I 6 Diffusion

7/27/2019 I 6 Diffusion

http://slidepdf.com/reader/full/i-6-diffusion 1/30

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-y2dy erfc (z) ≡≡ 1 - erf (z)

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

erf (z) ≈ 2π

z for z <<1 erfc (z) ≈ 1π

e-z2

zfor z >>1

d erf(z)

dz= -

d erfc(z)

dz=

2

πe

2-z

d2

erf(z)

dz2 = -

4

πz e

2-z

⌡⌠ 0

z

erfc(y)dy = z erfc(z) + 1π

(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 zand 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 gas

or seal surface with oxideSiO2

SiO2

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 Source BN 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][

:

,,

2cm D

constant diffusion D x

t xC Dt x J

=

⋅−=

∂

∂

J

C(x)

x




Using the Continuity Equation

( )

Equation Diffusion

x

C D

x x

J

t

C

t x J t

t xC

=−=⇒

=⋅∇+

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂0,

,




If D is independent of C

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

( ) ( )∂

∂

∂

∂

C x t

t D C x t

x

, ,=2

2

Concentration Independent Diffusion Equation




Temperature Dependence of D

.,

/106.8

0

5

0

tabulated are E D

kelvineV

constant Boltzmank

eV inenergyactivation E

e D D

A

A

kT A E

−

−

×=

==

=




kT E

O

A

e D D −

=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

2

00

2

2

21,

2

C Dt

Dt

x erfc C

dy e C t x C Dt

x

y

ππ

C 0

t 3>t 2

t 2>t 1

x=0

xt 1

Characteristic distance for diffusion.

Surface Concentration (solid solubility limit)




( ) ( )

Dt x

e Dt

Co

x

C

t Dt C

dxt xC t Q

42

2

,

0

0

−−=

∝⋅

=

= ∫ ∞

π∂

∂

π

[1] Predeposition dose

[2] Conc. gradient




B. Drive-in Profile

( )

( )( )

⋅==

•

=

=∞=•

=

Dt x erfc Co t x C

Conditions Initial

X

C

t x C

Conditions Boundary

x

20,

:

0

0,

:

0∂∂

∂∂

t

C(x)

x=0

Predep’s (Dt)




( ) ( ) xQt xC δ⋅≈= 0,

C(x)

At t=0

x

Shallow Predep Approximation:

( ) ( )

( )

C x t

Q

Dt edrive in

x Dt

drive in, = −

−

−π

2

4C(x)

x

t 1 t 2

( )

Q

C Dt predep

=

⋅0 2

π

Solution of Drive-in Profile :




indrive

predep

Dt

Dt R

−

=

x

Approximation over-estimates conc. here

Approximation

under-estimates

conc here

Good

agreement

C(x)/C 0

R=1

R=0.25

Exact solution

Delta function

Approximation

How good is the δδ(x) approximation ?




Summary of Predep + Drive-in

( ) 22

2

42

1

22

110

2

2

1

1

2 t D x

et D

t DC xC

t

D

t

D

−

=

==

==

π

Diffusivity at Predep temperature

Predep 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



EE143 Lecture #9Professor Nathan Cheung, U.C. Berkeley24

The exact solutions with∂C

∂x

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

can be constructed by adding another full gaus sian placed at -R p [Method of

Images].

C(x, t) =φ

2π (∆R 2

p+ 2Dt)1/2

⋅ [e-

(x - R p

)2

2 (∆R 2

p+ 2Dt) + e

-(x + R

p)2

2 (∆R 2

p+ 2Dt) ]

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

C(x,t) → φe

- x2 /4Dt

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

Diffusion of Gaussian Implantation Profile (arbitrary R p)




The Thermal Budget

Dopants will redistribute when subjected to various thermal cycles of IC processing steps. If the diffusion constants at each step are independent of

dopant concentration, the diffusion equation can be written as :

∂C

∂t= D(t)

∂2C

∂x2

Let β (t) ≡ ⌡⌠

0

tD(t’)dt’

∴ D(t) =∂ β∂t

Using∂C

∂t =∂C

∂β •∂ β∂t

The diffusion equation becomes:∂C

∂β •∂ β∂t

=∂ β∂t

•∂2C

∂x2or

∂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.

Example

Consider 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 to be minimized and we need low thermal budget processes.




T(t)

time

∑=i

ieffective Dt Dt

Budget Thermal

)()(

well

drive-in

step

S/D

Anneal

step

* 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) ,

x j

(junction depth),

NB

(background concentration),

R S

(sheet resistance),

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

Irvin’s Curves

p-type Erfcn-type Erfc

p-type half-gaussian

n-type half-gaussian

* 4 sets of curves

See 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=x j 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 R S can be integrated numerically from:

4) The Irvin’s Curves are plots of No versus ( R s• x j ) for various NB.

( ) ( )[ ]∫ −⋅=

j x B dx N xC xq

Rs

0

1

µ

Motivation to generate the Irvin’s CurvesBoth NB(4-point-probe), R S (4-point probe) and x j (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,x j, and R S

Documents

I 6 Diffusion