Strained Silicon, Electronic Band Structure and … · Review for Electronic Band Structure Ab initio vs Empirical methods A strain example: Si on SiGe buffer Lower Dimension Aproximation

®

c Bandsues.

odeling6 Crolles

nd H. Jaouen

Q&TPS

Strained Silicon, ElectroniStructure and Related Is

STMicroelectronics, Q&TPS, Device M850 rue Jean Monnet, BP 16, F-3892

CEDEX, France

D. Rideau, F. Gilibert, M. Minondo, C. Tavernier a

D. RIDEAU

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8 APRIL 2005 MOS-AK STRASBOURG

Q&TPS

OUTLOOK (1/4)

Current & Capacitances

STRAIN Matrix

INPUT

DESIRED VALUES

What to do?

Strain matrix

Ansys ...

?

D. RIDEAU

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Q&TPS

Electronic Structure upon Strain

Empirical: TB KP EPMAbinitio: LDA RPA GW

Dispersion relation and Gaps

OUTLOOK (2/4)

D. RIDEAU

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ng Times: “Fermi Golden rule”“modified” GR Algorithm

Q&TPS


Empirical: TB KP EPM

Integration over Brillouin Zone

ScatteriDensity of States

Carrier DensityMean Carrier Energy

Linear Response Theory

GR Algorithm

Fermi Dirac Statistics

Kubo-Greenwood formula

Mobility

Abinitio: LDA RPA GW


G. Gilat and J. Raubenheimer, PR 144, 390 (1966)

OUTLOOK (3/4)

D. RIDEAU

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ng Times: “Fermi Golden rule”

MC

“modified” GR Algorithm

Mobility

µαI


Q&TPS


Empirical: TB KP EPM

Integration over Brillouin Zone

ScatteriDensity of States

Carrier DensityMean Carrier Energy

Linear Response Theory

Poisson Schrodinger

GR Algorithm

Fermi Dirac Statistics

Kubo-Greenwood formula


Mobility

Semiconductor Equation

Abinitio: LDA RPA GW



Compact Models

OUTLOOK (4/4)

D. RIDEAU

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ethod

Q&TPS

Review for Electronic Band Structure

Ab initio vs Empirical methods

A strain example: Si on SiGe buffer

Lower Dimension Aproximation

6X6 KP and Effective mass Hamiltonian

1Review for Electronic Band Structure M

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Block Functions

itting parameters

in DamoclesY.M. Niquet et al, PRB 62 5109 (2000) and

(includes SO)

C. Tserbak et al, PRB 47 7104 (1993)

KPEPM

eigenvalues

od

Q&TPS

Troullier-Martins psp from fhi98PPHartwingsen psp including SO

Schrodinger Equation

Wave function BasisPlane WavesGaussian-orbital

Matrix ElementsEvaluation

FAb initio

Kohn-Sham Scheme GW correction

HKS Ec= Vion VHF+ + … Σ ψnk⋅ r3

d∫+

C.Hartwingsen et al, PRB, 58 3641 (1998)

EPM “Best” TB “Best” in

KP (UTOX)

DFT + LDA

TB

Self-consistentevaluation

Simple eigenvalueproblem

Electronic Band Structure, Overlap integral...

Hψnk Enkψnk=

Review for Electronic Band Structure Meth

D. RIDEAU

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K G

ABINIT V4.4.3

3641 (1998))

Q&TPS

L G X W K’,U L W X

−10

−8

−6

−4

−2

0

2

4

6E

NE

RG

Y (

eV)

WAVE VECTOR

GWLDA KSS

PSP: Hartwingsen psp (C.Hartwingsen et al, Phys. Rev. B, 58

Ab Initio: vs

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K G

HYS. REV. B 14, 556 (1976)

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−10

−8

−6

−4

−2

0

2

4

6

EN

ER

GY

(eV

)

WAVE VECTOR

EPM (local)

“NON LOCAL EFFECT”

UTOX AFTER J.R. CHELIKOWSKY AND M.L. COHEN, P

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KPGW

K G

Q&TPS

KPUTOX


−10

−8

−6

−4

−2

0

2

4

6

EN

ER

GY

(eV

)

WAVE VECTOR

D. RIDEAU

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40

(110)

ass Hamiltonian

Structure

Q&TPS

0 20

k(108m−1)

−20 0 20−900

−800

−700

−600

−500

−400

−300

−200

−100

0

100

(111) k(108m−1) (100)

Ek (m

eV)

Valence Bands: KP and Effective M

KPMASS

Effective mass approximation for Electronic Band

D. RIDEAU

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Mass Hamiltonian

Q&TPS

Hole “curvature mass” for Effective

0 20 40 60 800.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

wafer in−plane orientation (deg.)

Hol

e C

ondu

ctio

n m

ass

(m0

units

)

hhlhsh

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e Mass Hamiltonian

Q&TPS

−10 −5 0 5 10

50

100

150

200

250

300

(100) k(108 m−1) (001)

Ek

(me

V)

∆2∆4

Conduction Bands: KP and Effectiv

MASSKP

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0 0.1 0.2 0.3 0.40

02

04

06

08

01

12

14

16

x

Si

ε = Strain

Q&TPS

0.0

0.0

0.0

0.0

0.

0.0

0.0

0.0

exx

SiGe lattice larger than Si lattice

STRAIN (STUDIED CASE)

D. RIDEAU

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G

EQUIVALENT VALLEY

Q&TPS

L G X,Z G Y,X−3

−2

−1

0

1

2

3

4

ENER

GY

(eV)

RELAXED SILICON

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G

Q&TPS

L G X,Z G Y,X−3

−2

−1

0

1

2

3

4

ENER

GY

(eV)

2% TENSILE

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AIN

CONDUCTION BANDS

VALENCE BANDS

Q&TPS

ENERGY SHIFT VS . STR

0.97 0.98 0.99 1 1.01 1.02 1.03−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

a||/a

0

EN

ER

GY

(eV

)∆2

∆4

LH

HH

SH

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(RELAXED )

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CONDUCTION BANDS

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(1% TENSILE)

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CONDUCTION BANDS

D. RIDEAU

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(1% COMPRESSIVE)

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CONDUCTION BANDS

D. RIDEAU

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LAXED )

Q&TPS

VALENCE BANDS (RE

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TENSILE)

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VALENCE BANDS (1%

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COMPRESSIVE)

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VALENCE BANDS (1%

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1% Compressive

0 20 40

k(108m−1) (110)

−20 0 20−900

−800

−700

−600

−500

−400

−300

−200

−100

0

100

(111) k(108m−1) (100)

Ek

(meV

)
Q&TPS
VALENCE BANDS

1% Tensile

0 20 40

k(108m−1) (110)

−20 0 20−900

−800

−700

−600

−500

−400

−300

−200

−100

0

100

(111) k(108m−1) (100)

Ek

(meV

)

0 20 40

k(108m−1) (110)

−20 0 20−900

−800

−700

−600

−500

−400

−300

−200

−100

0

100

(111) k(108m−1) (100)

Ek

(meV

)

Relaxed

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ZONE

Q&TPS

Density Of States and DOS masses

Scattering Rates

Carrier Density

1

2

3

INTEGRATION OVER BRILLOUIN

Electronic Structure



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DENSITY OF STATES

ρE E( ) δk BZ∈

∑ E En k( )–[ ]n∑=

W

L

K

X

U

1/48 1/8


DENSITY OF STATES

SYMMETRIES

Electronic Structure



INTEGRATION

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−4 −2 0 2 4 6E (eV)

Q&TPS

−12 −10 −8 −60

1

2

3

4

5

g E (

1022

cm

−3 e

V−

1 )

DENSITY OF STATES

−12 −10 −8 −6 −4 −2 0 2 4 60

1

2

3

4

5

g E (

1022

cm

−3 e

V−

1 )

E (eV)

EPM (LINES)KP (LINES)

GW (DASHED LINES)GW (DASHED LINES)

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EFF MASSES)

CTRONS

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DENSITY OF STATES (FB VS.

ELEHOLES

D. RIDEAU

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ture masses and valence band

EPMd

1.170.21

0.916

4.270.315

1.3867

Q&TPS

MASSES

Table 1 Experimental and theoretical band gap, conduction band curvaLuttinger parameters for Silicon.

Exp.a

k.p GWc

(eV) 1.170 1.17 1.1

(mo)0.191 0.194 0.191

(mo)0.916 0.916 0.921

4.27 4.27b 4.27 0.32 0.315b 0.315

1.458 1.386b 1.386

Eg

mt

ml

γ1

γ2

γ3

a Ref.[11]; b Fit for the 6-level k.p;c with ABINIT V4.3.3 [3]; d Ref. [17].

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ALENCE BANDS)

0 200 400 600 8000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

msh

(D

OS

)

T (K)

x=0x=0.1x=0.2x=0.3

Q&TPS

DOS MASSES INSI/SIGE (V

0 200 400 600 8000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

m(D

OS

)

T (K)

x=0x=0.1x=0.2x=0.3

0 200 400 600 8000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

mlh

(D

OS

)

T (K)

x=0x=0.1x=0.2x=0.3

KP (UTOX)

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IN :

.1 0.2 0.3 0.4 0.5−E

F (eV)

Q&TPS

CARRIER DENSITY VS. STRA

−0.2 −0.1 0 010

15

1016

1017

1018

1019

1020

1021

1022

EV

p (/

cm3)

FB 1.5% tensileOB 1.5% tensileFB relaxed SiOB relaxed Si

−0.2 −0.1 0 0.1 0.2 0.3 0.4 0.510

15

1016

1017

1018

1019

1020

1021

1022

EF−E

C (eV)

n (/c

m3)

FB 1.5% tensileOB 1.5% tensileFB relaxed SiOB relaxed Si

UTOX AFTER M. V. FISCHETTI ET AL . IN DAMOCLES

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ENTS)

0 0.5 1 1.5VG (V)

PMOS

x=0.2

relaxed

Q&TPS

CAPACITANCE (MEASUREM

−1.5 −1 −0.5 0 0.5 1

0.006

0.008

0.01

0.012

0.014

0.016

0.018

VG (V)

C (

µF/m

m2)

−1 −0.5

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

C (

µF/m

m2)

NMOS

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D CURVES)

0 0.5 1 1.5VG (V)

Q&TPS

−1.5 −1 −0.5 0 0.5 1

0.006

0.008

0.01

0.012

0.014

0.016

0.018

VG (V)

C (

µF/m

m2)

CAPACITANCE (SIMULATE

Charge Sheet Model Density Gradient

x=0.2

relaxed

−1 −0.50.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

C (

µF/m

m2)

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iile

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CURRENT (LOW FIELDS )

1 1.2 1.4 1.6 1.8 2 2.2

10−11

10−10

10−9

10−8

10−7

10−6

10−5

VG−VFB(V)

ID(A

/µm

2)

UTOX relaxed SUTOX 1.5% tens

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re

Q&TPS

CONCLUSIONS

❍ Methods for Band Structure

❍ STRAINED SILICON Band Structu

❍ DOS and Scattering times

❍ Capacitances

Documents

Strained Silicon, Electronic Band Structure and … · Review for Electronic Band Structure Ab initio vs Empirical methods A strain example: Si on SiGe buffer Lower Dimension Aproximation