International Workshop of Computational Electronics Purdue University, 26th of October 2004

Treatment of Point Defects in Nanowire Treatment of Point Defects in Nanowire MOSFETs Using the Nonequilibrium Green’s MOSFETs Using the Nonequilibrium Green’s

Function Formalism Function Formalism

M. Bescond, J.L. Autran*, N. Cavassilas, D. Munteanu, and M. Lannoo

Laboratoire Matériaux et Microélectronique de Provence UMR CNRS 6137 - Marseille/Toulon (France) - www.l2mp.fr

* Also Institut Universitaire de France (IUF)

OutlineOutline Introduction

MOSFETs downscaling: statistical fluctuations of doping impurity positions

3D Quantum simulation of point defects in nanowire transistors Nonequilibrium Green Function formalism: Mode-Space approach Treatment of point defects

Results: influence of the impurity location and type Energy subbands Transverse modes Current characteristics

Conclusions and perspectives

Dimensions of nanowire MOSFETsVG

VDVS

L

DRAINSOURCE CHANNEL

VG

Gates

GatesOxide

W

XZ

Y

VG

VDVS

L

DRAINSOURCE CHANNEL

VG

Gates

GatesOxide

W

XZ

Y

XZ

Y

Si TSi

SiO2

TOX

Si TSi

SiO2

TOX

Si TSi

SiO2

TOX

Si TSi

SiO2

TOX

a) b)

VG

VDVS

L

DRAINSOURCE CHANNEL

VG

Gates

GatesOxide

W

XZ

Y

VG

VDVS

L

DRAINSOURCE CHANNEL

VG

Gates

GatesOxide

W

XZ

Y

XZ

Y

Si TSi

SiO2

TOX

Si TSi

SiO2

TOX

Si TSi

SiO2

TOX

Si TSi

SiO2

TOX

a) b)

WSiVG

VDVS

L

DRAINSOURCE CHANNEL

VG

Gates

GatesOxide

W

XZ

Y

VG

VDVS

L

DRAINSOURCE CHANNEL

VG

Gates

GatesOxide

W

XZ

Y

XZ

Y

Si TSi

SiO2

TOX

Si TSi

SiO2

TOX

Si TSi

SiO2

TOX

Si TSi

SiO2

TOX

a) b)

VG

VDVS

L

DRAINSOURCE CHANNEL

VG

Gates

GatesOxide

W

XZ

Y

VG

VDVS

L

DRAINSOURCE CHANNEL

VG

Gates

GatesOxide

W

XZ

Y

XZ

Y

Si TSi

SiO2

TOX

Si TSi

SiO2

TOX

Si TSi

SiO2

TOX

Si TSi

SiO2

TOX

a) b)

WSi

p-type channel region: Volume = LWSiTSi=1053=150 nm3

If doping concentration=1019 cm-3 1.5 impurity on average.

Discrete distribution and statistical location.

Effect of the impurity type and location.

Source and drain region: continuous doping of 1020 cm-3.

Dimensions: L=8 nm, WSi=3 nm, and TSi=3 nm, TOX=1 nm.

Channel region: discrete doping of 1019 cm-3, with 1 impurity on average.

Nonequilibrium Green’s function Nonequilibrium Green’s function formalismformalism

Point Defect Treatment

The 3D Mode Space Approach*

The 3D Schrödinger = 2D (confinement) + 1D ( transport)

VG

VDVS

L

DRAINSOURCE CHANNEL

VG

Gates

GatesEOT

W

VG

VDVS

L

DRAINSOURCE CHANNEL

VG

Gates

GatesEOT

W

2D (confinement)

1D (transport)

Si TSi

SiO2

TOX

Si TSi

SiO2

TOX

Si TSi

SiO2

TOX

Si TSi

SiO2

TOX

Si TSi

SiO2

TOX

Si TSi

SiO2

TOX

Si TSi

SiO2

TOX

Si TSi

SiO2

TOX

i,nψ

ith eigenstate of the nth atomic plan

Hypothesis: n,i is constant along the transport axis.

* J. Wang et al. J. Appl. Phys. 96, 2192 (2004).

3 2

3

2

11

YX

Z

3 2

3

2

11

3 2

3

2

11

YX

Z

YX

Z

3 2

3

2

11

YX

Z

3 2

3

2

11

3 2

3

2

11

YX

Z

YX

Z

Y

X

Z

Y

X

Z

The 3D Mode Space Approach Electron distribution along subbands (valley (010)):

Simplified tight-binding approach: cubic lattice with ax, ay, az.

- 1 orbital/atom: : position z=laz, y=may, x=nax.

- Interactions between first neighbors.

1st

2nd

3rd

y

0 5 10 15 20-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

EFD

EFS

L=8 nmV

DS=0.4 V

VG=0 VP

oten

tial e

nerg

y (e

V)

X (nm) 1st

2nd

3rd

y

0 5 10 15 20-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

EFD

EFS

L=8 nmV

DS=0.4 V

VG=0 VP

oten

tial e

nerg

y (e

V)

X (nm)

i,nψ

i,nε

( )εGi

: Transverse eigenstate

: Transverse eigenvalue

: 1D Green function

For each subband i:

i=1

i=2

i=3

n,m,l

Point defect description

Treated as a macroscopic variation

Included in the self-consistent mode-space approach without coupling the

electron subbands

Treated as a localized variation = Chemical structure

Included in the real space approach based on the Dyson equation.

[G=(I-G0V)-1G0]

Point Defect = On-site Potential + Coulomb Tail

Energy

Impurity

r

erV

Si

Lr

Coul

c

0

/

4

1

dU

Treatment of on-site potential After achieving self-consistence including the Coulomb Tail, the device is subdivided at the point defect location:

Source Drain

S1 S2

X

Y

Z

Y

Source Drain

S1 S2

X

Y

Z

Y

Calculation of the Green’s functions of the surfaces S1 and S2:

( ) ( ) ',',',,=',',',, 111,','1

,,1111111111 1111∑ nmlψψεGψψnmlnmlεGnml ininii

ininS

( ) ( ) ',',',,=',',',, 222,','2

,,2222222222 2222∑ nmlψψεGψψnmlnmlεGnml ininii

ininS

Treatment of on-site potential Ud is then included using the Dyson equation:

2

10

'0

0

S

SS

G

GG

0

0

21

12

H

HV

2

1-

22 -' SSS GUGIG

DS ffVNVNTrd

eI 1121

0221211

0111

4

00 Im

1SGN

100

VVGGI SS

Intra-atomic potential matrix

Retarded Green function of the uncoupled system:

Calculation of the current*:

* M. Bescond et al., Solid-State Electron. 48, 567 (2004).

ResultsResults

Influence of point defect

Simulation results Electronic subbands: Effect of the Coulombic potential (valley (010))

Source Drain

1 nmX

Y

Z

Y

Source DrainSource Drain

1 nmX

Y

Z

Y

Subband profile is affected by the impurity. Subbands are still independant: justification of the mode-space approach. Acceptor impurity increases the channel barrier.

Simulation results Evolution the 1st confinement eigenstate (valley (010)):

Highest variations of the eigenstate with centered impurity.

Scalar product : weak variations.95.0≈ψψcent,1free,1

Defect free Centered defect

2

cent,1ψ

2

free,1ψ

z

Defect in the corner

2

nocent,1ψ

Simulation results First subband profile and current characteristics:

Defect in the corner: weak influence on the subband profile. Defect free: highest current. Centered defect: lowest current. Defect in the corner: intermediate behavior: current decrease of 50%. Variation of the subthreshold slope.

5 10 15-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

Defect free Centered defect Non centered defectF

irst s

ubba

nd p

rofil

e (e

V)

Source-drain axis (nm)

ZZ

ZZ

5 10 15-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

Defect free Centered defect Non centered defectF

irst s

ubba

nd p

rofil

e (e

V)

Source-drain axis (nm)

ZZ

ZZ

5 10 15-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

Defect free Centered defect Non centered defectF

irst s

ubba

nd p

rofil

e (e

V)

Source-drain axis (nm)

ZZ

ZZ

0.0 0.2 0.4 0.6 0.8

10-11

10-9

10-7

10-5

I D (

A)

VG (V)

Defect free Defect in the center Defect in the corner

VG=0 V VDS=0.4 V

Ud=2 eV

Simulation results Influence of Coulomb potential:

On-site potential defect does not affect the total current. Coulombic potential has the most significant impact. Electrons can be transmitted through the unperturbated neighboring atoms.

VDS=0.4 V

Conclusion

Modeling of electron-ion interaction based on the NEGF formalism.

Study of the effect the acceptor impurity in terms of physical properties.

Centered impurity involves a significant degradation of the current.

Not only a shift of the current but rather a subthreshold slope variation.

The Coulomb potential has a prevalent rule compared to the on-site potential of the impurity.

Treatment of donor impurities in source and drain.