Properties of Error Function erf (z) And …jmk/papers/ERF01.pdfProfessor Nathan Cheung, U.C....

EE143 Lecture #9Professor Nathan Cheung, U.C. Berkeley 1

Properties of Error Function erf(z)And Complementary Error Function erfc(z)

erf (z) =2

ππ ⌡⌠ 0

z

e-y2 dy erfc (z) ≡≡ 1 - erf (z)

erf (0) = 0 erf( ∞) = 1 erf(-∞ ) = - 1

erf (z) ≈ 2

π z for z <<1 erfc (z) ≈

1

π e-z2

z for z >>1

d erf(z)

dz = - d erfc(z)

dz = 2

π e

2-z

d2 erf(z)dz2 = -

4

π z e

2-z

⌡⌠ 0

z

erfc(y)dy = z erfc(z) + 1

π (1-e-z2

) ⌡⌠ 0

∞

erfc(z)dz = 1

π


The value of erf(z) can be found in mathematical tables, as build-in functions in calculators andspread sheets. If you have a programmable calculator, you may find the following approximation useful (it is

accurate to 1 part in 107): erf(z) = 1 - (a1T + a2T2 +a3T 3 +a 4T4 +a5T 5) e-z2

where T = 1

1+P z and P = 0.3275911

a1 = 0.254829592 a2 = -0.284496736 a3 = 1.421413741 a4 = -1.453152027 a5 = 1.061405429

z

0.0000001

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6

erfc(z)

exp(-z^2)


Dopant Diffusion

(1) Predeposition dopant gas

SiO2SiO2

Si

* dose control

(2) Drive-in

Turn off dopant gasor seal surface with oxideSiO2SiO2

Si

SiO2

Doped Si region

* profile control(junction depth;concentration)

Note: Dopant predeposition by diffusion can also be replaced a shallow implantation step


Dopant Diffusion Sources

(a) Gas Source: AsH3, PH3, B2H6

(b) Solid SourceBN Si BN Si

SiO2

(c) Spin-on-glass SiO2+dopant oxide


(d) Liquid Source.


Diffusion Mechanisms

(a) Interstitial (b) Substitutional


Diffusion Mechanisms : ( c) Interstitialcy


Diffusion Mechanisms : (d) Kick-Out, (e) Frank Turnbull


Mathematics of Diffusion

Fick’s First Law:

( ) ( )

sec][

:

,,

2cmD

constantdiffusionDx

txCDtxJ

=

⋅−=∂

∂

J

C(x)

x


Using the Continuity Equation

( ) ( )

EquationDiffusion

x

CD

xx

J

t

C

txJt

txC

=−=⇒

=⋅∇+

∂∂

∂∂

∂∂

∂∂

∂∂

0,,


If D is independent of C (i.e., C is independent of x).

( ) ( )∂∂

∂∂

C x t

tD

C x t

x

, ,=

2

2

Concentration Independent Diffusion Equation


Temperature Dependence of D

.,

/106.8

0

5

0

tabulatedareED

kelvineV

constantBoltzmank

eVinenergyactivationE

eDD

A

A

kTAE

−

−

×=

=

=

=


kTE

O

A

eDD−

=Diffusion Coefficients of Impurities in Si


A. Predeposition Diffusion Profile

( )( )

( )

•

•

= = =

= ∞ =

= =

Boundary Conditions

Initial Condition

C x t C solid solubility of the dopant

C x t

C x t

:

:

,

,

,

0

0

0 0

0


Solid Solubility of Common Impurities in Si


( )

≡=

⋅=

−⋅= ∫ −

0

0

200

2

2

21,

2

C

Dt

Dt

xerfcC

dyeCtxC Dt

xy

ππ

C0

t3>t2

t2>t1

x=0x

t1

Characteristic distance for diffusion.

Surface Concentration (solid solubility limit)


( ) ( )

Dtx

eDt

Co

x

C

tDtC

dxtxCtQ

42

2

,

0

0

−−=

∝⋅

=

= ∫∞

π∂∂

π

[1] Predeposition dose

[2] Conc. gradient


B. Drive-in Profile

( )

( )( )

⋅==

•

=

=∞=•

=

Dt

xerfcCotxC

ConditionsInitial

XC

txC

ConditionsBoundary

x

20,

:

0

0,

:

0∂∂∂∂

t

C(x)

x=0

Predep’s (Dt)


( ) ( )xQtxC δ⋅≈= 0,

C(x)

At t=0

x

Shallow Predep Approximation:

( )( )

( )C x tQ

Dte

drive in

xDt drive in, =

−

−−

π

2

4C(x)

x

t1t2

( )Q

C Dtpredep=

⋅0 2

π

Solution of Drive-in Profile :


indrive

predep

Dt

DtR

−

=

x

Approximation over-estimates conc. here

Approximationunder-estimatesconc here

Goodagreement

C(x)/C0

R=1

R=0.25

Exact solution

Delta functionApproximation

How good is the δδ(x) approximation ?


Summary of Predep + Drive-in

( ) 22

2

42

1

22

110

2

2

1

1

2 tDx

etD

tDCxC

t

D

t

D

−

=

==

==

π

Diffusivity at Predep temperaturePredep time

Diffusivity at Drive-in temperature

Drive-in time


Semilog Plots of normalized Concentration versus depth

Predep Drive-in


Diffusion of Gaussian Implantation Profile

Note: φ is the implantation dose


The exact solutions with ∂C∂x

= 0 at x = 0 (.i.e. no dopant loss through surface)

can be constructed by adding another full gaussian placed at -Rp [Method of

Images].

C(x, t) = φ

2π (∆R2p + 2Dt)1/2

⋅ [e-

(x - Rp)2

2 (∆R2p + 2Dt) + e

- (x + Rp)2

2 (∆R2p + 2Dt) ]

We can see that in the limit (Dt)1/2 >> Rp and ∆Rp ,

C(x,t) → φe

- x2 /4Dt

(πDt)1/2 (the half-gaussian drive-in solution)

Diffusion of Gaussian Implantation Profile (arbitrary Rp)


The Thermal Budget

Dopants will redistribute when subjected to various thermal cycles of ICprocessing steps. If the diffusion constants at each step are independent ofdopant concentration, the diffusion equation can be written as :

∂C∂t

= D(t) ∂2C

∂x2

Let β (t) ≡ ⌡⌠

0

t D(t’)dt’

∴ D(t) = ∂ β∂t

Using ∂C∂t

= ∂C∂β •

∂ β∂t

The diffusion equation becomes: ∂C∂β •

∂ β∂t

= ∂ β∂t

•∂2C

∂x2 or ∂C∂β =

∂2C

∂x2


When we compare that to a standard diffusion equation with D being time-

independent: ∂C

∂ (Dt) = ∂2C

∂x2, we can see that replacing the (Dt) product in

the standard solution by β will also satisfy the time-dependent D diffusionequation.

ExampleConsider a series of high-temperature processing cycles at { temperature T1,time duration t1} ,{ temperature T2, time duration t2}, etc. The

corresponding diffusion constants will be D1, D2,... . Then, β = D1t1+D2t2+..... = (Dt)effective

** The sum of Dt products is sometimes referred to as the “thermal budget”of the process. For small dimension IC devices, dopant redistribution has tobe minimized and we need low thermal budget processes.


T(t)

time

∑=i

ieffective DtDt

BudgetThermal

)()(

welldrive-instep

S/DAnnealstep

* For a complete process flow, only those steps with high Dt values are important

Examples: Well drive-in and S/D annealing steps


Establish the explicit relationship between:

No

(surface concentration) ,

xj(junction depth),

NB

(background concentration),

RS

(sheet resistance),

Once any three parameters are known, the fourth one can be determined.

Irvin’s Curves

p-type Erfc n-type Erfcp-type half-gaussiann-type half-gaussian

* 4 sets of curvesSee Jaeger text


Approach

1) The dopant profile (erfc or half-gaussian )can be uniquely determined if one knows the concentration values at two depth positions.

2) We will use the concentration values No at x=0 and NB at x=xj to determine the profile C(x).(i.e., we can determine the Dt value)

3) Once the profile C(x) is known, the sheet resistance RS can be integrated numerically from:

4) The Irvin’s Curves are plots of No versus ( Rs• xj ) for various NB.

( ) ( )[ ]∫ −⋅=

jx

B dxNxCxqRs

0

1

µ

Motivation to generate the Irvin’s CurvesBoth NB(4-point-probe), RS (4-point probe) and xj (junction staining) can be conveniently measured experimentally but not No (requires secondary ion mass spectrometry). However, these four parameters are related.


Figure illustrating the relationship of No,NB,xj, and RS

Properties of Error Function erf (z) And …jmk/papers/ERF01.pdfProfessor Nathan Cheung, U.C....

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Transcript of Properties of Error Function erf (z) And …jmk/papers/ERF01.pdfProfessor Nathan Cheung, U.C....