IEE5328 Nanodevice Transport Theory and Computational Tools

Prof. Ming-Jer ChenDept. Electronics EngineeringNational Chiao-Tung UniversityMarch 18, 2013

Lecture 3A:

A Self-Consistent Solver of Poisson-Schrodinger Equations in a MOS System

(Advanced Device Physics with emphasis onhands-on calculations)

1IEE5328 Prof. MJ Chen NCTU

Double-gate MOSFET Simulator:MOS Electrostatics

Student: Ting-Hsien Yeh 葉婷銜 Advisor: Dr. Ming-Jer Chen

National Chiao Tung University NEP Lab

3

Structure• Schematic double-gate n-MOSFET and its MOS band diagram.

• In this work, we set up a simulator called DG-NEP to deal with a symmetrical double-gate n-MOSFET structure.

tox

Body(p-type)N+

Oxide

Oxide

DS

Gate

Gate

tbody

Vg

N+

tox

Evacuum

tSi (or t

body)

tox

p-Substrate

OxideOxide

Metal-gate

Efp

Ev

Ec

Efm

Efm

~~

~~

~~

~~

Evacuum

Metal-gate

m

Si

z-scaletox

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4

Start

Setting the environment and physics parameters.

Calculate Ef at equilibrium,and set Ev=0.

Use Poisson’s equation to solve potential(V0).

Use V0 to solve Schrodinger equation to obtain wave function and subband occupancy.

Use updated concentration to get new potential by using Poisson’s equation. If |Vn+1-Vn|<1.0 × 10-12 eV

Calculate charge density,voltage…

.Yes

No

Flowchart for DG-NEP simulator Without Penetration Effect

National Chiao Tung University NEP Lab

5

Schrödinger and Poisson Self-consistent of DG-NEP

• The three-dimensional carriers (both electrons and holes) density:

• Poisson Equation:

,2

3 ,2,

( ) ln 1+e ( )f i jE Ei

DOS kTD i B i j

i j

mn z g k T z

20 [ ( ) ( ) ( )]( ) A

si

q N z n z p zd V z

dz

National Chiao Tung University NEP Lab

6

Physical Model in DG-NEP

Nano Electronics Physics Lab @ NCTU 6

•The two-dimensional electron density

•The total inversion layer charge density

,,

inv i ji j

N n•The average inversion layer thickness

•The flat band voltage

•The gate voltage

•The oxide voltageox Si s

oxox

t FV

,

, 2ln 1+e

f i jE EiDOS kT

i j i B

mn g k T

2

0 2

02

0

( ) 2( )

( )

Si

Si

Si

t

t

av tinv

zn z dz qZ zn z dz

Qn z dz

ln( )Vfb m Si g B

A

NV E k T

N

g s ox fbV V V V

• The transverse effective field: 2

2

( ) ( )

( )

Si

ox

Si

ox

t

teff t

t

E z n z dzE

n z dz

National Chiao Tung University NEP Lab

7

Subband Energy and Wave-function

• For Tsi=30nm:

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5-200

-100

0

100

200

300

400T=300K N

sub=1x1015cm-3 T

ox=5nm T

Si=30nm

mox=0.5m

0 Metal workfunction=4.05eV

Schred DG-NEP w/o P w/i P E E E E E E E

f

Subban

d ener

gy (m

eV)

Vg (V)-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

1011

1012

1013

T=300K Nsub

=1x1015cm-3 Tox=5nm T

Si=30nm

mox=0.5m

0 Metal workfunction=4.05eV

Schred's sim. DG-NEP w/o P DG-NEP with P

Ninv (cm

-2)

Vg (V)

0 10 20 30

0.0

0.5

1.0

tox=5nm t

Si=30nm N

sub=1x1015cm-3

Metal-work function=4.05eVm

ox=0.5m

0 Vs=1.02V

Energ

y (eV)

Depth (nm)

Ec

E2,1

E2,2

E2,3

E2,4

E4,1

E4,2

Dash line:wave-function

We can find that our DG-NEP simulation results without penetration effect match Schred's ones.

National Chiao Tung University NEP Lab

8

Subband Energy and Wave-function

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5-200

-100

0

100

200

300

400T=300K N

sub=1x1015cm-3 T

ox=5nm T

Si=10nm

mox=0.5m

0 Metal workfunction=4.05eV

Schred DG-NEP w/o P w/i P E E E E E E E

f

Subban

d ener

gy (m

eV)

Vg (V)-5 0 5 10 15

0.0

0.5

1.0

Dash line: wave-function

tox=5nm t

Si=10nm N

sub=1x1015cm-3

Metal-work function=4.05eVm

ox=0.5m

0 Vs=1.02V

Ener

gy

(eV)

Depth (nm)

Ec

E2,1

E2,2

E2,3

E2,4

E4,1

E4,2

• For Tsi=10nm:

National Chiao Tung University NEP Lab

9

Subband Energy and Wave-function

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0

500

1000

1500

2000

2500

3000

3500 T=300K mox=0.5m

0

Nsub

=1x1015cm-3

Tox=5nm T

Si=1.5nm

Schred DG-NEP w/o P w/i P E E E E E E E

f

Subban

d ener

gy (m

eV)

Vg (V)

-5.0 -2.5 0.0 2.5 5.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

mox=0.5m

e Vs=1.02V

tox=5nm t

Si=1.5nm N

sub=1x1015cm-3 Metal-work function=4.05eV

Dash line:wave-function

Ener

gy (eV)

Depth (nm)

Ec

E2,1

E2,2

E2,3

E2,4

E4,1

E4,2

• For Tsi=1.5nm:

National Chiao Tung University NEP Lab

10

The Comparison of Potentials and Electron Density Distributions with Those of Shoji, et al.

• In this paper , ml=0.98m0 , mt=0.19m0 , mox=0.5m0 , Nsub=1x1015cm-3

[10] M. Shoji and S. Horiguchi, “Electronic structures and phonon limited electron mobility of double-gate silicon-on-insulator Si inversion layers,” J. Appl. Phys., vol. 85, no. 5, pp. 2722–2731, Mar. 1999.

0 5 10 15 20 25 30-0.2

-0.1

0.0

0.1

0.2

0.3

Symbol:Shoji'sLine:This Work

Nsub

=1x1015cm-3 tSi=30nm E

eff=5x105 V/cm

tox=2nm

Ener

gy (eV)

Distance (nm)

0

5

10

15

20

25

30

Electro

n D

ensity (10

18cm

-3)

0 1 2 3 4 5

-0.2

-0.1

0.0

0.1

0.2

0.3

Ener

gy (eV)

Distance (nm)

Symbol:Shoji'sLine:This Work

0

5

10

15

20

25

30

35

Electro

n D

ensity (10

18cm

-3)

Nsub

=1x1015cm-3 tSi=5nm E

eff=5x105 V/cm

tox=2nm

(a) Tsi=30nm: (b) Tsi=5nm

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• For thick tSi, two of each subbands have almost the same energy due to the upper and lower inversion layers sufficiently separated as a distinct bulk inversion layer. As tSi decreases, the barrier between two inversion regions becomes lower and making the subband energies split.

0 10 20 30 40 500.0

0.1

0.2

0.3T=300K N

sub=1x1015 cm-3

tox=2nm m

ox=0.5m

0

Eeff

=1x105 V/cm

Line & Dash :This WorkSymble:Shoji's

E4,4

E4,3

E4,2

E4,1

E2,6

E2,5

E2,4

E2,3

E2,2

Energ

y (eV)

tSi (nm)

E2,1

0 10 20 30 40 50-0.1

0.0

0.1

0.2

Line & Dash :This WorkSymble:Shoji's

T=300K Nsub

=1x1015 cm-3

tox=2nm m

ox=0.5m

0

Eeff

=5x105 V/cm

E4,4

E4,3

E4,2

E4,1

E2,6

E2,5

E2,4

E2,3

E2,2

Energ

y (eV)

tSi (nm)

E2,1

The Comparison of Subband Energies with Those of Shoji, et al.

(a) Eeff=1 × 105 V/cm (b) Eeff=5 × 105 V/cm

National Chiao Tung University NEP Lab

12

Comparison with Gamiz, et al.

[11] F. Gamiz and M. V. Fischetti, “Monte Carlo simulation of double-gate silicon-on-insulator inversion layers: The role of volume inversion, ” J. Appl. Phys., vol. 89, no. 10, pp. 5478–5487, May 2001.

104 105 1060

20

40

60

80

100

tox

=5nm Nsub

=1x1015

cm-3

T=300K mox=0.5m

0

Metal-work function=4.05eV

tb=20nm 10nm 7.5nm 5nm 4nm 3nm 1.5nm

Gamiz's sim. This work

Rela

tive P

opula

tion (%

)

Effective Field (V/cm)

Non-primed subbands

104 105 106

0

20

40

60

80

100

Primed subbands

tox

=5nm Nsub

=1x1015

cm-3

T=300K mox=0.5m

0

Metal-work function=4.05eV

tb=20nm 10nm 7.5nm 5nm 4nm 3nm 1.5nm

Gamiz's sim. This work

Rela

tive P

opula

tion (%

)Effective Field (V/cm)

(a) Non-primed subbands (b) Primed subbands

National Chiao Tung University NEP Lab

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• Energy separation for two different body thicknesses

1010 1011 1012 10130

20

40

60

80

100

120

tb=3nm 7.5nm

Gamiz's sim. This work

tox

=5nm Nsub

=1x1015

cm-3 T=300K m

ox=0.5m

0

Metal-work function=4.05eV

E' 0-E

0 (m

eV)

Inversion Charge (cm-2)

Comparison with Gamiz, et al.

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-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.20.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Gate

Capacitance (F

/cm

2 ) Nsub

=1x1018cm-3 Tox=1.5nm

Metal workfuction=4.19eVT=300K

Vg (V)

Alam's:TSi=10nm , 25nm

w/o P with PThis Work:T

Si=10nm , 25nm

w/o P with P(m

ox=0.5m

0)

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.20.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0 Nsub

=1x1017cm-3 Tsi=10nm

Metal workfuction=4.19eVT=300K

Gate

Capacitance (F

/cm

2 )

Vg (V)

tox=1.5nm , 2.5nm

Schred's: Alam's(with P): This Work(w/o P): (with P):

The Comparison of C-V with Alam, et al. and Schred.(a) Different substrate thickness (b) Different oxide thickness

Si

ox

t

3 3-t Q=q ( ) ( ) ( ) .

oxt

g D D ag

dQC where p z n z N z dz

dV

[14] M. K. Alam, A. Alam, S. Ahmed, M. G. Rabbani and Q. D. M. Khosru, “Wavefunction penetration effect on C-V characteristic of double gate MOSFET, ” ISDRS 2007, December 12-14, 2007, College Park, MD, USA.

National Chiao Tung University NEP Lab

15

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.20.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Gate

Capacitance (F

/cm

2 ) Nsub

=1x1018cm-3 Tox=1.5nm

Metal workfuction=4.19eVT=300K

Vg (V)

Alam's:TSi=10nm , 25nm

w/o P with PThis Work:T

Si=10nm , 25nm

w/o P with P(m

ox=0.5m

0)

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.20.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0 Nsub

=1x1017cm-3 Tsi=10nm

Metal workfuction=4.19eVT=300K

Gate

Capacitance (F

/cm

2 )

Vg (V)

tox=1.5nm , 2.5nm

Schred's: Alam's(with P): This Work(w/o P): (with P):

The Comparison of C-V with Alam, et al. and Schred.(a) Different substrate thickness (b) Different oxide thickness

Si

ox

t

3 3-t Q=q ( ) ( ) ( ) .

oxt

g D D ag

dQC where p z n z N z dz

dV

[14] M. K. Alam, A. Alam, S. Ahmed, M. G. Rabbani and Q. D. M. Khosru, “Wavefunction penetration effect on C-V characteristic of double gate MOSFET, ” ISDRS 2007, December 12-14, 2007, College Park, MD, USA.