analytical modeling of nano mosfets in the quasi ballistic … · 2021. 3. 15. · mos-ak workshop...

MOS-AK Workshop 20101

Analytical modeling of nano MOSFETsin the quasi ballistic regime :

beyond the drift diffusion approximation

IMEP-LAHCGrenoble Institute of Technology

France

DIEGMUniversity of Udine

Italy

R. Clerc, G. Ghibaudo

P. Palestri, L. Selmi


Transport in Nano MOSFET (L < 20 nm) challenges Compact Modeling

« Sophisticated computer simulations using techniques such as full band Monte Carlo and full quantum transport approaches are being used to explore the physics of the ultimateMOSFET, but circuit models continue to be based on concepts and approaches developedin the 1960's. »

M. Lundstrom, Int. SOI Conference, 2006

Since the pioneering work of Natori (1994) and Lundstrom (1996), the quasi ballistic regime of transport in nano MOSFETs

has been extensively investigated.

However, existing compact model are still based on Drift Diffusion and saturation velocity concepts

Let us examine the applicability of Quasi Ballistic theories to compact modeling


Outline

The “orthodox” Lundstrom theory

Beyond Lundstrom’s theory

Ballisticity extraction from experiments


The “orthodox” Lundstrom theory


Performance are no longer limited by transport along the channel,

but by injection at the source end

Transistor in ballistic regimeSource Drain

L

Energy

Flu

x of

car

rier (In quasi equilibrium)

Virtual Source

Channel

Net current

Gate(In quasi equilibrium)

Vd

injiBAL d V Q WI ≈

Transistor in diffusive regimeSource Drain

L

Energy

Flu

x of

car

rier

(In quasi equilibrium)

Virtual Source

Channel

Gate(In quasi equilibrium)

Vd

Scattering )y(dy

E d Q W)y(I Fn

effid µ=

Concept of Ballistic Limit


Concept of Backscattering Coefficient

Perfect non degenerated

reservoir of carriers“Source like”

x = Lx = 0

thermalisation

thermalisation

rs js+

(1 – rs) js+

(1– rd) jd-

rd jd-

Perfect non degenerated

reservoir of carriers“Drain like”

Distance

jd-

js+

Using Mc Kelvey flux theory of Transport :

J. P. McKelvey, R. L. Longini and T. P. Brody, « Alternative approach to the solution of added carrier transport problems in semiconductors », Phys. Rev., vol123 pp. 51-57 (1961).

Assuming an Inversion Charge Qi at the virtual source controlled by the gate :(like in the Natori’s model)

thid

dd vQ r)(1 kT)/ eV( exp)r1(

r)(1 kT)/ eV( exp)r1(

W

I

−−++−−−−=

thiSat d vQ

r1

r1

W

I

+−=

kT

V e v

2

Q )r1(

W

I dth

iLin d −=

A. Rahman and M. S. Lundstrom, « A compact scattering model for the nanoscale double-gate MOSFET », IEEE. TED, vol. 49 p 481 - 489 (2002)


λL

Lr

kT

kTHF +

=

intuited as a generalization of rLF

Gate

Source

Drain

F+

F-kT

LkT

L

rHF = Back Scattering Coefficient at High Field (Saturation Regime)

M. Lundstrom Z. Ren, IEEE TED 49 p.133 (2002)

empirical

λ+=

L

LrLF

M. Lundstrom, « Fundamentals of Carrier Transport », second edition, Cambridge university press, 2000

λ+=

L

LrLF (using Boltzmann or Fermi Statistics)assuming a constant isotropic mean free path :

rLF = Back Scattering Coefficient at Low Field (Ohmic Regime)

When L >>λ, this is consistent with the Drift Diffusion model :

dieffDD Lin d V Q µ

L

1

W

I =e

kT

v

µ 2

th

eff=λif

M. Lundstrom IEEE EDL 22 p. 293 (2001)

kT

V e v

2

Q )r1(

W

I dth

iLin d −=L >>λ

Concept of Backscattering Coefficient


In the MOSFET model, we have : withthiSat d vQ

r1

r1

W

I

+−=

λL

Lr

kT

kTHF +

=

When L >>λ, this is consistent with the Drift Diffusion model :

2

)V(VC µ

L

1

W

I2

Tgoxeff

DD Sat d −=

e

kT

v

µ 2

th

eff=λ

if

(same expression than in ohmic regime)

and TgSAT d

kT VV

e/kT 2L

V

e/kT 2LL

−==

which is in fact equal to the LkT layer calculated from the Drift Diffusion potential profile !

1

10

100

1 10 100 1000 10000

Channel Length L (nm)

1000

Dra

in C

urre

nt(µ

A/µ

m)

λ

Id Lin

Id Lin DD Id Lin BAL

Id Sat BAL

Id Sat

Id Sat DD

µ = 200 cm2V-1s-1

V inj = 1.2 × 105 m/sNinv = 1.45×1013 cm-2

M. S. Lundstrom and J. H. Rhew Journal of Computational Electronics, vol. 1 pp 481 - 489 (2002).

• In ohmic regime, Quasi ballistic transport occurs when L ∼ λ• In saturation, Quasi ballistic transport occurs when LkT ∼∼∼∼ λλλλ i.e when L ∼∼∼∼ 10 λλλλ

Backscattering Coefficient Modeling


rHF extracted is in fact a function of LkT

λL

Lr

kT

kTHF +

= ??

100 200 300 400 500

0

10

20

0

MC low field mobility (cm2V-1s-1)

Ext

ract

edλ D

ev(n

m)

DG 10 nm

DG 4 nmBulk

Strained Bulk

µev

kT 2

th0 =λ

2

λλ 0=

DevkT

kTMC L

Lr

λ+=

??λ extracted is in fact

proportional to µ

µev

kT 2

th0 =λ

IEDM 2006« Multi Subband Monte Carlo investigation of the mean free path and of the kT layer in degenerated quasi ballistic MOSFETs »P. Palestri, R. Clerc, D. Esseni, L. Lucci, L. Selmi

Validation by Monte Carlo Simulation


Strain Silicon : double advantage in QB devices

0

0.5

1

1.5

2

2.5

Bulk DG 12

Inje

ctio

n V

eloc

ity 1

07 cm

/s

Analytical Model

0

0.1

0.2

0.3

0.4

0.5

Bulk DG 12

Bac

k S

catte

ring

rth

effeff

kT

kT

v

kT/e )E(µ 2

2

1

L

Lr

×=λ

λ+=

Multi SubbandMonte Carlo Simulations

Ph. AcousticPh. OpticalSurf. RoughnessTsi Fluctuation

D. P

onto

net

.al.

Pro

c. E

ssde

rc20

06 p

. 166

Application to device performance modeling


(An increase of Vinj may also degrade r : is subband engineering a viable strategy ?)

Vth and rHF are valid in non degenerated inversion layers

• How to generalize LkT and λ in degenerated inversion layer ?

λL

Lr

kT

kTHF +

= • is only an empirical formula : where does it come from ?

• How to evaluate LkT ? (V(x) is not known)

(In particular the impact of velocity overshoot on V(x), and thus LkT is not self consistently taken into account)

Limits of Lundstrom’s Model

µv

kT/e 2λ

th

= • OK in low field

µv

kT/e 2λ

th

α=2

1≈αwith • But in high field rather equal to


Beyond Lundstrom’s Model

R. Clerc, P. Palestri, L. Selmi« On the Physical Understanding of the kT-Layer Concept in Quasi-Ballistic Regime of Transport in Nanoscale Devices »IEEE TED 53, p 1634 – 1640 (2006)


vx

x

B

F1 F2

F4 F5

F3x + ∆x

λΦ−Φ−−=Φ ++

−− )x()x(

m

F e )f(x,0

dx

d LE

)x(dv v)vf(x, F

0

xxx1+

∞

Φ== ∫

BA231 CFFF →+=+

λΦ−Φ∆=

λ−∆=

++∞

→ ∫)x()x(

x dv /v

)v,x(f)vf(x, xC LE

0

xx

xLExBA

1D Balance Equation in the phase space

f+(x,v)

f-(x,v)

Collision Integral (relaxation length approximation ) :

A

Balance Equation for the flux of carrier along the device

Assumption on the shape of f(x,vx) needed

1D flux conservation in the relaxation length approximation


)kT 2

vm( exp (x)n

kT π2

m2) v,x(f

2

−= ±±

The quasi ballistic Drift Diffusion

Assuming local Maxwellian distribution function (at each point) :

- 4 - 2 0 2 4Dis

trib

utio

n F

unct

ion

(a.u

)

velocity v/vth

This work

Drift Diffusion Approximation

Application to the backscattering coefficient calculation:

)β1(Lλ

)β(1 Lr

kT

kT

−+−= β = exp( − L / LkT)

0.01 0.1 1

0.2

0.4

0.6

0.8

1

λ+L

L

λ+kT

kT

L

LL = λ / 5

L = 50 λ

This model

Drain Voltage Vd (V)

Bac

kSca

tterin

gC

oeffi

cien

t r

This equation includes both :

kTHF

kT

Lr

λ L=

+LF

Lr

λ L=

+

Similar to Drift Diffusion


x thth x th x

λ vd nq (n n ) v λ v 2n F

dx kT / q+ − −Φ = − = − −

Using this approach, if we derive the current flux :

x th

µµ '

1 µF / v=

+th x

d n(n n ) v D' n µ ' F

dx+ −Φ = − = − −

We thus have now a new formalism which includes :

Thermal velocity limitation at the source (Ballistic limit)

Same backscattering coefficient in high and low field (Ballistic mobility)

With no need to calculate LkT …

An alternative to Drift Diffusion ?

n n n+ −= +Using :

Except for the boundary conditions, it is equivalent to Drift Diffusion, including saturation velocity !

Idem for H. Wang, G. Gildenblat, “Scattering matrix based compact MOSFET model”, in IEDM Tech. Dig., pp. 125–128 (2002)

The quasi ballistic Drift Diffusion


A more suitable approximated distribution function (at each point) :

m

U(x)2

Ballistic Electron

Velocity

Dis

trib

utio

n F

unct

ion

(a.u

.)

Backscattered Electron

MB

From MC simulations

)kT 2

vm( exp (x)n

kT π2

m2) v,x(f

2

−= ±±

Non Thermal Approach

What is wrong with quasi ballistic Drift Diffusion ?


)L,L(A L 2

)L,L(A Lr

kTkT

kTkT

+λ= du)

kT

)u(U1(

2

)u(2 )L,L(A

1L/L

0

kT

kT−

∫

+λ

λ=Backscattering formula :


1 10 1000.01

0.1

1

kT layer length LkT (nm)

Bac

ksca

tterin

gC

oeffi

cien

t r

MC

This model

Thermal

2 λth = 64 nm

2 λth = 18 nm

2 λth = 36 nm

Linear Pot.

1 10 1000.01

0.1

1

Bac

ksca

tterin

gC

oeffi

cien

t r Parabolic Pot.

2 λth = 64 nm

2 λth = 18 nm

2 λth = 36 nm

kT layer length LkT (nm)

MC

This model

Thermal



Distance (nm)0 2 4 6 8 10 12 14

0

0.5

1

1.5

2

2.5

3

VV+

V−

Linear Pot.

vth

Symbols : MCLines : Model

LkT = 2 nm2 λth = 36 nm

Vel

ocity

(10

5m

/s)

V thermal

vth

Symbols : MCLines : Model

0 5 10 15 20

Distance (nm)

Linear Pot.LkT = 2 nm

2 λth = 36 nm

V+

V

V−

V thermal

0.5

1

1.5

2

2.5

Vel

ocity

(10

5m

/s)

Velocity profile in high field in a non self consistent linear potential profile

With absorbing drain : With real drain (emitting) :



Is there any experimental confirmation of the quasi ballistic nature of transport ?

Let us show some results :

Not clear yet !

• let us consider only the ohmic regime (more simple : you do not need to know the charge at the virtual source)

• a standard low field mobility extraction has been performed on :

65 like nm Bulk and undoped Fully Depleted SOI technology

featuring physical gate length down to 40 nm

• according quasi ballistic theory :

d Lin dith app i d

I e VQ 1(1 r) v µ Q V

W 2 kT L= − = app dd

Lµ µ

L=

+ λ

Apparent mobility should decrease with L

in the quasi ballistic regime


Bulk nMOS

0.01 0.1 1 10200

300

400

500

600

700

Low

Fie

ld M

obili

tyµ e

xp(c

m2 /

Vs)

Channel length L (µm)

µexp

µdd

Undoped FD-SOI nMOS

0.01 0.1 1 1080

100

120

140

160

180

200


µdd

µexp

Low

Fie

ld M

obili

tyµ e

xp(c

m2 /

Vs)


Bulk 65nm CMOS technologydoped channel (≈1017/cm3) with halos,

SiON gate oxide (CET=2.2nm)polysilicon gate.

Undoped Fully depleted SOIBody thickness of 10 nm

Metal gate TiN, 2.5 nm of HfSiON dielectricRaised source and drain


0.01 0.1 1 100.6

0.7

0.8

0.9

1


Bac

ksca

tterin

gco

effic

ient

r

Bulk nMOS

0.85

0.9

0.95

1

FD-SOI nMOS

0.01 0.1 1 10Channel length L (µm)

Bac

ksca

tterin

gco

effic

ient

r

Bulk nMOS Undoped FD-SOI nMOS


rLF ~ 0.7 rLF ~ 0.85

Extracted value of r are very large … Neutral defects ?

app

bal

µr 1

µ= −The backscattering coefficient has been extracted using


Conclusions


Conclusions

The Lundstrom backscattering theory provides a powerful guidelineto analyse qualitatively device performance in the quasi ballistic regime

However, It is not a compact model (How to compute LkT ?)

If you try to generalize this approach along the channel → quasi ballistic drift diffusion → which is similar to drift diffusion with saturation velocity in high field regime

However, experiments shows a more complex picture : we are not sure yet to operate in the quasi ballistic regime (neutral defects ?)

You need to account for the ballistic distribution of carriers


Thank you for your attention !


Dra

in C

urre

nt (

a.u)

1 10 100 1000 10000Channel Length (nm)

Ballistic enhancement factor

Drift Diffusion1/L

Drift Diffusion Analytical Model (Velocity Saturation)

Mean FreePath

Ballistic limitMore Exact Quasi Ballistic Model


0

500

1000

1500

2000

2500

3000

3500

1 10 100 1000 10000

Bulk Undoped UTB

Strained Undoped UTB

I on

Cur

rent

(µ

A/µ

m)

Channel Length (nm)

DDvsat=105 m/s

BULKµ = 130 cm2V-1s-1

V inj = 1.2 × 105 m/sNinv = 1.45×1013 cm-2

Undoped UTBµ = 200 cm2V-1s-1

V inj = 1.2 × 105 m/sNinv = 1.45×1013 cm-2

Strained Undoped UTBµ = 370 cm2V-1s-1

V inj = 1.3 × 105 m/sNinv = 1.45×1013 cm-2

2 possible strategies to improve Ion :

• improving Vinj

(subband engineering)by DOS reduction

• improving λλλλ, which mean improving µµµµ= effective field mobility

like in pure Drift Diffusion model !

Device Optimisation in the Quasi Ballistic Regime

Still no clear experimental evidence

Application to device performance enhancement


Re-investigation of the kT layer Concept

λL

Lr

kT

kTHF +

= has been derived using Quasi Ballistic Drift Diffusion.

key assumptions on which this formula is based :

• Boltzmann statisticsin the contacts• Non Self consistentpotential • use of the relaxation length approximationwith a constant λ

• the population of backscattered carriers (f−) has an equilibrium Maxwellian distribution: at each point x, regardless of the channel length and of the magnitude of the electric field

λ should be energy dependent (especially at high field)

reasonable approximation Only close to the virtual sourceOr along the channel when many collisions are involved

analytical modeling of nano mosfets in the quasi ballistic … · 2021. 3. 15. · mos-ak workshop...

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