diffusion - international islamic university...

Diffusion

Some introductory comments...

• Diffusion the redistribution of material in response to a

concentration gradient (more accurately: a chemical

potential gradient)

• The effect of diffusion is usually (but not always!) to

reduce the magnitude of a concentration gradient

• We will view diffusion from two different standpoints:

1. a unit manufacturing process for changing chemical

composition with depth important in modern

semiconductor manufacturing (but somewhat difficult

to achieve submicron geometries)

2. a parasitic effect that accompanies any thermal

process very important in semiconductor

manufacturing (alters desired concentration profiles)

Spin-on doped glass

Solid

Liquid (bubbler)

Gaseous

Ion Implant

Planar Sources

solid

silica tube

Carrier

gasSource

boatwafers

gas

dopant

Heated

bath

liquid

Planar sources

Fick’s First Law

where D is the diffusion coefficient and J is the net flux (cm-2

sec-1) of the diffusing species (note: negative sign shows that a positive flux will decrease the concentration profile)

J DC z t

z

( , ) xy

z

Problem: How do we observe or measure the flux

within a solid material?

Answer: We usually can’t.

Solution: Use a different equation!

Fick’s Second Lawx

y

z

dz

J1

J2

In the volume with length dz,

J

z

J J

z

2 1

J2 = flux leaving

J1 = flux entering

The concentration in the volume element

will be changing with time (J2 J1), and this

can be expressed as:

A J J AdzJ

zAdz

C

t2 1

or

C z t

t

J

z zD

C

z

,

C z t

tD

C z t

z

, ,

2

2

2nd order differential equation needs 2 independent boundary conditions

Analytical solutions of Fick’s Law (2)Case 2: “drive-in diffusion” an initial amount QT is deposited onto

the surface and allowed to diffuse into the bulk

C z C t

d C t

d zC z t QT

, ,

,,

0 0 0

00

0

Boundary conditions:

constant

Solution: C z tQ

Dte tT z Dt, ,

2 4 0

C tQ

Dt

T0,

The surface concentration decreases with time:

Analytical solutions of Fick’s Law (1)

Case 1: constant surface concentration (“pre-deposition diffusion”):

C z

C t C

C t

s

,

,

,

0 0

0

0

Boundary conditions:

in words: the surface concentration

is always Cs; the concentration at

t=0 everywhere or z= anytime = 0

Solution: C z t Cz

DttS, ,

erfc

20

where “efrc u” is the complimentary error function (i.e. 1-erf u) or

erfc u e dxx

u

2 2

“characteristic

diffusion length”Dt

Sidebar -- Useful expressions to the Error Function

erfc u e dxx

u

2 2

ux dxeu

0

22 erf

erfc u = 1 - erf u erf (0) = 0 erf () = 1

12

erf uuu

11

erfc

2

uu

eu

u

Analytical solutions of Fick’s Law (3)In both cases (pre-deposition and drive-in diffusion), the dose is the

total amount of materials diffused into the substrate

Q t C z t dz C t DtT ( ) , ,

02

0

Pre-deposition diffusion:

Drive-in diffusion:

C z t dz QT,0

dose increases with time

dose = constant

Diffusion doping for junction formation

• The junction depth xj is the depth where the concentration of a diffused dopant (n or p) equals the bulk concentration CB (p or n)

• CB will be either the intrinsic carrier concentration ni or the bulk doped concentration, which ever is larger, where

for Si (pre-factor of 4.21014 for GaAs)

1

10-1

10-2

-310

-410

CS

CB

0 xj

CS

C

distance

junction

depth

kTE

igeTn

22315)103.7(

The diffusion coefficient D

• The diffusion coefficient is a strong function of temperature:

where Eio is the activation energy (usually between 1 to 5 eV)

and Dio is a pre-exponential factor ( temperature independent)

that typically ranges from 10-2 to 10+2

• If diffusion occurs by vacancies, then the diffusion coefficient is

the sum of all possible diffusion coefficients, weighted by their

probability of existence:

kTE

ioiioeDD

3

3

2

2

3

3

2

2

Dn

pD

n

pD

n

p

Dn

nD

n

nD

n

nDD

iii

iii

i

Typical diffusion calculations (1)

The problem:

A p+-n junction is made by diffusing boron into silicon with a background n-type concentration of 1016 cm-3. A constant boron surface concentration Cs = 1020 cm-3 is maintained. For boron in silicon Do= 0.76 cm2/s and Ea = 3.46 eV. Calculate the time required to form the junction at a depth of 1.0 mm if the diffusion temperature is 1050°C.

The solution:

Calculate the diffusion coefficient at 1050°C and solve for C(z,t) under pre-deposition conditions

Solution to (1)

1. Calculate D:

o 5

14 2 1

3.46(0.76)exp

8.617 10 1050 273

5 10 cm s

aE

kTD D e

2. Determine depth where n = p:

416 20

14

4

10( , ) erfc 10 10 erfc

2 2 (5 10 )

223.610 erfc 6612

s

zC z t C

Dt t

tt

seconds (~1 hr 50 min)

For the exclusive use of adopters of the book Introduction to

Microelectronic Fabrication, Second Edition by Richard C. Jaeger. ISBN0-

201-44494-1.

© 2002 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all

copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means,

without permission in writing from the publisher.

DiffusionProfile Comparison

Complementary Error Function and

Gaussian Profiles are Similar in Shape

erfc z 1 erf z

erf z 2

exp x 2 dx

0

z

Typical diffusion calculations (2)The problem:A two-step boron diffusion is performed into n-type silicon with an impurity concentration of 1016 cm-3. The pre-deposition diffusion is conducted at 950°C for 15 minutes with a surface concentration of Cs = 1021 cm-3. The drive-in temperature is 1100°C. 1. How long should the drive-in last to form a junction 2.0 mm deep?2. What is the impurity concentration at the surface after the drive-in diffusion?The solution:• Calculate the diffusion coefficients at 950°C and 1100°C • Determine the pre-deposition dose QT(t)• Use this dose as input to the drive-in diffusion time calculation

Solution to (2)1. Calculate D’s:

o 5

15 2 1 o 13 2 1 o

3.46(0.76)exp

8.617 10 950 273

4 10 cm s (950 C); 1.5 10 cm s (1100 C)

aE

kTD D e

2. Determine the pre-deposition dose:

21 15

15 2

2 2pre-dep. 0, 10 (4 10 )(15 60)

2.14 10 cm

TQ C t Dt

3. Calculate time for the drive-in diffusion profile to reach the background:

22 2

4

2

15 2 4 219

16 2 13 13

2( , ) ln ln

4( , )

(2.14 10 ) 2(2 10 ) 1ln ln 8200s; 3.44 10

(10 ) (1.5 10 ) 4(1.5 10 )

z DtT T

S

Q Q zC z t e t

DtDt C z t D

t t Ct

Typical diffusion calculations (3)

The problem:

At high dopant concentrations the diffusion process may be altered by the effect of doping on the Fermi level position. What is the diffusion coefficient of arsenic in silicon doped with 11020 As/cm3 at 1400K?

The solution:

Calculate the diffusion coefficient at 1050°C and solve for C(z,t) under pre-deposition conditions

Solution to (3)1. Determine mechanism: according to Table 3.2, As diffuses according to both

uncharged (Di) and –1 (D- )charged vacancies. Using the data in the Table,

the diffusion coefficient is written as:

3.44 4.05(0.066) (12.0)kT kT

As i

i i

n nD D D e e

n n

2. Determine the effect of doping on the bandgap and hence ni:

5

2 4 2

0

23 2 15 3 2 0.715 2(8.617 10 )(1400) 19

0

2014 14

19

14 2 1

(4.73 10 )(1400)1.17 0.715 eV

636 1400

(7.3 10 )(1400) 2 10

1 105 (2.7 10 ) (5 3.2 10 )

2 10

1.9 10 cm s

g

g g

E kT

i i

As

i

TE E

T

n n T e e

nD

n



201-44494-1.




Diffusion CalculationExample - Boron Diffusion

• A boron diffusion is used to form the base region of an npn

transistor in a 0.18 W-cm n-type silicon wafer. A solid-

solubility-limited boron predeposition is performed at 900o

C for 15 min followed by a 5-hr drive-in at 1100oC. Find

the surface concentration and junction depth (a) after the

predep step and (b) after the drive-in step.



201-44494-1.




Diffusion CalculationExample - Boron Diffusion

Predeposition step is solid - solubility limited.

T1 = 900oC 1173KNO 1.1x1020/cm3

D1 10.5exp 3.69eV

8.614x105eV /K 1173K

1.45x1015cm2 /sec

t1 15 min 900 sec D1t1 1.31x1012cm2

N x 1.1x1020erfcx

2.28x106cm

/cm

3

Dose : Q 2NOD1t1

1.42x1014 /cm2

T2 =1100oC 1373K

D2 10.5exp 3.69eV

8.614x105eV /K 1373K

2.96x1013cm2 /sec

t2 5 hr 18000 sec D2t2 5.33x109cm2

N2 x 1.42x1014 /cm2

5.33x109cm2 exp

x

2 D2t2

2

1.1x1018 exp x

1.46x104

2

/cm3



201-44494-1.




Diffusion CalculationExample (cont.)

N1 x 1.1x1020erfcx

2.28x106cm

/cm

3

x j1 2 D1t1erfc1 No

NB

2.28x106cm erfc1 3x1016

1.1x1020

2.28x106cm erfc1 2.73x104

x j1 2.28x106cm 2.57 5.86x106cm 0.058

N2 x 1.1x1018 exp x

1.46x104

2

/cm3

x j2 1.46x104cm ln1.1x1018

3x1016

2.77x104cm 2.77mm



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Diffusion CalculationExample (cont.)

Starting Wafer : n - type 0.18 W cm

n - type 0.18 W cm ND 3 x 1016/cm3



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Two Step Diffusion

• Short constant source diffusion used to

establish dose Q (“Predep” step)

• Longer limited source diffusion drives

profile in to desired depth (“drive in”

step)

• Final profile is Gaussian



201-44494-1.




Sheet Resistance

Irvin’s Curves

• Irvin Evaluated this Integral and Published a Set of Normalized Curves Plot Surface Concentration Versus Average Resistivity

• Four Sets of Curves

– n-type and p-type

– Gaussian and erfc

1

1

1

x j x dx

0

x j

RS

x j

1

x dx0

x j

RS qmN x dx0

x j

1

RSx j



201-44494-1.




Sheet Resistance

Irvin’s Curves (cont.)



201-44494-1.




Two Step DiffusionSheet Resistance - Predep Step

Initial Profile

No 1.1x1020 /cm3

NB 3x1016 /cm3

x j 0.0587 mm

p type erfc profile

Square/ 850 0587.0

- 32

- 32

WW

W

m

mR

mxR

S

jS

m

m

m



201-44494-1.




Two Step DiffusionSheet Resistance - Drive-in Step

Final Profile

No 1.1x1018 /cm3

NB 3x1016 /cm3

x j 2.73 mm

p type Gaussian profile

RSx j 700 W -mm

RS 700 W -mm

2.73 mm 260 W /Square

Some basic concepts in diffusion

mechanisms

• Diffusion is a thermally activated process -- energy is required to

displace an atom from a given site in a crystal

• Impurities can occupy either interstitial or substitutional sites

• Interstitial impurities

– atoms that don’t bond readily with host atoms

– can diffuse rapidly via interstitial sites

– do not contribute to doping (electrically inactive)

• Substitutional impurities

– bond with the host atoms

– slower diffusion

– contribute to doping (electrically active)

Atomistic models of diffusion (1)

Direct exchange -- requires multiple bonds to be broken

for the host and impurity atoms to exchange


Vacancy exchange -- requires fewer bonds to be broken

and is a dominant mechanism for substitutional impurities

but

the mechanism requires the presence of vacancies


Interstitialcy mechanism -- a host-atom interstitial

displaces the impurity, which can then diffuse as an

interstitial to another lattice site; occurs simultaneously

with vacancy diffusion


Mechanisms for fast-diffusing impurities to reincorporate

into the lattice:

Kick-out mechanism -- direct replacement of a host atom

Frank-Turnbull mechanism -- impurity combines with a

host atom vacancy

diffusion - international islamic university...

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