Effect of Discrete Dopant Distribution on MOSFETs Scaling into the Future

Yoshio Ashizawa and Hideki Oka Fujitsu Laboratories Ltd.

50 Fuchigami, Akiruno, Tokyo, 197-0833, Japan

E-mail: ashizawa.yoshio@jp.fujitsu.com

Abstract - Device characteristics fluctuation due to

discrete dopant distribution is one of the serious

problems for future device scaling. We have evaluated

the potential field on the substrate by applying 2D FFT.

It is found that specific wave length range, 20 nm to 25

nm, is dominant to device characteristics. Moreover, if

device size scales less than this specific wave length, there

will be size effect caused by local potential fluctuation.

I. INTRODUCTION

Device characteristics fluctuation induced by random

dopant distribution has been investigated using combination

of Synopsys Taurus Process Atomistic simulator[1] and

SELETE HyDeLEOS device simulator[2]. 2D FFT is

applied to the doping concentration and potential fields at the

Si surface to characterize the fluctuations. It is found that the

larger fluctuation will be predicted among transistors when

device size becomes smaller than specific the wave length

determined by spatial potential fluctuations.

II. FLUCTUATION INDUCED BY DISCRETE DOPANT

DISTRIBTUTION

Fig. 1 shows the simulation procedure and device

structure. Initial boron is given at constant concentration in

the substrate. After annealing, boron field is converted to

discrete atomistic distribution by DADOS. As dopant

distribution in the substrate is dominant to device

characteristics fluctuation in our preliminary works, all

dopant profiles are the same except for substrate, in which

dopant distribution is atomistically simulated.

• Process Simulator

– TAURUS Process Atomistic

– Initial boron distribution• Uniform

– Anneal• Kinetic Monte Carlo

• DADOS integrated into TPA

– Discrete Boron distribution

• Device Simulator– SELETE HyDeLEOS

– Drift diffusion

– Long-range part of Coulomb potential• Sano model

90nm Node Structure

W=100nm100nm

50nm

B Na=5×1018cm-3

S

G

DTox=1.2nm

B

AsAnalytic

Atomistic

L=50nm

Atomistic

90nm Node Structure

W=100nm100nm

50nm

B Na=5×1018cm-3

S

G

DTox=1.2nm

B

AsAnalyticAs

Analytic

Atomistic

L=50nm

Atomistic

32

3

)(

)cos()()sin(

2)(

rk

rkrkrkkr

c

cccclong −=π

ρ

Fig. 1: Simulation procedure and device structure.

After process simulation, atomistic boron positions are

applied to drift diffusion device simulator, in which Sano

model[3] is used for the long-range part of Coulomb

potential of individual impurities.

Table 1 is the summary of target device parameters for

scaling nMOSFETs, where Na is the acceptor concentration

in the substrate.

Table 1: Target parameters for scaling nMOSFETs.

0

5

10

15

20

25

30

0 0.01 0.02 0.03 0.04 0.05

ITRS 2003Mizuno et. alToriyama et. al

σVth

(mV)

65nm

90nm

L=W=10nm

Tox=0.4nm

L=W=25nm

Tox=0.8nm

L=W=100nm

Tox=3nm

32nm Node

45nm

LWTox / ( )

Na=1018cm-3

0

5

10

15

20

25

30

0 0.01 0.02 0.03 0.04 0.05

ITRS 2003Mizuno et. alToriyama et. al

σVth

(mV)

65nm

90nm

L=W=10nm

Tox=0.4nm

L=W=25nm

Tox=0.8nm

L=W=100nm

Tox=3nm

32nm Node

45nm

LWTox / ( )

Na=1018cm-3

Fig. 2: σVth vs. dimensionless size factor with scaling.

Fig. 2 shows the relation between calculated and reported

[4][5] σVth and dimensionless size factor LWTox , which

reflects the degree of geometrical scaling. Because vertical

scaling is behind horizontal scaling, dimensionless size

factors are increased with scaling.

As for the slope between the plot and the origin, it is a

concentration factor, which represents the contribution of

concentration Na to σVth. Generally, the slope is considered

to get steeper with increasing Na. But all our results for these

roadmap parameters are on the same line even though each

impurity concentration is different. However, we got the

same results with others when Na is decreased to 1018 cm-3 as

is not shown in this figure. This means that the concentration

effect becomes saturated over 5 x 1018 cm-3.

We have extended scaling parameters to make sure if the

concentration factor actually saturates with other parameters.

Fig. 3 shows the response surface for the slope, in which

F(Na) is calculated from σVth divided by the dimensionless

size factor for each technology node and acceptor

concentration Na.

2x10191.4x10198x10185x1018Na(cm -3)

0.91.01.11.2Vdd(V)

0.60.70.91.2Tox(nm)

12182636L(nm)

32nm Node45nm Node65nm Node90nm Node

2x10191.4x10198x10185x1018Na(cm -3)

0.91.01.11.2Vdd(V)

0.60.70.91.2Tox(nm)

12182636L(nm)

32nm Node45nm Node65nm Node90nm Node

31

2-4

The surface is mildly curved at lower Na but getting evener

at higher Na not only along the roadmap parameters but also

with other parameters.

Fig. 3: Response surface of the concentration factor with

various target parameters.

III. POWER SPECTRUM ANALYSIS

It is greatly helpful to characterize dopant distribution or

potential field by the spatial frequency space because spatial

frequency space is closely related with real space (Table 2).

There is only 2D random noise observed for phase angle in

spatial frequency space because discrete dopant is randomly

and disorderly distributed. Its effect is considered to be the

same level as baseline noise for all calculations. Therefore,

only power is discussed in this work although phase angle is

also important.

Table 2: Correspondence between real space and spatial

frequency space.

Phase AnglePosition

In teraction

Power SpectrumConcentra tion

Potentia l

Spatial Frequency SpaceReal Space

Phase AnglePosition

In teraction

Power SpectrumConcentra tion

Potentia l

Spatial Frequency SpaceReal Space

2

i

2 ),(),(),( vuFvuFvuP r +=

),(

),(tan 1

vuF

vuF

r

i−=ξ

2D FFT1

0

1

0

)(2

2),(

1),(

N

x

N

y

vyuxN

j

eyxfN

vuFπ π

θ

θ2

0

),()( fPfQ

1D Trans.

Real Space Spatial Frequency Space

x

y

22 ),(),(

),(

vuFvuF

vuP

ir

v),( vuF

1D Power Spectrum

)1log(

log)(log

α　

α

fk

fkfQ

log f

log Q

Slope= -α

2D Complex F(x,y)

0

),( yxf

100nm x 100nm

2D Field

Integral

Contribution

Na(x,y)f(x,y)=

ϕ(x,y)

22 vuf

P(f,θ)

u0

2D FFT1

0

1

0

)(2

2),(

1),(

N

x

N

y

vyuxN

j

eyxfN

vuFπ π

θ

θ2

0

),()( fPfQ

1D Trans.

Real Space Spatial Frequency Space

x

y

22 ),(),(

),(

vuFvuF

vuP

ir

v),( vuF

1D Power Spectrum

)1log(

log)(log

α　

α

fk

fkfQ

log f

log Q

Slope= -α

2D Complex F(x,y)

0

),( yxf

100nm x 100nm

0

),( yxf

100nm x 100nm

2D Field

Integral

Contribution

Na(x,y)f(x,y)=

ϕ(x,y)

Na(x,y)f(x,y)=

ϕ(x,y)

22 vuf

P(f,θ)

u0

Fig. 4: Application of 2D FFT to f(x,y) and transformation to

1D power spectrum.

Fig. 4 shows the schematic procedure to apply FFT.

F(u,v) is calculated by 2D FFT against 2D function f(x,y),

which corresponds to concentration or potential field.

Power P(u,v) is defined as the square norm of the complex

function of F(u,v), where Fr(u,v) and Fi(u,v) represent the

real and imaginary part of F(u,v) respectively.

Because P(u,v) is a 2D function in spatial frequency space,

it is still difficult to compare or characterize it directly.

Therefore, P(u,v) is transformed to 1D power spectrum Q(f)

by integrating P(f,θ) along all circumferences. 1D power

Q(f) represents the contribution of each spatial frequency.

Moreover, the slope characterizes the degree of the

fluctuation.

0.0001

0.01

1

100

104

1 10 100

1e18(cm-3)

5e18(cm-3)

8e18(cm-3)

1.4e19(cm-3)

2e19(cm-3)

Power Spectrum (arb. unit)

Spatial Frequency (1/100nm-1)

Na

S DG

(a) Acceptor Conc. Field

90nm Node

L=36nm

W=100nm

Up

Slope= - 2

0.0001

0.01

1

100

104

1 10 100

1e18(cm-3)

5e18(cm-3)

8e18(cm-3)

1.4e19(cm-3)

2e19(cm-3)

Power Spectrum (arb. unit)

Spatial Frequency (1/100nm-1)

Na

S DGS DG

(a) Acceptor Conc. Field

90nm Node

L=36nm

W=100nm

Up

Slope= - 2

(b) Potential Field

1000

104

105

106

107

108

109

1010

1 10 100

1e18(cm-3)5e18(cm-3)8e18(cm-3)1.4e19(cm-3)2e19(cm-3)

Power Spectrum (arb. unit)

Spatial Frequency (1/100nm-1)

Na

S DG

90nm Node

L=36nm

W=100nm

Slope= - 2

(b) Potential Field

1000

104

105

106

107

108

109

1010

1 10 100

1e18(cm-3)5e18(cm-3)8e18(cm-3)1.4e19(cm-3)2e19(cm-3)

Power Spectrum (arb. unit)

Spatial Frequency (1/100nm-1)

Na

S DGS DG

90nm Node

L=36nm

W=100nm

Slope= - 2

Fig. 5: 1D power spectrum for acceptor concentration field

(a) and potential field (b) at the Si surface.

Fig. 5 (a) shows power spectrum for acceptor

concentration field. Power is also observed to be saturated at

higher acceptor concentration, which is similar in Fig. 2 and

Fig. 3. Fig. 5 (b) shows the case of the potential field.

Both fields obey the 1/f2-rule generally because these fields

are calculated from the second order of PDE for diffusion or

potential. However, there is less noise in potential field than

in concentration field. In potential field, high spatial

frequency wave is eliminated from concentration field by

solving the Poisson equation because the radius of the

long-range part of Coulomb potential by an acceptor has

LWT

VNANF

ox

thxaa

σ)(

LWT

VNANF

ox

thxaa

σ)(

32

wider distribution than that of concentration, as is discussed

later.

There is almost no concentration dependence among

potential fields except for the specific frequency range of

5/100 nm-1 to 4/100 nm-1. This spatial frequency range

corresponds to the wave length of 20 nm to 25 nm.

IV. POTENTIAL FIELD WITH SPECIFIC WAVE

LENGTH RANGE

There are two reasons why FFT is adopted to characterize

fluctuation. One reason is to find what wave length range

characterizes the fluctuation and the other reason is to

reconstruct the original field with the specific wave length

range by applying FFT using a window function.

Fig 6 (a) shows the original surface potential field with

random dopant distribution. Fig. 6 (b) shows the

reconstructed potential field by inverse FFT with waves

above 20 nm wave length only.

Inverse FFT

λ≧20nm

90nm Node L=36nm

W=100nm

(a) Surface Potential Field (Original)

(b) Reconstructed Field with Low

Spatial Frequency Component

S

D

D

S

Na=5 ×1018cm-3

Inverse FFT

λ≧20nm

90nm Node L=36nm

W=100nm

(a) Surface Potential Field (Original)

(b) Reconstructed Field with Low

Spatial Frequency Component

S

D

D

S

Na=5 ×1018cm-3Na=5 ×10

18cm-3

Fig. 6: Original surface potential field and reconstructed

field by inverse FFT with waves above 20 nm wave length.

Comparing Fig. 6 (a) with Fig. 6 (b), the original potential

field is almost reconstructed and the specific wave length

takes a dominant role in fluctuation of Vth. Moreover, the

Poisson equation behaves as a low-pass filter for

concentration field because potential filed doesn’t reflect the

long-range part of individual impurities. Fig. 7 (a) shows

relative acceptor concentrations for the long-range part of

Sano model by an acceptor at the origin. Fig. 7 (b) shows the

case of potential field. As Na is increased, the relative

acceptor concentration increases but the radius shrinks

gradually. However, as for potential field, the peak value at

the origin goes down but the radius stays almost constant in

contrast to the concentration field.

Because potential field is described by overlapping these

single waves, specific wave length is observed by Fourier

analysis, whose range corresponds to the wave length of 20

nm to 25 nm.

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20 25

1e17(cm-3)

1e18(cm-3)

5e18(cm-3)

8e18(cm-3)

1.4e19(cm-3)

2e19(cm-3)

Relative Acceptor Conc. ( )

Radius Distance (nm)

Na

(a) Relative Acceptor Concentration

Shrink

Up

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20 25

1e17(cm-3)

1e18(cm-3)

5e18(cm-3)

8e18(cm-3)

1.4e19(cm-3)

2e19(cm-3)

Relative Acceptor Conc. ( )

Radius Distance (nm)

Na

(a) Relative Acceptor Concentration

Shrink

Up

-80

-70

-60

-50

-40

-30

-20

-10

0

0 5 10 15 20 25

1e17(cm-3)

1e18(cm-3)

5e18(cm-3)

8e18(cm-3)

1.4e19(cm-3)

2e19(cm-3)

Potential (mV)

Radius Distance (nm)

Na

(b) Potential Distribution

Down

Half of Specific

Wave Length

-80

-70

-60

-50

-40

-30

-20

-10

0

0 5 10 15 20 25

1e17(cm-3)

1e18(cm-3)

5e18(cm-3)

8e18(cm-3)

1.4e19(cm-3)

2e19(cm-3)

Potential (mV)

Radius Distance (nm)

Na

(b) Potential Distribution

Down

Half of Specific

Wave Length

Fig. 7: Relative acceptor concentration (a) and its potential

field (b) by an acceptor at the origin.

V. DISCUSSION

Assuming that the specific wave length in the potential

field, there will be a new fluctuation caused by size effect

between wave length and device size (Fig. 8).The potential

field is fluctuated within a device for large scale devices.

Scaling the device size, if Na in the substrate is increased

accordingly, potential is still averaged within a device

(Center). But if the device size becomes smaller than

specific wave length of 20 nm to 25 nm, the fluctuation is

determined by the local potential field directly (Right). In

such case, the frequency of Vth distribution will obey

32

3

)(

)cos()()sin(

2)(

rk

rkrkrkkr

c

cccclong −=

πρ

33

Poisson distribution rather than symmetric normal

distribution. Moreover, this will cause a fundamental

problem to engineers in designing circuits with conventional

3σ criteria because Poisson distribution has broader

distribution even if σ is the same level with that of normal

distribution.

Geometry scaling is aggressive, while electrical properties,

such as unit charge, dielectric, long-range part of Coulomb

potential and so on, are kept almost constant. This unbalance

for scaling causes the size effect.

ρε

φeDopant

Normal dist.

Radius by the

long-range part of

Coulomb potential

Potential

Normal dist. Poisson dist.

Potential

25nm

DopantPoisson Equation

Wave length

(Left)

Averaged

(Right)

Local difference(Center)

Still averaged

Device size

Device size

Scaling Scaling

ρε

φeDopant

Normal dist.

Radius by the

long-range part of

Coulomb potential

Potential

Normal dist. Poisson dist.

Potential

25nm

DopantPoisson Equation

Wave length

(Left)

Averaged

(Right)

Local difference(Center)

Still averaged

Device size

Device size

Scaling Scaling

Fig. 8: Size effect by relative relation between device size

and wave length of the potential fluctuation.

G

Narrow

ShrinkStay

almost

const.

0 50 100 150

90nm Node

65nm Node

45nm Node

32nm Node

32nm Node

32nm Node

⊿Potential (max) (mV)

W=22nm

W=10nm

W=36nm

Maximum Potential Difference in a Device

GG

NarrowNarrow

ShrinkShrinkStay

almost

const.

0 50 100 150

90nm Node

65nm Node

45nm Node

32nm Node

32nm Node

32nm Node

⊿Potential (max) (mV)

W=22nm

W=10nm

W=36nm

0 50 100 150

90nm Node

65nm Node

45nm Node

32nm Node

32nm Node

32nm Node

⊿Potential (max) (mV)

W=22nm

W=10nm

W=36nm

Maximum Potential Difference in a Device

Fig. 9: Scaling rules and maximum potential difference in a

device.

0

5

10

15

20

200 300 400 500

W=72nmW=36nmW=22nmW=10nm

Freq

uenc

y ( )

Vth (mV)

32nm NodeL=12nm

Na=2×1019(cm-3)

Fig. 10: Histogram of Vth Frequency distribution at 32 nm

node with narrowing device.

Fig. 9 shows scaling rules to see the size effect and

maximum potential difference in a device. There is a

remarkable jump at 32 nm node with narrowing width.

Fig. 10 is the frequency of Vth distribution at 32 nm node

for various device widths. Distribution is getting broader and

lower with narrowing device.

Fig. 11 shows “normal probability plot”, in which narrower

width’s curve deviates from others and it is the proof of

departing from symmetric normal distribution. This means

the increase of fluctuation should be notified because of the

broader distribution tail due to Poisson distribution.

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

200 250 300 350 400 450 500

W=72nmW=36nmW=22nmW=10nm

Nor

mal

Dev

iatio

n R

atio

to σ

Vth

( )

Vth (mV)

32nm NodeL=12nm

Na=2×1019(cm-3)

Poisson

Dist.

Fig. 11: Normal probability plot of σVth at 32 nm node with

narrowing device.

VI. CONCLUSION

We have evaluated the device characteristics fluctuation

against the technology roadmap. Absolute value of the

fluctuation will be increased with scaling but concentration

factor will be saturated for high acceptor concentration over

5 x 1018 cm-3. Fourier analysis shows that power of spatial

frequency generally obeys the 1/f2-rule and concentration

dependence is less sensitive in potential filed than in

concentration field.

This work suggests that it becomes hard to fabricate

hundreds million transistors with uniform characteristics in a

chip if device size is close to the specific wave length, 20 nm

to 25nm, of the potential fluctuation especially.

REFERENCES

[1] Taurus-Process Reference Manual, Version W-2004.09,

(Synopsys Inc., Mountain View, CA, 2004).

[2] Unpublished (http://www.selete.co.jp/).

[3] N. Sano et al., IEDM Tech. Dig., pp. 275-278, 2000.

[4] T. Mizuno et al., IEEE Trans. Electron Devices, vol. 41,

pp. 2216-2221, 1994.

[5] A. Asenov et al., IEEE Trans. Electron Devices, vol. 50, pp. 1837-1852, 2003.

34