials (Ge, III-V semiconductors)  Non classical CMOS devices (FinFETs etc.)

7 Strained Silicon  Strained Silicon has been adopted in all advanced logic technologies  Scalable to future generations  Easily incorporated in existing processes  Enhances performance even in the ballistic regime due to effective mass reduction 90nm INTEL Technology node transistor with process induced uniaxial stress [Thompson 04]

8 How is strain added to silicon ? Uniaxial stress is induced in the following ways  SiGe source-drain for PMOS  Tensile nitride capping layer for NMOS

9 How is strain added to silicon ? Biaxial stress is induced by epitaxialy growing a silicon layer on relaxed SiGe. The lattice mismatch induces biaxial tensile stress in the silicon layer.

10 Outline  Brief history of MOSFET scaling and need for Strained Silicon.  Understanding Strain.  Si valence band structure calculation using k.p method.  Si valence band structure calculation using tight binding method.  Strain effects on Si valence and conduction band – qualitative picture.  Summary

11 Understanding Strain Stress (  ) : Strain (  ) :

12 Understanding Strain Elastic Stiffness Coefficients Elastic Compliance Coefficients

13 Outline  Brief history of MOSFET scaling and need for Strained Silicon.  Understanding Strain.  Si valence band structure calculation using k.p method.  Si valence band structure calculation using tight binding method.  Strain effects on Si valence and conduction band – qualitative picture.  Summary

14 Silicon valence band using k.p The form of the Schrodinger equation when written in terms of u nk (r) near a particular point k 0 of interest.

15 Silicon valence band using k.p Luttinger-Kohn’s model: k.p method for degenerate bands  Mainly for silicon valence bands  Consider the heavy hole, light hole and split-off bands as class A and rest of the bands as class B  Use 2 nd order degenerate perturbation theory

16 Luttinger-Kohn Hamiltonian

17 Valence Band structure

18 Valence Band structure

19 Valence Band structure

20 Outline  Brief history of MOSFET scaling and need for Strained Silicon.  Understanding Strain.  Si valence band structure calculation using k.p method.  Si valence band structure calculation using tight binding method.  Strain effects on Si valence and conduction band – qualitative picture.  Summary

21 Silicon Valence band using tight-binding method pxpx pypy pzpz  sp3s* tight binding picture used  20x20 Hamiltonian including spin-orbit interaction considered  Silicon valence band predominantly composed of p- bonding states which are degenerate at the  point

22 Tight binding Band structure

23 Tight binding Band structure

24 Outline  Brief history of MOSFET scaling and need for Strained Silicon.  Understanding Strain.  Si valence band structure calculation using k.p method.  Si valence band structure calculation using tight binding method.  Strain effects on Si valence and conduction band – qualitative picture.  Summary

25 Strain effects on silicon valence band  Splits the degeneracy of the valence band at the  point  The bands are no longer just HH or LH due to the strong coupling between the two, but either HH-like or LH-like  Biaxial stress does not warp the bands much due to the presence of only a hydrostatic component in the strain matrix which maintains the crystal symmetry.  Uniaxial stress warps the bands causing a reduction in the effective mass due to the presence of a shear term which destroys the crystal symmetry

26 Summary  k.p method is emperically based and treats the band structure with precision  k.p is useful for calculating band structure only for k values close to the band edge which is generally the region of interest  Tight-binding on the other hand considers the microscopic interatomic interactions and hence gives a good physical insight into the strain effects on the band structure  We see differences in the exact band structures computed by the two methods but they show similar trends under the application of strain  Computing more accurate band structures with the tight-binding method involves consideration of up to 10 orbitals (sp3d5s*) along with spin which gets very complicated when the strain effect is added Thank You