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Precise 2D Compact Modeling of NanoscaleDG MOSFETs Based on Conformal

Mapping Techniques

T. A. Fjeldly, S. KolbergUniversity Graduate Center (UNIK)

Norwegian University of Science and Technology, Kjeller, Norway

B. IñiguezDepartament d'Enginyeria Electronica, Electrica i Automatica,

Universitat Rovira i Virgili, Tarragona, Spain

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Overview

Objective:Establish framework for precise, compact 2D modeling of short-channel nanoscale MOSFETs

Here: Double Gate (DG) MOSFET

• Part 1: Subthreshold and near thresholdconditions where capacitive coupling between contacts dominant electrostatics

• Part 2: Strong inversion where effects of electrons dominant electrostatics. Must betreated self-consistently.

• Determine 2D barrier topography• Calculate electron distribution• Find short-schannel effects including DIBL• Calculate current – drift-diffusion and ballistic

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Modeling procedure• Include gate oxide in extended device

body:

• Map body into upper half plane oftransformed (u,iv)-plane using conformal mapping (Schwartz-Christoffel transform).

• Subthreshold: Solve Laplace’s equation in(u,iv)-plane (low-doped body)

2D potential distribution for capacitivecoupling. Map back to (x,y)-plane

• Near threshold: Determine total potentialdistribution self-consistently by includingelectrostatic effects of electrons.

• Strong inversion: Use long-channel solution. Treat 2D capacitive coupling as perturbation.

• Calculate current (drift-diffusion and ballistic)

• Verify against 2D numerical simulations.

oxSioxox tt εε='

Schwartz-Christoffel transformation( )

)(,

2 kKivukFLiyxz +

=+=

( )( ) )1,()(,'1'1

'),(0 222

kFkKwkw

dwwkFw

=−−

= ∫

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Mapping of the body boundary

( ))(

,2 kK

ukFLiyxz =+=

Real part

Imaginary part

G1 G2G2 SD

2xL

y _ t’Si + 2tox

-

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Mapping along symmetry lines

Gate-to-gate symmetry line

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛

+−

−

+=

=

22

2 1,1

1

'2

0

v

vkFkK

tty

x

oxSi

( )( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+

=

+=

kkF

kkFLx

tty oxSi

12cos,

12

2

22 '

θ

Source-to-drain symmetry line

2xL

Device center

-

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DG MOSFET and regimes of operation

SINANO template device:

• Double gate n-channel SOI MOSFET with aluminum gates

• Specifics: L = 25 nm, tsi = 12 nm, tox = 1.6 nm, εox = 7, body doping Nd =1015 cm-3, contact doping 1019 cm-3,extended body thickness: tsi + 2t’ox

Operating regimes:→ Part 1: In subthreshold, dopant and carrier charges can be neglected

in electrostatics when n or Nd ≤ 1018 cm-3. Potential and electron distributions dominated by the 2D capacitive coupling between the contacts (source, drain and gates)

→ Part 2: Above threshold requires self-consistent analysis- 2a) Near threshold: Use approximate barrier profile- 2b) Strong inversion: Electrons dominate, treat capacitive coupling

as a perturbation

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Part 1: Subthreshold – body potential

Solution of Laplace’s equation in (u,iv)-plane (thin oxide approx):

( ) ( )( )

( )

( )

( ) ⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +

−⎟⎠⎞

⎜⎝⎛ +

++

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

−⎟⎠⎞

⎜⎝⎛ −

+

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +

+⎟⎠⎞

⎜⎝⎛ −

−+

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +

−⎟⎠⎞

⎜⎝⎛ −

−−

=

+−=

−−

−−

−−

−−

∞

∞−∫

vu

kvkuVV

vu

kvkuV

vu

vuVV

kvku

kvkuVV

duvuu

uvvu

DSbi

bi

FBGS

FBGS

1tan1tan

1tan1tan

1tan1tan

1tan1tan

1

''

',

11

11

111

112

22

π

π

ϕπ

ϕ

Vds = 0.5 VVGS = -0.45 V

Vds = 0 V

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Comparison with numerical simulations

Subthreshold potential variation along the

symmetry axes

VGS = -0.47 V, VDS = 0 V

Charge density distribution– gate-to-gate symmetry line

• Classic electron distribution:

• QM electron distribution (deep well):

Combine subband states from parabolic well (lowest subbands) and square well potentials (higher subbands).

( ) ( )[ ]⎟⎟⎠

⎞⎜⎜⎝

⎛ −+−⎟

⎠

⎞⎜⎝

⎛=Tk

yqETkmyn

B

SSbgBnc

ϕϕ

π

2exp

22

23

2h

( ) ( ) 2

1ynyn j

l

jsqjq ψ∑

==

VGS = -0.47 V, VDS = 0 V

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Threshold voltage and DIBL

’Classical’ threshold voltage definition: Gate voltage VT where band bending at barrier is 2qφb at VDS = 0 V

For symmetric gate bias: VT = -0.47 V

Drain induced barrier lowering (DIBL):Shift in VT with applied VDS

Asymmetric gate biasing:Apply fixed negative bias at Gate 2 relative to Gate 1. Pro/con effects:

+ reduced short channel effect+ adjustable thereshold voltage- effective channel width halved- added small-signal capacitance

Part 2: Strong inversion – self-consistency

Consider energy barrier at gate-to-gate symmetry axis

φ1(y): 1D potential from electrons

φ2(y): 2D capacitive coupling potential between contacts

Superposition principle: ( ) ( ) ( )yyy 21 ϕϕϕ +=

( ) ( )FBGSbiFBGS VVVvkv

VVv +−⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛+−= −− 1tan1tan2 11

2 πϕ

( ) ( )

( )( )

( )( )

( )( ) ( )⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

+<≤⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛

<⎟⎟⎠

⎞⎜⎜⎝

⎛

×−=−=

∫∫

∫

∫

∫

+

+

+

2'2'for''''exp'

''exp'

'for''exp

2'2

''

2'2

'

2'2

'

0 11

oxSioxtt

yth

y

t

tt

tth

ox

oxtt

tth

oy

ttytdyV

ydy

dyV

yt

tydyV

yy

KyEy

oxSi

ox

oxSi

ox

oxSi

ox

ϕ

ϕ

ϕ

ϕ

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UniK/NTNU

2a Near operational threshold

( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+

−−+−=≈2

'2211

oxSimFBGSp tt

yVVyy ϕϕϕ

Assume that the total gate-to-gate potential φ(y) has a parabolic form, which agrees very well with numerical simulations (Atlas):

Parameter φm is determined by:

• substituting φp(y) into expression for φ1(y)

• performing integrations in expression for φ1(y)

• substituting φ1(y) into total potential φ(y)

• obtaining self-consistent expression for φm by considering device center

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Parameter φm

( ) ( )

( ) ( ) ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡−⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

−−+⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎥⎦

⎤⎢⎣

⎡+

−×

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−+−+⎟⎠⎞

⎜⎝⎛−⎥

⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛ −+−= −

1'2

'21exp'2

'21erf)(

2exp

4'2

211tan4

2

223

21

oxSi

ox

th

m

m

th

oxSi

ox

th

m

m

thm

th

gbFBGSbioxSiBnFBGSbim

ttt

VV

ttt

VVsign

VEVVVttqTkm

kVVV

ϕϕ

ϕϕπϕ

ϕππ

ϕh

VDS = 0 V

2b Strong inversion

Gate-to-gate barrier:

Use long-channel potential φ1(y) for electrons from 1D Poisson’s equation (Taur 2001), adjusted to include effects of body doping:

Treat capacive coupling potential φ2(y) as a perturbation. Find self-consistent total potential by first-order correction using 1D Poisson’s equation

Describes how DIBL-effect fades in strong inversion

( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−⎟⎟

⎠

⎞⎜⎜⎝

⎛ −−= 2'

2exp

2cosln21 Siox

th

bc

ths

ithc tty

VVqnVy ϕϕε

ϕϕ

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Current modeling – subthreshold and near threshold

Drift-diffusion transport:Use barrier shape derived self-consistently. Assume that the current is ’small’. Then we have:

nso(x): surface concentration of electrons in all cross-sections from source to drain.

Ballistic transport:

• Current consists of free-flight electrons with sufficient energy to fly over barrier • Electrons ’filtered’ through available quantum states at barrier position• Current limited by ability of source and drain to supply high-energy carriers fast enough

Natori formalism (1994, 2002):

IBT = q(F+-F–) where

( ) ( ) ( )( )∫

−−=−=

L

so

VVthnF

snDD

xndx

eVqWdx

xdVxnqWIthDS

0

1µµ

( )∑ ∑ ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛ −−=

±±

valleys j B

ijFsi

B

TkqVEE

FmTk

F 2122

232hπ

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Current modeling – strong inversion

Current flows primarily near the Si/SiO2 interfaces.Use classical long-channel expressions for IDD.

Strong inversion threshold voltage (Taur et al. 2002, adjusted for body doping) :

Simulated (Atlas) strong inversion potential contour plot VGS = 0.1 V, VDS = 0 V

( )⎟⎟⎠

⎞⎜⎜⎝

⎛ −+=+=

iox

GSSioxBmsBT qnt

VVtq

TkVVV2

ln2where2 000

εφϕ

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DD and ballistic current – comparision with simulations

Drift-diffusion current Ballistic current

O : simulations: model

O : simulations: model

Observations:

• Good correspondence between modeled and simulated currents• Quite similar results for drift-diffusion and ballistic current• Some more work needed to model ’knee’ region)

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SummaryCompact 2D modeling framework established for short-channel, nanoscale DG MOSFETs by means of conformal mapping techniques

Precise description of:• 2D device barrier topography• Short-channel effects including DIBL• Self-consistency at moderate and high electron densities• Fading of DIBL in strong inversion• Drift-diffusion and ballistic transport

Model verified by numerical simulations