Mobility in UTB-SOI PFETS: Local Coordinate-Based Modeling with the - - PowerPoint PPT Presentation

Mobility in UTB-SOI PFETS:

Local Coordinate-Based Modeling with the Density Gradient Method

Daniel Connelly, Acorn Technologies

• D. E. Grupp, Acorn Technologies

Daniel Yergeau, Paco Leon: Mixed Technology Associates ACORN Technologies Inc ACORN Technologies Inc ACORN Technologies Inc ACORN Technologies Inc ACORN Technologies Inc ACORN Technologies Inc ACORN Technologies Inc ACORN Technologies Inc

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FDSOI Mobility Modeling

Current practice: Use Eperp or Eeff to model µ reduction with increasing inversion layer density. Technology Trend: Length scaling of FDSOI ever- decreasing tSi. Problem: Eperp or Eeff don’t predict µ reduction with decreasing tSi. Ballisticity ∝ µ1/2. Result: Excessively optimistic scaling forecasts by simulation engineers. Unsatisfying Kludge: “Manually adjust” global µAPS scale factor based on theoretical or experimental data. Better Approach: A local model which includes structural effects.

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1

surface mobility

= 1

coulombic mobility

+ 1

APS mobility

+

Mobility Composition

1

surface roughness scattering mobility

+ 1

thickness variation scattering mobility

new term, after Uchida et al, IEDM 2002 use Eeff-based model new approach, using Agostinelli et al, TED 1991 Agostinelli et al, TED 1991

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Confinement and APS

Bulk: electric field predicts confinement. FDSOI: structural information needed.

confinement k-space dispersion APS

• bulk channel

FDSOI channel

SiO 2

E F

confinement due to electric field confinement due to structure

E F

SiO 2 SiO 2

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APS Modeling Philosophy

New idea: Use position to model confinement effects on mobility. Advantage: Structural confinement effects can be modeled. Efficiency: Calculation done only at start of simulation, not after each iteration. New Requirement: An accurate determination of carrier distributions near interfaces. Tool for the Job: The Density Gradient Method.

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Obligatory Density Gradient Validation Slide

mh*Ndim= 0.48

0.5 1 1.5 2 2.5 1 2 3 4 1 2 3 0.5 1 1.5 2 1 2 3 4 5 0.5 1 1.5 2 2 4 6 0.5 1 1.5 2 Prophet Schred 3 5 7 10 20

silicon thickness [nm]

0.0 0.2 0.4 0.6 0.8 1.0

∆VG [volts]

1012 1013

Nholes (tSi/2) [cm-2]

1 2 3 5

hole centroid (tSi/2) [nm]

tSi=5nm tSi=7.5nm tSi=10nm tSi=15nm

position from oxide interface (nm) hole concentration (10

19/cm 3)

Nholes=5×1012/cm2 all curves : 1×1012 2×1012 5×1012 1×1013 2.5 3 4 5 7.5 10 15 20 Nholes [cm-2] tSi [nm] tox=1.5nm

Density Gradient (Prophet) vs. Schrödinger-Poisson (Schred)

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Methodology Development

Basis: µAPS proportional to inversion layer thickness. Challenge: defining and calculating “inversion layer thickness” in arbitrary structures is difficult. Observation: For planar “bulk” channel, inversion layer thickness is roughly proportional to mean distance from Si/SiO2. Planar “Bulk” Channel: Consider APS mobility proportional to distance from SiO2. FDSOI Channel: use 1/zeff

2 = 1/z1 2 + 1/z2 2.

Extension: treat arbitrary distribution of Si, SiO2.

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2 3 5 7 10 20 30 50

Neff (1011/cm2)

50 60 70 80 90 100 120 150 200

effective hole mobility [cm2/volt-sec]

8.9×10

1 5

1.6×10

1 6

5.1×10

1 6

2.7×10

1 7

6.6×1017

ND [cm-3]

γ-based Eeff-based

2 3 5 7 10 20 30 50

Neff (1011/cm2)

50 60 70 80 90 100 120 150 200

2 2 3 K 297K 343K 393K 443K

variable T, ND=7.8×1017/cm3 variable ND, T=300K

Bulk PFET Mobility Modeling

simulated: NF=3×1010/cm2

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Ns (1011/cm2)

50 60 70 80 90 100 120 150 200

effective hole mobility [cm2/volt-sec]

43Å 54Å 68Å 111Å 175Å 500Å

2 3 5 7 10 20 30 50

Ns (1011/cm2)

50 60 70 80 90 100 120 150 200 43Å 54Å 68Å 111Å 175Å 500Å

Prophet: NF=3×1011/cm3 from Ren et. al.

FDSOI PFET Mobility Model vs. Data

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Extension to Multiple Dimensions: Acoustic Phonon Scattering

a heuristic approach

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APS Mobility Determination

Convert γ to an effective field:

Eγ = (q/εSi)(kγ γ)3

where here: kγ = 25.1µm1/3 Heuristic approach: Use Eγ in place of Eeff in Agostinelli et al nonlocal µeff(Eeff) APS term to yield a local APS µ field.

Result: µ roughly proportional to 1/γ, proportional to distance from SiO2.

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2-D Integration: Acoustic Phonon Scattering

• SiO2

SiO2

integration region 2 integration region 1 point integration region 3

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γ γ γ γ-Field at Convex or Concave Corners

• 10
• 5

5 10

x coordinate (au)

• 10
• 5

5 10

y coordinate (au)

20

10 2 4 2

concave corner convex corner interface

γ -1

4 6 8 6

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Remaining Problem: Thickness Variation Scattering

OBSERVATION: Mobility in ultra-thin files (tSi < 5 nm) falls

• ff much faster than expected from APS.

BASIS: Local ground state energy depends on local thickness. CONCLUSION: Thickness variation means moving holes see time-variable potential, which yields scattering. tSi DEPENDENCE: Uchida et al show scattering proportional to 1/tSi

6.

CHALLENGE: Extend locally to multiple dimensions.

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Thickness Variation vs. Surface Roughness

• less thickness variation

more thickness variation

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Comparison of 1-d σ

σ σ σt Model to Data

0.05 0.1 0.2 0.3 0.5

effective field [MV/cm]

30 40 50 60 70 80 90 100 120 150 200

effective hole mobility [cm2/v-sec]

from Uchida model

3 4 5 7 10 20 30 50

silicon thickness [nm]

40 50 60 70 80 90 100 120

from Uchida Eeff-based model γ-based model added σtsi model

5.49nm 60nm tSi=2.72nm E

e f f

• b

a s e d γ-based

(a)

3.57nm

(b)

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Extension to Multiple Dimensions: Thickness Variation Scattering

independent unit-angle resonator assumption

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σ σ σ σt Mobility Determination

Convert γt to a mobility term:

µσt = (kγt γt)-6 cm2/v-sec

Match to Uchida et al: kγt = 1.36 nm

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2-D Integration: Thickness Variation Scattering

• SiO2

SiO2

integration region 2 i n t e g r a t i

• n

r e g i

• n

1 point

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γ γ γ γt-Field Examples

thick-thin transition square cylinder

8 2 10 4 9 6 µσt

• 1 (au)
• xide

reflection plane

semi-infinite silicon silicon

7 µ

σt

(au)

5 10 15 20

• xide

reflection plane reflection plane

(a) (b)

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Application: Cylindrical vs. Planar Gate NMOS

2-D confinement lower mobility

2 3 5 7 10 20 30 50

silicon thickness [nm]

50 100 150 200 250

effective hole mobility [cm2/V-sec]

0.75 0.8 0.85 0.9 0.95

ratio

ratio: C/2G C 1G,2G

Neff=1011/cm2

1G : one-gate SOI 2G : dual-gate C : cylindrical

C

• SiO2

tSi

tox

silicon Gate

cylindrical device

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Remaining Challenges

Surface Roughness Scattering: Surface roughness scattering may also depend on structural confinement. Electrons: Band splitting in (001) electrons complicate mobility

• modeling. (111) electrons may prove easier.

Multidimensional Validation: All validation in this work is 1-

• dimensional. Extensions to multiple dimensions need to be

validated, an experimental challenge. High Energy Transport: This work treats only low-field mobility. High- field effects are treated after bulk planar MOSFET models. Do these apply to UTB FDSOI? Other Materials: Different material interfaces than Si/SiO2 are expected to behave differently. Implementation: Model has not yet been implemented into a simulator; all calculations here are post-processed.

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Summary

Acoustic Phonon Scattering: Modeling µ as a function of position is a promising approach to APS modeling in nanoscale Si structures, matching “universal mobility” relationships for planar bulk channels, while predicting the correct trends in UTB Si devices. Thickness Variation Scattering: An extension of a 1- dimensional “thickness variation scattering” relationship to multiple dimensions was proposed.

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Acknowledgements

Thanks to Stockton Gaines, Joe Daniele, Tom Horgan, Jan Lewis, and Carl Faulkner at Acorn Technologies, and to Robert Dutton at Mixed Technology Associates, for making this project possible.