BULK Si (100) VALENCE BAND STRUCTURE UNDER STRAIN
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BULK Si (100) VALENCE BAND STRUCTURE UNDER STRAIN
Sagar Suthram
Computational NanoelectronicsClass Project - 2006
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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
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MOSFET Scaling History
Si MOSFET first demonstrated in SSDRC in 1960.
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).
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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
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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
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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.)
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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]
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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
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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.
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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
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Understanding Strain
0A
FLimA
Stress () :
Strain () : 0
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Understanding Strain
11 12 12
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C C C 0 0 0C C C 0 0 0C C C 0 0 0 0 0 0 C 0 0 0 0 0 0 C 0 0 0 0
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S S S 0 0 0S S S 0 0 0S S S 0 0 0
2 0 0 0 S 0 0 0 0 0 0 S 02 0 0 2
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C Elastic Stiffness Coefficients
S
Elastic Compliance Coefficients
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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
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Silicon valence band using k.p
ruer nkrik
nk.
rumkkErurVpk
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The form of the Schrodinger equation when written in terms of unk(r) near a particular point k0 of interest.
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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 2nd order degenerate
perturbation theory
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Luttinger-Kohn Hamiltonian
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Valence Band structure
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Valence Band structure
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Valence Band structure
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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
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Silicon Valence band using tight-binding method
px
py
pz
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
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Tight binding Band structure
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Tight binding Band structure
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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
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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
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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
addedThank You