Compact model for nanoscale MOSFETs in an intermediate regime - - PowerPoint PPT Presentation

Giuseppe Iannaccone Università di Pisa

Compact model for nanoscale MOSFETs in an intermediate regime between ballistic and drift-diffusion transport

• G. Mugnaini, G. Iannaccone

Dipartimento di Ingegneria dell’Informazione Università di Pisa, Italy

Giuseppe Iannaccone Università di Pisa

Outline

! Motivation ! Ballistic DGMOSFET in a closed form ! Vertical electrostatics and local equilibrium ! Ballistic Segmentation of a drift-diffusion channel ! Effects of non-linear ballistic transport on the mobility ! Compact macromodel for the ballistic chain ! Bulk MOSFET under the effects of Fermi-Dirac statistics ! Conclusion

Giuseppe Iannaccone Università di Pisa

Motivation

Existing compact models do not provide a description for the transition from drift-diffusion to ballistic transport regime. Far from equilibrium transport is commonly described by the introduction of electron heating, which is not valid for totally ballistic MOSFETs. We expect that the partially ballistic transport regime will be important in nanoscale devices and it is already important for high mobility FETs. The Fermi-Dirac statistics is not typically considered in MOSFET models, but can be important for very thin devices or low temperatures.

Giuseppe Iannaccone Università di Pisa

A remark

EKV-like models = linearized charge models and

• pposite current fluxes

! EKV ! ACM ! USIM ! UCCM ! Maher-Mead model ! and so on!...

Giuseppe Iannaccone Università di Pisa

Ballistic DG MOSFETs

Giuseppe Iannaccone Università di Pisa

Ballistic DG MOSFET in a closed form

Starting from the Natori theory [Natori,1994] of ballistic FETs, two hemi-maxwellian carrier populations are present on the peak of the barrier:

( )

        + = − − + −

− −

t d c t s c

V V c b c m g g

e e qN Q V C

φ φ φ φ

φ χ φ 2 2

        − = − =

− −

t d c t s c

V V th c r f ds

e e v qN I I I

φ φ φ φ

2

Giuseppe Iannaccone Università di Pisa

If the anisotropy of the effective mass tensor is considered, the unidirectional thermal velocity is found to vary slowly with the silicon thickness: Nc is the effective density of electron states in all conduction subbands. Remarkably vth does not depend on bias in the case of rectangular confinement.

∑ ∑ ∑ ∑

       − +        − =        −         + +        − =

t t n t n t l n l n c c t t n l t t n t l n t l n th

N N N N m kT m kT N m kT N v φ ε φ ε φ ε π π φ ε π

, , , , , , , ,

exp 2 exp exp 2 2 exp 2

Effect of the anisotropy of the effective mass tensor

Giuseppe Iannaccone Università di Pisa

Charge based vertical electrostatics(I)

Interestingly the vertical electrostatics can be written in a form that is similar to the EKV-like electrostatics: where: plays the role of an equivalent Fermi potential, mean of Vs and Vd

          + =

− −

t d t s

V V t m

e e V

φ φ

φ 2 log

        + =           + − −

− − c m t g m V V V t g

qN Q C Q e e V V

m t d t s

log 2 2 log φ φ

φ φ

! ! ! " ! ! ! # $

Giuseppe Iannaccone Università di Pisa

Charge based vertical electrostatics(II)

Analytic solution through the Lambert W-function [Corless,1993]: Where the normalisation mobile charge: It is useful to define a threshold voltage

                    + −         + − − =

− − −

2 2

t d t s t T g

V V V V t g b m g c

e e e W C Q V

φ φ φ

φ χ φ φ

t g n

C Q φ 2 =

        + + − ≡

n c t g b m T

Q qN C Q V log 2 φ χ φ

Giuseppe Iannaccone Università di Pisa

Charge based vertical electrostatics(III)

! Mobile charge density upon the peak ! Total Current

                    + =

− − −

2

t d t s t T g

V V V V n m

e e e W Q Q

φ φ φ

                            + =

− − − t ds V V V V th n ds

V e e e W v Q I

t d t s t T g

φ

φ φ φ

2 tanh 2

Giuseppe Iannaccone Università di Pisa

Gummel symmetry test

The proposed MOSFET model is similar to Natori-Rahman- Lundstrom model. Some benefits: ! Explicit equations ! fully symmetrical form. The Natori-Rahman-Lundstrom model does not pass the Gummel simmetry test:

and ) ( ) (

2 2

=       − = −

=

x

V x x x

dV I d V I V I

Giuseppe Iannaccone Università di Pisa

The vertical ballistic electrostatics is fully consistent with local equilibrium transport (i.e. drift-diffusion transport with uniform mobility). If Vs=Vd=VFn (VFn the local quasi Fermi potential):

! Same electrostatics of EKV-like models

Vertical electrostatics and local equilibrium

        + = − −

c m t g m Fn T g

qN Q C Q V V V log 2 φ

Giuseppe Iannaccone Università di Pisa

Dissipative MOSFETs

Giuseppe Iannaccone Università di Pisa

Ballistic Segmentation of a drift- diffusion channel(I)

If N is the number of MOSFETs in the chain, current continuity imposes N equations for the N unknowns:

• Consistently with the

approach of Buttiker probes [Buttiker,1986], a drift-diffusion MOSFET can be interpreted as along enough chain of ballistic

• transistors. Fermi potentials are

defined only at the -th contact.

Giuseppe Iannaccone Università di Pisa

! For the k-th ballistic MOSFET: ! If N is large enough and is small enough: ! (discrete) (continuous) ! (nonlinear) (linear) ! (discrete) (continuous)

λ λ λ λ Is the length of a ballistic transistor

Ballistic Segmentation of a drift- diffusion channel(II)

1

• 0..N

k with 2 tanh 2

1

1

=         −                     + =

+ − − −

+

t k k V V V V th n ds

V V e e e W v Q I

t k t k t T g

φ

φ φ φ

Fn k

V V →

x x → ) tanh(

d N s

V V V V = =

k k

V V −

+1

Fn k k

V V V ∇ → −

+

λ

1

Giuseppe Iannaccone Università di Pisa

Ballistic Segmentation of a drift- diffusion channel(III)

With aboveseen simplifications, it is: It is natural to define a low field mobility:

! Mobility in terms of mean free path λ

λ λ λ

! The length of a ballistic transistor is the mean

free path

Fn t th Fn m t k k th k m

V v V Q V V v Q ∇ →         −

+

φ λ φ 2 ) ( 2 tanh

1 ,

t th n

v φ λ µ 2 =

Giuseppe Iannaccone Università di Pisa

Drift-diffusion limit(I)

! By means of current continuity along the channel,

for a long channel drift-diffusion MOSFET:

              =

− −

dx dV e e W v Q I

Fn V V V t th n ds

t Fn t T g

φ φ

φ 2

Vd Fn V s V Fn V t Fn t T g t Fn t T g Fn m t Fn t T g

V V V V V V t n th L Fn V Q V V V n t th ds

e e W e e W Q v dx dx dV e e W Q v L I

= =

                        +         = =               =

− − − − − −

∫

φ φ φ φ φ φ

φ φ 2 2 2

2 ) (

! ! ! " ! ! ! # $

Giuseppe Iannaccone Università di Pisa

Finally it is:

! EKV-like model for non-degenerate long DGMOSFETs

subject to rectangular quantum confinement!

Drift-diffusion limit (II)

        − + − =         + = − −         + = − −

md ms n md ms t n ds c md t g md d T g c ms t g ms s T g

Q Q Q Q Q L I qN Q C Q V V V qN Q C Q V V V 2 log 2 log 2

2 2

φ µ φ φ

Giuseppe Iannaccone Università di Pisa

Ballistic Segmentation of the channel(I)

! A long enough ballistic chain (N>>1) describes a drift-

diffusion MOSFET rigorously.

! If N=1, the ballistic chain reduces to a ballistic transistor ! For intermediate N the ballistic chain describes the

transition from drift-diffusion to ballistic transport

! N depends on the low-field mobility and on the length L

N=L/λ λ λ λ

Giuseppe Iannaccone Università di Pisa

Ballistic Segmentation of the channel(II)

0.0 0.1 0.2 0.3 0.4 0.5 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 N=30 Drift diffusion Ids[A/m]

Vds[V]

Giuseppe Iannaccone Università di Pisa

Ballistic Segmentation of the channel(II)

0.0 0.1 0.2 0.3 0.4 0.5 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 N=30 Drift diffusion Ballistic chain Ids[A/m]

Vds[V]

Giuseppe Iannaccone Università di Pisa

Ballistic Segmentation of the channel(II)

0.0 0.1 0.2 0.3 0.4 0.5 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 N=10 N=30 Drift diffusion Ballistic chain Ids[A/m]

Vds[V]

Giuseppe Iannaccone Università di Pisa

Ballistic Segmentation of the channel(II)

0.0 0.1 0.2 0.3 0.4 0.5 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 N=5 N=10 N=30 Drift diffusion Ballistic chain Ids[A/m]

Vds[V]

Giuseppe Iannaccone Università di Pisa

Ballistic Segmentation of the channel(II)

0.0 0.1 0.2 0.3 0.4 0.5 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 N=5 N=4 N=10 N=30 Drift diffusion Ballistic chain Ids[A/m]

Vds[V]

Giuseppe Iannaccone Università di Pisa

Ballistic Segmentation of the channel(II)

0.0 0.1 0.2 0.3 0.4 0.5 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 N=5 N=4 N=3 N=10 N=30 Drift diffusion Ballistic chain Ids[A/m]

Vds[V]

Giuseppe Iannaccone Università di Pisa

Ballistic Segmentation of the channel(II)

0.0 0.1 0.2 0.3 0.4 0.5 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 N=5 N=4 N=3 N=2 N=10 N=30 Drift diffusion Ballistic chain Ids[A/m]

Vds[V]

Giuseppe Iannaccone Università di Pisa

Ballistic Segmentation of the channel(II)

0.0 0.1 0.2 0.3 0.4 0.5 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 N=1 (Ballistic limit) N=5 N=4 N=3 N=2 N=10 N=30 Drift diffusion Ballistic chain Ids[A/m]

Vds[V]

Giuseppe Iannaccone Università di Pisa

Effects of non-linear ballistic transport

• n the mobility

Giuseppe Iannaccone Università di Pisa

Compact macromodel for the ballistic chain

The complete model for the ballistic chain is too computational expensive to be introduced in a circuit-level simulator (N-1 internal nodes) When non-linear transport emerges, it shows its effects mainly in the last ballistic transistor of the chain.

2 4 6 8 10 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Vg'

Discrete Fermi level Vk Virtual reservoir at L/λ

λ λ λ

Giuseppe Iannaccone Università di Pisa

Comparison of the macromodel with the ballistic chain

Giuseppe Iannaccone Università di Pisa

Comparison of the macromodel with the ballistic chain

Giuseppe Iannaccone Università di Pisa

Comparison of the macromodel with the ballistic chain

Giuseppe Iannaccone Università di Pisa

! The model can be improved if the weakly non-linear

transport in the drift-diffusion section is considered.

! The mobility

is introduced in the DD section without any clamping function!

! The ballistic section acts as the smoothed clamping in

traditional models.

Horizontal degradation of mobility

Giuseppe Iannaccone Università di Pisa

Vertical degradation of mobility

m n n

Q θ µ µ + = 1

( )

     − ≈ threshold above threshold below

T g g n n n

V V C θ µ µ µ

Then: Model analogous to [Iniguez,1996] Symmetrical behavior in terms of Vgs and Vgd

Giuseppe Iannaccone Università di Pisa

Short channel effect and DIBL(I)

■ The effects of the 2D

geometrical capacitances can be included in the ballistic section, through:

( )

        + = − + + − + −

− −

t d c t s c

V V c b d d s s c m g g

e e qN Q V C V C V C

φ φ φ φ

φ χ φ 2 2

effects horizontal

! ! " ! ! # $

Giuseppe Iannaccone Università di Pisa

! The horizontal contributions is included also in the Drift-diffusion section ! Such capacitances are the main responsible for DIBL below threshold

Short channel effect and DIBL(II)

( )

t s cs V

c b d d s s cs m g g

e qN Q V C V C V C

φ φ

φ χ φ

−

= − + + − + − ! ! " ! ! # $

effects horizontal

' ' 2

( )

t d cd V

c b d d s s cd m g g

e qN Q V C V C V C

φ φ

φ χ φ

−

= − + + − + − ! ! " ! ! # $

effects horizontal

' ' 2

Giuseppe Iannaccone Università di Pisa

! If the quantum confinement is considered field-dependent

(as in the bulk case) the eigenvalues are not rectangular

! Pragmatic eigenvalues (f is a fitting parameter): ! the charge based electrostatics: ! Effective gate capacitance subject to field dependent

quantum confinement:

Effect of field-dependent quantum confinement

m t n t n

fQ + =

R , ,

ε ε

R ,t n

ε

f C C

g g

+ → 2 1 2 1

          + = − −

−

t m

fQ R c m t g m Fn g

e qN Q C Q V V V

φ

φ log 2

Giuseppe Iannaccone Università di Pisa

Experimental curves of a FinFET [Yu et al.,2002] with L=80 nm

Comparison with experimental curves

Giuseppe Iannaccone Università di Pisa

Bulk MOSFET under the effects

• f Fermi-Dirac statistics

! Bulk as another gate. ! DD section-> degenerate

extension of EKV-like models [Enz,1995][Cunha,1999]

( )

b c t k n d c t k n s c t k n m c g

V F V F qN V φ φ γ φ ε φ φ ε φ φ χ φ φ − +               − − +       − − = − − −

∑

, , ,

2

( )

b c t k n Fn c t k n m c g

V F qN V φ φ γ φ ε φ φ χ φ φ − +       − − = − − −

∑

, ,

2

Ballistic section Drift-diffusion section

Giuseppe Iannaccone Università di Pisa

Macromodel for degenerate bulk MOSFETs(I)

             − − =               − − −         − − =

∫ ∑ ∑

−

d s

V V Fn t k n Fn c t k n k n ds t k n Fn c t k n Fn c t k n k n ds

dV V F L v qN I V F V F v qN I φ ε φ φ λ φ ε φ φ ε φ φ

, 2 / 1 , , , 2 / 1 , 2 / 1 , ,

2 2

∫ ∑

      − − =

−

d s

V V Fn t k n Fn c t k n k n ds

dV V F L v qN I φ ε φ φ λ

, 2 / 1 , ,

2

dx dV V F v qN I

Fn t k n Fn c t k n k n ds

      − − =

−

∑

φ ε φ φ λ

, 2 / 1 , ,

2

The com pact macrom odel agrees pretty well with the com plete ballistic chain. A correction has been found in the nondegenerate lim it. The DD+ B m odel No Non-linear transport in the DD section Current continuity!

Giuseppe Iannaccone Università di Pisa

Macromodel for degenerate bulk MOSFETs(II)

0.0 0.2 0.4 0.6 0.8

• 3
• 2
• 1

1 2 3 Proposed model Experiment

Vgs[V] Ids[A/m]

0.0 0.2 0.4 0.6 0.8 50 100 150 200 250 300 350 400 450 500 550 Proposed Model Experiment Vgs=0.4V Vgs=0.5V Vgs=0.8V Vgs=0.6V Vgs=0.7V

Ids[A/m] Vds[V]

Bulk MOSFET L=30nm[Doyle et al.,2002]

Giuseppe Iannaccone Università di Pisa

Model for onedimensional transistors (1DEG)

! Adaptation to 1D structures (1DEG).

2 )) ( ) ( 2 ) ( ) ( ) ' (

, , , , , 2 / 1 , 2 / 1 , k n d c k n s c k n k n ds k n d c k n s c k n m c g g m

V F V F v qN I V F V F qN Q V C Q ε φ ε φ ε φ ε φ φ − − − − − = − − + − − = − =

∑ ∑

− −

The proposed model respects the Gummel symmetry Test

• 0.2
• 0.1

0.0 0.1 0.2

• 60.0µ
• 40.0µ
• 20.0µ

0.0 20.0µ 40.0µ 60.0µ 80.0µ

Violation of Simmetry

Ids[A] Vds[V]

Jimenez et al. Proposed ballistic model

Comparison with the model [Jimenez et al,2003] for ballistic nanowire transistors

Giuseppe Iannaccone Università di Pisa

■FD statistics degrades the

mobility of the DD section

■If a uniform λ

λ λ λ is assum ed along the channel:

Effects of degeneracy on mobility

1 1

T=70K,tsi=8nm

Third subband Second subband First subband

<µ

µ µ µn>/<µ µ µ µ

n>

Vg'

0.0 0.1 0.2 0.3 0.4 0.5 100000 200000 300000 400000 500000

T=70K,tsi=8nm Mobility degradation Thermal velocity enhancement

Vg'=0.8V Vg'=0.6V Vg'=0.4V Vg'=0.2V Vg'=0V

Vdrift[m/s] Vds[V]

[ ] [ ]

k n k n t k n k n

E F E F v

, , 2 / 1 , ,

2

−

= φ λ µ For 1DEG: Degradation of the subband- averaged mobility given by the degeneracy in a Drift-Diffusion wire(T=70k).

■FD statistics improves the thermal

velocity of the B section (saturation velocity)

[ ] [ ]

k n k n t k n k n

E F E F v

, 2 / 1 , 1 , ,

2

− −

= φ λ µ

For 2DEG:

Giuseppe Iannaccone Università di Pisa

! Effective mass approximation for carbon nanotube

Adaptation to carbon nanotubes transistors

nm d m d m d Egap 1 / 37 . 11 ) (

*

= = γ

Second subband First subband E /2

E E k

Comparison with output characteristics of CNTFET Details in: Mugnaini,ESSDERC 2005

ds V [V]

0.8V 0.6V 0.4V 0.3V 0.5V 0.7V

I [uA] ds

−7 −6 −5 −4 −3 −2 −1 −1.2 −1 −0.8 −0.6 −0.4 −0.2

Giuseppe Iannaccone Università di Pisa

Conclusion

! We have proposed an analytical model with the following

features:

"Description of transport from ballistic to drift-diffusion "Inclusion of Fermi-Dirac statistics "Different device architectures "Alternative interpretation and implementation of the

saturation velocity effect.

! Future developments: "Modeling of excess noise in nanoscale device "Quasi-static model

• G. Mugnaini, G. Iannaccone, IEEE-TED, 2005