Università degli Studi di Pisa Giuseppe Iannaccone G. Iannaccone Dipartimento di Ingegneria...

Università degli Studi di PisaGiuseppe Iannaccone

G. IannacconeDipartimento di Ingegneria dell’Informazione, Università degli Studi di Pisa

Via Diotisalvi 2, I-56122, Pisa, [email protected]

IWCE 2004

Analytical and Numerical Investigation of Noise in nanoscale

Ballistic Field Effect Transistors

Acknowledgments:

Support from EU (SINANO), MIUR (FIRB), Fondazione CRP


Motivation

1. Effects typical of mesoscopic devices can be observed in “mundane” MOSFETs at room temperature as the ballistic component of the drain current increases

2. Suitable noise models are required by circuit designers, especially for analog and mixed signal applications

Here we focus on the limit of ballistic transport.


Noise of the drain current

Transition from Thermal to shot in as the ballistic limit is approached

Shot noise of the gate current

Plus contributions due to defects (not considered here)

Noise in nanotransistors

INTEL - Prototype 20 nm MOSFET

NMOS Gate delay 0.6 ps

Shot noise of the gate tunnel current

Noise of the drain current


Shot Noise Noise is an extremely sensitive probe of electron-electron

interaction. No interactions Poissonian process

Interaction introduce “coordination” in the collective motion of electrons, making the process non Poissonian ( S Sfull).

Interaction: Pauli Exclusion and Coulomb Repulsion

2qISfull

Full shot noise

S: power spectral density of the noise current


Ballistic transport in MOSFETs (I)

Density of states in the first subband in the channel:

Electron density at the subband peak in the channel:zy

tzyD

EEh

mEEN

22 2),(

kTEEDS DFSeEf /)()( )(1

1)(

)()(2 2

0 0

2 MzyDMzySDzyD EEEfEEEfNdEdEn

FSE

FDE

ME

y

0, zy EE

Source Drain


Charge fluctuations in MOSFETs

Fluctuations of n2D as function of fS and fD:

fluctuations electrostatics

in the contacts

Subband maximum EM depends on n2D via electrostatics

Electrostatic effects are included in a single capacitance per unit area CC

M

D

M

SDzyMDSDzyD E

f

E

fNdEdEEffNdEdEn 2

0 0

2

0 0

2 22

C

DM C

nqE 2

2


Equivalent circuit

Add quantum capacitance towards the source and the drain.

Equivalent circuit:

M

SDzyQS E

fNdEdEqC 2

0 0

22

M

DDzyQD E

fNdEdEqC 2

0 0

22

FSEFDE

ME

y

C


Barrier modulation Fluctuation of channel barrier

Current density is modulated by barrier height !

QDQSC

DSDzy

M CCC

ffNdEdEq

E

2

0 0

22

)()(2 2

0 0

MzyDMzySDyzy EEEfEEEfNvdEdEqI

Total

capacitance

yyy mEv /2 Longitudinal velocity


Current fluctuations depend on fluctuations of the occupation factors and of the channel barrier:


in the contacts

Current fluctuations expressed as a function of contact fluctuations

Current Fluctuations

M

D

M

SDyzyMDSDyzy E

f

E

fNvdEdEEqffNvdEdEqI 2

0 02

0 0

22

DyQDQSC

QSSQDDS

yQDQSC

QDDQSS

Dyzy

fvCCC

CvCvf

vCCC

CvCv

NvdEdEqI

11

11

2 2

0 0


Shot noise power spectral density Power spectral density:

Far from equilibrium, if fD = 0 , we have CQD = 0, and

SSS fff 1δ 2 DDD fff 1δ 2

DDyQDQSC

QSSQDDSS

yQDQSC

QDDQSS

Dyzy

ffvCCC

CvCvff

vCCC

CvCv

NvdEdEqS

11

111

1

4

22

2

0 0

2

SSyQSC

QSSDyzy ff

vCC

CvNvdEdEqS

11

14

2

2

0 0

2

SDyzy fNvdEdEqI 2

0 0

2

SyQSC

QSS fvCC

Cv

qI

S

1

11

2

2


Noise suppression factor (Fano factor)

Fano factor is always < 1 and

If CC is very large (e.g., large gate capacitance) then Coulomb interaction is completely screened:

For Maxwell-Boltzmann statistics (e.g. below threshold), fS << 1

SyQSC

QSS fvCC

Cv

qI

S

1

11

2

2

Effect of Pauli

Exclusion

Effect of Coulomb Interaction

1CC

Sf 1

SC f 1

22

11

QSC

C

yQSC

QSS

CC

C

vCC

Cv


25 nm “Well tempered” MOSFET

Doping Profile of the 25 nm “well tempered” MOSFET (D. Antoniadis)

Effective channel length 25 nm Super-halo doping in the channel

minimizes charge sharing effects

Lowest subband profile from 2D PS solver (G. Fiori et al., APL 81, 3672 (2002))

With Vg=1V, Vds=0.1 V, 96.5% of current is carried by the 1st subband


Subband Maximum and Source Quantum Capacitance CQS


Shot noise suppression in well tempered MOSFETs

VDS = 0.5 V VDS = 1 V


Noise in the partially ballistic MOSFETs (I)(with G. Mugnaini)

the first N-1 MOSFETs can be aggregated in an equivalent drift-diffusion MOSFET. (G. Mugnaini et al., submitted to IEEE-TED).

The channel of an arbitrary MOSFET is decomposed in a chain of ballistic MOSFETs of length the mean free path.


Noise in the partially ballistic MOSFETs (II)(with G. Mugnaini - preliminary)

Thermal noise source + shot noise source As the ratio between the device length and the mean free path is reduced,

Noise has a transition THERMAL SHOT Presently including the effect of electrostatics on noise

N=L/


Gate currents: Fresh and stressed oxidesExperimental results by F. Crupi –

From G. Iannaccone et al. IEEE-TED 50, 1363 (2003)

Stress voltage 7.8 V(8 V is the breakdown voltage) SILCs should introduce alter also the noise properties

Cu

rre

nt (

A)

2 3 4 510

-13

10-12

10-11

10-10

10-9

10-8

10-7

SILC

Fresh

Voltage (V)20 50 80 110 140 170 20010

-29

10-28

10-27

10-26

10-25

10-24

IDC = 300 nA

IDC = 30 nA

IDC = 3 nA

IDC = 300 pA

Full Shot Noise Experimental Data

Si [

A2 /H

z]

Frequency [Hz]

I-V characteristics6 nm oxide

Noise properties of thecurrent through fresh oxides:full shot noise at large currents


Current through fresh oxides(tunneling + native TAT)

theory and exp.Trap distribution is a gaussian centered at 1.8 eV below the oxide CB, with 0.1 eV standard deviation

3 4 5 610-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

tox

= 5.9 nm

Cu

rren

t D

en

sit

y (

A/c

m2 )

Vg (volt)

experimental data fresh native total

5 6 7 8 9 1010-1110-1010-910-810-710-610-510-410-310-2

tox

= 9.9 nm

Vg (volt)

Cu

rren

t D

en

sit

y (

A/c

m2 )

experimental data fresh native total

Exp. By F. Crupi


TAT modelG. Iannaccone et al. IEEE-TED 50, 1363 (2003

• Generation and Recombination rates2g f d

1r f d

• Trap occupation factor

1 2

1 2 1 2

'g g

fg g r r

1 1' 1 ' 'I qg f qr f

1 2

2

1 1 2 2

2' 2 ' 1 2 'SILC

g rS qI qI

g r g r

FANO Factor SILC

' , ' 'TATJ I E x dE dx

' , ' 'TATS S E x dE dx


Extraction of SILC trap distributionComparison with experiments

6 nm oxide Gaussian distribution centered -0.5 eV below Si CB, standard deviation 82 meV

2 3 4 5 610-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

2 3 4 5 60.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

Cu

rren

t d

ensi

ty (

A/c

m2 )

VG (volt)

Fan

o F

acto

rV

G (volt)


Extraction of SILC trap distribution (V)

4 5 6 7 8 910-10

10-9

10-8

10-7

10-6

10-5

10-4

experimental not assisted native SILCs total

Cu

rren

t D

en

sit

y (

A/c

m2 )

VG (volt)

For thicker oxides shot noise suppression is due to transitions through native traps

4 5 6 7 8 9 1010

-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

tox

= 9.9 nm

fresh after stress

0.5 V

Cu

rre

nt

de

ns

ity

(A

/cm2 )

VG (volt)

5 6 7 8 90.70

0.75

0.80

0.85

0.90

0.95

1.00

VG (volt)

Fa

no

fa

cto

r


Conclusion We have derived an analytical expression of noise in

ballistic MOSFETs with two well defined contributions from Pauli exclusion and Coulomb repulsion.

Noise properties can be computed from a numerical simulation of DC electrical properties.

Numerical results for “well tempered” MOSFETs operating in the ballistic regime have been shown, exhibiting room temperature suppression of shot noise, in typical operating conditions, down to 0.25.

Shot Noise of the gate current: contribution of native traps may be important also for noise properties (experiments here are still missing)

For thicker oxides a distribution of traps can be extracted that reproduces both DC and noise characteristcs


Current Fluctuations

Current fluctuations depend on fluctuations of the occupation factors and of the channel barrier:


in the contacts

We can introduce two average velocities vS and vD:

M

D

M

SDyzyMDSDyzy E

f

E

fNvdEdEEqffNvdEdEqI 2

0 02

0 0

22

M

SDyzyQSS E

fNvdEdEqCv 2

0 0

22

M

DDyzyQDD E

fNvdEdEqCv 2

0 0

22


Equilibrium and far from equilibrium If fS = fD, S reduces to 4KTG, as it must be, where

Far from equilibrium, if fD = 0 , we have CQD = 0, and

The noise suppression factor is a weighted average

SSyQSC

QSSDyzy ff

vCC

CvNvdEdEqS

11

14

2

2

0 0

2

SSDyzyS

Dyzy ffNvdEdEqdE

dfNvdEdEqG

122 2

0 0

22

0 0

2

SDyzy fNvdEdEqI 2

0 0

2

SyQSC

QSS fvCC

Cv

qI

S

1

11

2

2


Extraction of native and SILC trap Extraction of native and SILC trap distributiondistribution (I)

• Simulations with a distribution uniform in energy do not provide satisfactory results

' , ' 'TATJ I E x dE dx

,E x E

Trap distribution indipendent of position

• Integral equation with ,E x as the unknown

Hp:


Extraction of SILC trap distribution (II)

Electrons from cathode VB

• Model A: Riccò, Gozzi, Lanzoni, IEEE TED 45, 1998.

“mean” quantities: fluxes and capture cross section

• Model B: Ielmini, Spinelli, Lacaita, IEEE TED 47, 2000.

Transient SILC components

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5

0.1

1

B A

E-EV (eV)

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0

0.1

1

E-EV (eV)


Extraction of SILC trap distribution (III)

5.9 nm oxide Comparison with exp. performed

in Pisa

Other thicknesses Comparison with experiments

drawn from the literature (Ricco’ et al.)

Effect of surface traps for very low voltages

1 2 3 4 5 610-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

tox

= 5.9 nm

Em = 0.5

= 0.082

MAX

1

= 1e-15 m-1

MAX

2

= 1.7e-15 m-1

Cu

rren

t d

en

sit

y (

A/c

m2 )

VG (volt)

2.5 3.0 3.5 4.0 4.510-11

10-10

10-9

10-8

10-7

10-6

10-5

experimental simulation

tox

= 6.4 nmtox

= 5.3 nm

3.0 3.5 4.0 4.5 5.0 5.510-10

10-9

10-8

10-7

10-6

10-5 experimental simulation

VG (volt)

VG (volt)

J-V Curves


Extraction of SILC trap distribution (IV)

1 2 3 4 5 6

0.70

0.75

0.80

0.85

0.90

0.95

1.00

Fan

o F

acto

r

VG (volt)

experiment E

m = 0.5

Em = 0.4

Em = 0.18

Em = 0.01

Fano Factor

Stronger suppression for Em approaching

the silicon gap center


Understanding the nature of SILCs

Stress-induced leakage currents (SILCs) are the single most important limit to downscaling of non-volatile memories [read disturb, retention degradation]

SILCs are due to tunneling assisted by traps generated by electric field stress.

The energy distribution of traps is not known

We show that detailed modeling, coupled with DC and noise characterization, can provide enough information to extract information about the energy distribution of traps


Tunneling Current: fresh oxides

Electron effective mass in the oxide conduction band

Determination of the oxide thickness

Native traps are required for fitting the current at low fields

3.0 3.5 4.0 4.5 5.0 5.5 6.010-1110-1010-910-810-710-610-510-410-310-2

experiments m

e,ox= 0.6 m

0 t

ox= 6.1 nm

me,ox

= 0.6 m0 t

ox= 6 nm

me,ox

= 0.6 m0 t

ox= 5.9 nm

Tu

nn

elin

g C

urr

en

t (A

/cm

2 )

Vg (volt)

4 5 6 7 8 9 1010-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

experiments m

e,ox = 0.6 m

0 t

ox= 10 nm

me,ox

= 0.6 m0 t

ox= 9.9 nm

Tu

nn

elin

g C

urr

en

t (A

/cm

2 )

Vg (volt)


, ,, ' l lJ x T E E

• Transition probability

22,M h E E

2 2

1h E E

E E

• Capture cross section

, k h E E

TAT model (I)• Two-Step tunneling• Anelastic TAT

G. Iannaccone et al.,

IEEE-TED 2003


Tunneling Current: Model The electron density n(x) at

the Si-SiO2 interface is computed by solving the Schrödinger equation for the two-fold and four-fold degenerate conduction band minima.

1D Poisson and Schrödinger equations are solved iteratively.

Once the band profiles and charge densities are obtained, we can compute the tunneling current

i

itFit

tl

i

ilFil

t

kT

EEx

mmkT

kT

EEx

kTmxn

exp1ln)(4

exp1ln)(2

)(

22

22

)()()()(2

22

xExxExm ikilikCik

k

)()()()()( xNxNxnxpq AD

TkEE

TkEEmmTkTq

TkEE

TkEETmkTqJ

BFFGit

BFittlB

i it

it

BFFGil

BFiltB

i il

ilT

/)exp(1

/)exp(1ln4

/)(exp1

/)(exp1ln2

2

2


25 nm “Well tempered” MOSFET

Doping Profile of the 25 nm “well tempered” MOSFET (D. Antoniadis)

Effective channel length 25 nm Super-halo doping in the channel

minimizes charge sharing effects

Quantum confinement in the middle of the channel (z = 45 nm)

With Vg=1V, Vds=0.1 V, 96.5% of current is carried by the 1st subband


Subband profile and characteristics

2D simulation First subband profile in the longitudinal

direction for increasing Vds.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40

100

200

300

400

500

600

700

800Si super-halo doping

1.21.251.3

1.35

1.4

1.45

1.5VGS

[V] =

I DS [A

/m]

VDS

[V]


Papers on the subject Y. Naveh, A. N. Korotkov, K. K. Likharev, Shot noise suppression in

multimode ballistic Fermi conductors, Phys. Rev. B, 60 (1998), R2169-2172.

O. M. Bulashenko and J. M. Rubì, Shot-noise suppression by Fermi and Coulomb correlations in ballistic conductors, Phys. Rev. B, 65 (2001) 045307.

O. M. Bulashenko and J. M. Rubì, Self-consistent theory of current and voltage noise in multimode ballistic conductors, Phys. Rev. B, 66 (2002), 045310.

G. Gomila, I. R. Cantalapiedra, T. Gonzalez, L. Reggiani, Semiclassical Theory of shot noise in ballistic n+-i-n+ semiconductor structures: Relevance of Pauli and long-range Coulomb correlations, Phys. Rev. B, 66 (2002) 075302.


Far-from equilibrium transport in each ballistic MOSFET, gives a local shot noise [van der Ziel,1986]:

If the chain is long enough, local equilibrium holds in the whole sructure and then local shot noise reduces to conventional thermal noise:

Similarly to the aboveseen current macromodel, a noise macromodel for a device operating in intermediate tranposrt regime, is given by the series of a thermal noise generator with a shot noise generator.We expect that when the ratio between the device length and the mean free path reduces, a more pronounced far-from equilibrium behavior emerges both in the static current and in the noise.

Noise in the partially ballistic MOSFETs (II)(with G. Mugnaini)


Pauli and Coulomb interactions

In most cases interactions make the collective motion more regular

Limits density in real space

Limits density in phase space

Reduced fluctuations

Sub-poissonian process fullSS


Fully ballistic transport regime

Electrons with sufficient energy to overcome the barrier near the source reach the drain conserving energy and transversal momentum

Electron states originating from the source obey the Fermi-Dirac statistics with source Fermi Energy Efs

Electron states originating from the drain obey the Fermi-Dirac statistics with drain Fermi Energy Efd

This ensures continuity of current density per unit energy in each subband


Model

Poisson equation in 2D

The electron density n() in the quantum region is obtained from the solution of the Schrödinger equation with density functional theory

p(), ND+(), NA

-() and of n() out of the quantum region are given by the corresponding semiclassical expressions

)()()()()( DA NNpnqr

Discretization with the box integration method Newton-Raphson method with predictor-corrector iteration scheme


The Schrödinger equation must be solved twice: For the 2 minima along the vertical (x) direction

For the other 4 minima

The quantum electron density becomes

x

yz

lililiClil

yE~Ey,xxmx

1

2

2

tititiCtit

yE~Ey,xxmx

1

2

2

kx

kz

ky

kx

ky

kz

Mass anisotropy and electron density

i i

tili nnn 42

y

EE

FDiyFSiyylli

y

EE

FSiyyllili

dEEEEfEEEf

ED

dEEEEfEDn

ii

ii

~

2

~

0

2

max

max

2

~~

~


Model out of equilibrium

When the Poisson-Schrödinger equation is solved, and charge density and potential profiles are known, we compute the current density in the i-th subband

The total current density is

limaxi E~Ey

FDyliFSylitli dE

kT

EEE~F

kT

EEE~F

h

kTmqJ

21

2122

12

i i

ttii

ltili JJJJ 222


Examples of nanotransistors

INTEL, in production now oxide thickness 2 nm

INTEL test device In production by 2005 (ITRS 2002 update oxide thickness 0.8 nm


Summary

Motivation: VLSI devices are already nanoelectronic devices !!

Mesoscopic Noise in MOSFETs Shot Noise of the drain current in ballistic MOSFETs Shot Noise of the gate current in fresh oxides and in

the case of tunneling assisted by traps Conclusion

Acknowlegments:

F. Crupi, A. Nannipieri, G. Curatola, G. Fiori

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