Comparison of Non-Equilibrium Green’s FuComparison of Non-Equilibrium Green’s Function and Quantum-Corrected Monte Carlo nction and Quantum-Corrected Monte Carlo

Approaches in Nano MOS Simulation Approaches in Nano MOS Simulation

H. TsuchiyaA. Svizhenko

M. P. AnantramM. OgawaT. Miyoshi

Department of Electrical and Electronics Engineering Kobe University, Japan

NASA Ames Research Center

BACKGROUNDBACKGROUND

Nano-scale MOSFETs with Lch < 10nm have been realized in several research laboratories.

A quantum mechanical modeling of nano-scale MOSFETs involving carrier’s quasi-ballistic behaviors will be indispensable.

Non-equilibrium Green’s function approach (NEGF) Quantum-corrected Monte Carlo approach (QMC)

We present a joint study on comparison between the NEGF and QMC approaches for a nano-scale MOSFET.

Transport equation with the lowest-order quantum correction

Czyxi ii

i

t

ffUU

r

f

m

k

t

f

krQC

1

,,

*

Quantum-Corrected MC ApproachQuantum-Corrected MC Approach

Czyxi ii

i

t

ffUfU

r

f

m

k

t

f

)(24

11 3

,,* krkr

Quantum-corrected Boltzmann transport equation

Quantum correction of potential

QCUUdt

d

dt

d r

kv

r

1

,

Quantum-corrected equations of motion

SiO2

2-fold valleys 4-fold valleys

Quantum-Corrected MC SimulationQuantum-Corrected MC Simulation

SiO2

SiO2

SiO2

TSi = 5 nm

0

1

2

3

-1 0 1 2 3 4 5 6

2-fold valleys4-fold valleys

Car

rier

De

nsity

(10

19cm

-3)

x (nm)

k x

k y

k z

Drain(100) S

i surfa

ce

Equi-energy surfaces of Siconduction band

2-fold perpendicularvalleys

4-fold in-planevalleysk z

k y

Conduction band

Source

xy

z

m l

m t

1' 1

3

3'

2

2'

1D Green’s function equations along channel direction

),()(),(),(1

2

2

yxyEyxyxVdx

d

mdx

d

m nnnxx

1D Schrödinger equation at each cross-section

Quasi-2D NEGF ApproachQuasi-2D NEGF Approach

)'(),,',(),,,(

),,',()(1

22

111

222

yyEkyyGEkyydy

EkyyGyEdy

d

mdy

d

m

kE

zrnz

rn

zrnnn

ynz

z

),,',(),,,(),,',(),,,(

),,',()(1

22

1,

1,

1,

11

,222

EkyyGEkyydyEkyyGEkyydy

EkyyGyEdy

d

mdy

d

m

kE

znznznzrn

znnny

nz

z

S ource D rainS i

G ate

G ate

Simulation ModelSimulation Model

TSi = 3 nm Lch = 10 nm Tox = 1.5 nm Channel: undoped ND = 1020 cm-3

(source and drain)

x

y

Drain current as a function of right boundary of scattering, YR-Scatt is c

alculated.

Only electron-phonon scattering is considered.

Only the lowest quantized subbands for 2-fold and 4-fold valleys is considered in the NEGF method.

Drain voltage is 0.6 V. Gate voltage is adjusted so that the injection electron density at the source edge of the channel becomes identical.

YR-Scatt

Scattering

1.0

1.2

1.4

1.6

1.8

2.0

2.2

-20 -10 0 10 20

NEGFquantum-corrected MCclassical MC

YR-Scatt

(nm)

I D (

mA

/m

)

Lch

= 10nm

VDS

= 0.6V

TSi

= 3nm

Drain Current vsDrain Current vsYYR-ScattR-Scatt

There is a good agreement between the NEGF and QMC results, while the classical model underestimates the drain current.

Averaged Electron Velocity ProfilesAveraged Electron Velocity Profiles

Velocity (classical) < Velocity (quantum)

because an increase in the occupancy of the 2-fold valleys due to the energy quantizatization is not taken into account in the classical model.

0

1

2

3

-10 0 10

quantum-corrected MCclassical MC

Vel

ocity

(10

7cm

/s)

y (nm)

VDS

= 0.6 V

YR-Scatt = 20 nm

Concentrations and PotentialsConcentrations and PotentialsYR-Scatt = 20nm

0

1

2

3

4

5

-10 0 10

NEGFquantum-corrected MC

She

et E

lect

ron

De

nsity

(cm

-2)

y (nm)

-0.8

-0.6

-0.4

-0.2

0

0.2

-10 0 10

NEGFquantum-corrected MC

Ene

rgy

(eV

)

y (nm)

0

1

2

3

4

5

-10 0 10

NEGFclassical MC

She

et E

lect

ron

De

nsity

(cm

-2)

y (nm)

-0.8

-0.6

-0.4

-0.2

0

0.2

-10 0 10

NEGFclassical MC

Ene

rgy

(eV

)

y (nm)

NEGF vs QMC NEGF vs classical MC

Influence of Impurity and Plasmon ScatteringsInfluence of Impurity and Plasmon Scatterings

1.2

1.6

2.0

2.4

-20 -10 0 10 20

phonon scatt.phonon and impurity scatt.phonon, impurity and plasmon scatt.

YR-Scatt

(nm)

I D (

mA

/m

) quantum-corrected MC

Lch

= 10nm

VDS

= 0.6V

TSi

= 3nm

Impurity scattering in source region plasmon scattering at drain-end of channel

are important to estimate drain current of MOSFET.

SUMMARYSUMMARY

We have found that the non-equilibrium Green’s function and quantum-corrected MC approaches are equivalent in the quantum transport simulation of nano-scale MOSFETs.

This result may be applicable to quantum correction models such as effective potential and Bohm potential, if the subband splitting is adequately incorporated.

We have also demonstrated that the impurity scattering in the source region and the plasmon scattering at the drain-side of the channel are important to estimate the drain current accurately.