IEE5328 Nanodevice Transport Theory and Computational Tools

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 tbody) tox

p-SubstrateOxideOxide

Metal-gate

Efp

Ev

Ec

EfmEfm

~~

~~

~~

~~

Evacuum

Metal-gate

m

Si

z-scaletox


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


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 zdz


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 j

i j

N n•The average inversion layer thickness

•The flat band voltage

•The gate voltage

•The oxide voltageox Si s

oxox

t FV

,

, 2 ln 1+ef i jE Ei

DOS kTi j i B

mn g k T

20 2

02

0

( ) 2 ( )( )

Si

Si

Si

tt

av tinv

zn z dz qZ zn z dzQ

n z dz

ln( )Vfb m Si g B

A

NV E k TN

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


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 Nsub=1x10

15cm-3 Tox=5nm TSi=30nm mox=0.5m0 Metal workfunction=4.05eV

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

E

E

E

E E

Ef

Subb

and e

nergy

(meV

)

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 TSi=30nm

mox=0.5m0 Metal workfunction=4.05eV

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

N inv (c

m-2 )

Vg (V)0 10 20 30

0.0

0.5

1.0

tox=5nm tSi=30nm Nsub=1x1015cm-3

Metal-work function=4.05eVmox=0.5m0 Vs=1.02V

Ener

gy (e

V)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.


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 Nsub=1x10

15cm-3 Tox=5nm TSi=10nm mox=0.5m0 Metal workfunction=4.05eV

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

E

E E

E Ef

Subb

and e

nergy

(meV

)

Vg (V)-5 0 5 10 15

0.0

0.5

1.0Dash line: wave-function

tox=5nm tSi=10nm Nsub=1x1015cm-3

Metal-work function=4.05eVmox=0.5m0 Vs=1.02V

Ener

gy (e

V)Depth (nm)

Ec

E2,1

E2,2

E2,3

E2,4

E4,1

E4,2

• For Tsi=10nm:


9

Subband Energy and Wave-function

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50

500100015002000250030003500 T=300K mox=0.5m0

Nsub=1x1015cm-3

Tox=5nm TSi=1.5nm

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

E

E

E

E

E

Ef

Subb

and e

nergy

(meV

)

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=0.5me Vs=1.02V

tox=5nm tSi=1.5nm Nsub=1x1015cm-3 Metal-work function=4.05eV

Dash line:wave-function

Energ

y (eV

)Depth (nm)

Ec

E2,1

E2,2

E2,3

E2,4

E4,1

E4,2

• For Tsi=1.5nm:


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 Eeff=5x10

5 V/cmtox=2nm

Energ

y (eV

)

Distance (nm)

0

5

10

15

20

25

30

Electr

on De

nsity

(1018

cm-3 )

0 1 2 3 4 5

-0.2

-0.1

0.0

0.1

0.2

0.3

Energ

y (eV

)Distance (nm)

Symbol:Shoji'sLine:This Work

0

5

10

15

20

25

30

35

Electr

on De

nsity

(1018

cm-3 )

Nsub=1x1015cm-3 tSi=5nm Eeff=5x10

5 V/cmtox=2nm

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


11

• 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 Nsub=1x1015 cm-3

tox=2nm mox=0.5m0

Eeff=1x105 V/cm

Line & Dash :This WorkSymble:Shoji's

E4,4

E4,3

E4,2

E4,1

E2,6

E2,5

E2,4E2,3

E2,2

Ener

gy (e

V)

tSi (nm)

E2,10 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 mox=0.5m0

Eeff=5x105 V/cm

E4,4E4,3

E4,2

E4,1

E2,6

E2,5

E2,4

E2,3

E2,2

Ener

gy (e

V)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


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

=1x1015cm-3

T=300K mox=0.5m0 Metal-work function=4.05eV

tb=20nm 10nm 7.5nm 5nm 4nm 3nm 1.5nm Gamiz's sim. This work

Rela

tive

Popu

latio

n (%

)

Effective Field (V/cm)

Non-primed subbands

104 105 106

0

20

40

60

80

100

Primed subbands

tox

=5nm Nsub

=1x1015cm-3

T=300K mox=0.5m0 Metal-work function=4.05eV

tb=20nm 10nm 7.5nm 5nm 4nm 3nm 1.5nm Gamiz's sim. This work

Rela

tive

Popu

latio

n (%

)Effective Field (V/cm)

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


13

• 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

=1x1015cm-3 T=300K mox=0.5m0

Metal-work function=4.05eV

E' 0-E0 (

meV

)

Inversion Charge (cm-2)

Comparison with Gamiz, et al.


14

-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

Gat

e Ca

paci

tanc

e (

F/cm

2 ) Nsub=1x1018cm-3 Tox=1.5nmMetal workfuction=4.19eVT=300K

Vg (V)

Alam's:TSi=10nm , 25nm w/o P with PThis Work:TSi=10nm , 25nm w/o P with P(mox=0.5m0)

-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=10nmMetal workfuction=4.19eVT=300K

Gat

e Ca

paci

tanc

e (

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 dzdV

[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.


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

Gat

e Ca

paci

tanc

e (

F/cm

2 ) Nsub=1x1018cm-3 Tox=1.5nmMetal workfuction=4.19eVT=300K

Vg (V)

Alam's:TSi=10nm , 25nm w/o P with PThis Work:TSi=10nm , 25nm w/o P with P(mox=0.5m0)

-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=10nmMetal workfuction=4.19eVT=300K

Gat

e Ca

paci

tanc

e (

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 dzdV

[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.

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IEE5328 Nanodevice Transport Theory and Computational Tools