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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

0

a aa

xx

yy

zz

yz

zx

xy

2

22

xx

yy

zz

yz

zx

xy

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Understanding Strain

11 12 12

12 11 12

12 12 12

44

44

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

xx

yy

zz

yz

zx

xy

44

2

2 0 0 C 2

xx

yy

zz

yz

zx

xy

11 12 12

12 11 12

12 12 12

44

44

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

xx

yy

zz

yz

zx

xy

440 0 0 S

xx

yy

zz

yz

zx

xy

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

mmp

nknnk

0

22

00

2

2.

2

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