DiffusionDiffusion is a process of mass transport that involves the movement of

one atomic species into another. It occurs by random atomic jumps

from one position to another and takes place in the gaseous, liquid,

and solid state for all classes of materials.

partial mixing homogenization

time

water

adding dye

What is Diffusion? Diffusion is material transport by atomic

motion. Inhomogeneous materials can become homogeneous by

diffusion. For an active diffusion to occur, the temperature should

be high enough to overcome energy barriers to atomic motion.

Diffusion Mechanisms. There are two main mechanisms of diffusion

of atoms in a crystalline lattice:

•the vacancy or substitutional mechanism

•the interstitial mechanismAtoms move from

concentrated regions to

less concentrated regions.

Vacancy diffusion.

To jump from lattice site to

lattice site, atoms need

energy to break bonds with

neighbors, and to cause the

necessary lattice distortions

during jump. This energy

comes from the thermal

energy of atomic vibrations

(Eav ~ kT).

Interstitial diffusion:

Interstitial diffusion is

generally faster than

vacancy diffusion because

bonding of interstitials to

the surrounding atoms is

normally weaker and there

are many more interstitial

sites than vacancy sites to

jump to.

Requires small impurity

atoms (e.g. C, H, O) to fit

into interstices in host.

Materials flow (the atom) is opposite the vacancy flow direction.

Generation of Point Defects

Point defects are caused by:

1. Thermal energy

)](exp[kT

EC

n

nX defect

site

defect

defect −==

kTECX defectdefect /ln]ln[ −=

Ln[X]

1/T

Edefect/k

*

Example

If, at 400oC, the concentration of vacancies in aluminum is 2.3 x 10-5,

what is the excess concentration of vacancies if the aluminum is

quenched from 600oC to room temperature? What is the number of

vacancies in one cubic µm of quenched aluminum?

Given, Es = 0.62 eV

k = 86.2 x 10-6 eV/K,

rAl = 0.143 nm

Diffusion Flux

The flux of diffusing atoms, J, is used to quantify how fast diffusion

occurs. The flux is defined as either in number of atoms diffusing

through unit area and per unit time (e.g., atoms/m2-second) or in terms

of the mass flux - mass of atoms diffusing through unit area per unit

time, (e.g., kg/m2-second).

At

MJ =

t

M

AJ

δ

δ1=

(Kg m-2 s-1); where M is the mass of

atoms diffusing through the area A

during time t.

Area Ain out

Steady-State Diffusion

Flux is proportional to the concentration gradient and the diffusion coefficient, D (m2/s), by Fick’s first law:

x

C

δ

δ

•Negative sign indicates direction of

gradient

•It is the “driving force”

•[m2/s (kg/m3)/m] = kg/(m2 As)

x

CDJ

δ

δ−=

x

C

δ

δ

Flux does not change with time

Concentration profile – concentration gradient is

maintained constant.

Concentration is expressed in terms of mass of diffusing

species per unit volume of solid (kg/m3)

AB

AB

xx

cc

x

c

−

−=

δ

δ

Fick’s First Law of Diffusionx

CCDJor

dx

dCDJ

∆

−=−= 21

Where

J: the number of atom diffusing down the concentration gradient per second per unit area, unit: atoms/cm2⋅s

C: the concentration of molecules (or the number of diffused molecules per unit volume), unit: atoms/cm3

x: atomic jump distance

D: diffusion coefficient, unit: cm2/s

Ji units[ ] =g

s ⋅ cm2

J i units?[ ] = Dcm2

s

⋅

∂C

∂x

g

cm4

, i = x, y, z( )

Example: (Fick’s 1st Law) : A thin plate of BCC Fe, T=1000K

carbon concentration:

C1=0.2wt%; C2=0%

CO/CO2

Oxidizing

atmosphereFe

t=0.1cm

Density of Fe: ρ = 7.9g/cm3

D = 8.9×10-7 cm2/s at 1000K

Calculate: the number of carbon atoms transport to

back surface per second through an area of 1cm2

Solution:

The concentration of carbon (atoms/cm3): A

C

Fe NA

wtC ⋅

⋅=

ρρρρ%

scmatoms

cm

cmatomsscm

t

CCD

dx

dCDJ

C

cmatoms

molatomsmolg

cmgC

⋅×=

××=

−=−=

=

×=

×⋅×

=

−

215

3202721

2

320

233

1

/109.6

1.0

/1092.7/107.8

0

/1092.7

/10023.6/01.12

/9.7%2.0

Diffusivity -- the proportionality constant between flux and concentration gradient depends on:

� Type of bonding

� Diffusion mechanism. Substitutional vs interstitial.

� Temperature.

� Type of crystal structure of the host lattice. Interstitial diffusion easier in BCC

than in FCC.

� Type of crystal imperfections.

(a) Diffusion takes place faster along grain boundaries than elsewhere in

a crystal.

(b) Diffusion is faster along dislocation lines than through bulk crystal.

(c) Excess vacancies will enhance diffusion.

� Concentration of diffusing species.

Diffusion coefficient D

depends on the temperature

RT

Q

o

d

eDD−

=

T R

Q - D = D d

olnln

D is the Diffusivity or Diffusion Coefficient

(m2 / sec )

Dois the prexponential factor or Diffusion

constant (m2 / sec )

Qdis the activation energy for diffusion (joules /

mole )

R is the gas constant ( joules / (mole deg) )

T is the absolute temperature ( K in Kelvin )

– Q/R

Non Steady State Diffusion

Diffusion flux and the concentration gradient at some particular point in a solid vary

with time, with a net accumulation of depletion of the diffusing species resulting

•Fick’s second law apples (when D is independent of composition)

Fick’s 2nd Law

Chigh Clow

dxdA

Jin Jout

dV=dA⋅dx

dAJJdVt

Coutin )( −=

∂

∂

Fick’s 2nd Law:

The rate of change of the number of

atoms in the slice dVThe rate that atoms entering the slice –the rate that atoms leaving the slice

=

2

2

)(

x

CD

x

CD

x

x

J

dV

dAJJ

t

Coutin

∂

∂=

∂

∂−

∂

∂−=

∂

∂−=−=

∂

∂

⇒2

2

x

CD

t

C

∂

∂=

∂

∂

In words: The rate of change of composition at

position x with time, t, is equal to the rate of

change of the product of the diffusivity, D, times

the rate of change of the concentration gradient,

dCx/dx, with respect to distance, x.

2

2

x

CD

t

C

∂∂∂∂

∂∂∂∂====

∂∂∂∂

∂∂∂∂

�Solutions to the DE are possible when physically meaningful boundary conditions are specified

�Particularly important solution – semi-infinite solid in which surface concentrations are constant, diffusing species is usually a gas, and the partial pressure is maintained at a constant value

Second order differential equations are nontrivial and difficult to solve.

Consider diffusion in from a surface where the concentration of diffusing species

at the surface is always constant. This solution applies to gas diffusion into a solid

as in carburization of steels or doping of semiconductors.

Boundary Conditions

• For t = 0, C = Co

at 0 < x

• For t > 0 C = CS

at x = 0

and C = Co

at x =∞∞∞∞

Dt2

x erf - 1 =

C - C

C - C

os

ox

where

CS = surface concentration

Co = initial uniform bulk concentration

Cx = concentration of element at distance x from surface at time t

x = distance from surface

D = diffusivity of diffusing species in host lattice

t = time

erf = error function = erf (x/2ooooDt) is the Gaussian error function – this is like a

continuous probability density function from 0 to x/2ooooDt

�The equation below demonstrates the relationship between concentration, position, and time

� Cx being a function of the dimensionless parameter x/2ooooDt may be determined at any time and position if the parametes Co, Cx, and D are known

Dt2

x erf- 1 =

C- C

C- C

os

ox

Special Case

Desired to achieve some specific

concentration of solute, C1in an alloy,

then

constant= C- C

C- C

os

ox

constant=Dt2

x

Example

The carburization of a steel gear at a temperature of 1000oC in gaseous CO/CO2

mixture, took 10hours. How long will take to carburize the steel gear to attain

similar concentration conditions at 1200oC?

For C in γγγγ – iron D = 0.2 exp{ - 34000 / 2T} cm2/s

Example: (Fick’s 2nd Law)

Determine the time it takes to obtain a carbon concentration of 0.24% at depth 0.01cm beneath the surface of an iron bar at 1000oC. The initial concentration of carbon in the iron bar is 0.20% and the surface concentration is maintained at 0.40%.

The Fe has FCC structure and the diffusion coefficient is

D = 2××××10-5 m2/s ⋅⋅⋅⋅exp( ).

Known: T=1000oC, depth x = 0.01cm, CX = 0.24%

CO = 0.2%, CS = 0.4%

D=2××××10-5 m2/s ⋅⋅⋅⋅exp( )

R = 8.314 J/K

Find: time t = ?

RT

molJ /000,142−

RT

molJ /000,142−

Solution:

D1273K = 2×10-5 m2/s ⋅exp

D1273K = 2.98 ×10-11 m2/s= 2.98 ×10-7 cm2/s

)1273314.8

000,142(

×−

−===

−

−=

−

−

Dt

xerf

CC

CC

OS

OX

212.0

2.0

04.0

2.04.0

2.024.0

⇒ erf(z) = 0.8, where z = Dt

x

2erf(z) = 0.8

12

1

12

1

)()(

)()(

zz

zz

zerfzerf

zerfzerf

−

−=

−

−

90.095.0

90.0

7970.08209.0

797.08.0

−

−=

−

− z

⇒ z = 0.906 ⇒ = 0.906Dt

x

2

⇒ t = [x / (2 × 0.906)]2/D = .min73.11041098.2

)812.1/01.0(7

2

==× −

s

t = 1.73min.

Effective penetration distance: xeff

(for 50% of concentration)

5.02/)(2

,2

),(

0

0

0

00

0

0

0

=−

−=

−

−+

=−

−

+=

CC

CC

CC

CCC

CC

CC

CCtxC

s

s

s

s

s

seff

Fick’s 2nd Law: )2(15.0

0

0

Dt

xerf

CC

CC eff

s

−==−

−

erf (0.5) ≈ 0.5 ⇒ xeff ≈≈≈≈ Dt

Effective penetration distance

In general, for most diffusion problems

xeff =

where γγγγ: a geometry-dependent parameter

γγγγ = 1 for a flat plate

γγγγ = 2 for cylinders

Dtγγγγ

Thermal Diffusion of Impurities into Silicon

The ability to modify the properties of a semiconductor through the

addition of controlled amounts of impurity atoms is an important aspect of

silicon device and IC manufacture.

There are two principal methods which are used to introduce impurities

into silicon, thermal diffusion and ion implantation.

We will discuss the basic equations describing the impurity profiles below

the surface of the wafer using the thermal diffusion method.

Thermal diffusion is a high temperature process where the dopant atoms

are deposited on to or near the surface of the wafer from the gas phase.

Wafers can be batch-processed in furnaces. The impurity profile or

distribution is determined mainly by the diffusion temperature and time,

and decreases monotonically from the surface. The maximum

concentration of a particular diffusing impurity is always found at the

surface.

The impurity concentration C(x,t) as a function of depth below the wafer surface, x,

and diffusion time, t is determined from Fick's diffusion law;

D is the diffusion coefficient and varies markedly from one impurity to another;

some impurities diffuse quickly through silicon (fast diffusants), while others move

more slowly (slow diffusants). of impurities in silicon.

D depends on the temperature of diffusion and can be expressed in the generalized

form as D(T) = Do exp(-EA / kBT) where Do is the diffusion coefficient extrapolated

to infinite temperature and EA is an activation energy (usually quoted in eV).

Thus, a plot of log D(T) (µm2 / hr) vs 1/T (K-1) will give a straight line with slope

EA.

Diffusion:

Smaller atoms diffuse more readily than big ones, and diffusion is

faster in open lattices or in open directions

Self-diffusion coefficients

for Ag depend on the

diffusion path. In general

the diffusivity if greater

through less restrictive

structural regions – grain

boundaries, dislocation

cores, external surfaces.

Example

(A)For an ASTM grain size of 6, approximately how many grains would there

be per square inch at a magnification of 100?

(B)The diffusion coefficients for copper in aluminum at 500 and 600oC are

4.8x10-14 and 5.3x10-13 m2s-1, respectively. Determine the approximate

time at 500oC that will produce the same diffusion results (in terms of

concentration of Cu at some specific point in Al) as a 10 hour heat

treatment at 600oC.

(C) For the problem (B) compute the activation energy for the diffusion of Cu

in Al.

(A) This problem asks that we compute the number of grains per square inch for

an ASTM grain size of 6 at a magnification of 100x. All we need do is solve for

the parameter N in the equation below, inasmuch as n = 6. Thus

N = 2n−1

= 26 −1

= 32 grains/in2

(B) Fick’s second law, as it is desired to achieve some specific concentration

conditions.

( )( )( )

hourssmx

hourssmx

D

tDt

tDtD

tconsDt

4.110108.4

10.103.5

tan

1214

1213

500

600600

500

600600500500

===

=

=

−−

−−

(C) Using the equation

−−

−

−

−

−

−−

===

==

500600

500

600

500

600

500600

500

600

500600 and

RT

Q

RT

Q

RT

Q

RT

Q

RT

Q

o

RT

Q

o

RT

Q

o

RT

Q

o

dd

d

d

d

d

dd

e

e

e

eD

eD

D

D

eDDeDD

( ) ( )

( ) ( )( )

( ) ( ) ( )[ ]

1

1213121411

500600

600500

500600500600

500600

500

600

.7.134

773

1

873

1

.103.5ln.108.4ln..31.8

11

lnln

11lnlnln

−

−−−−−−

=

−

−=

−

−=

−−=

−−=−=

molkJQ

KK

smxsmxKmolJQ

TT

DDRQ

TTR

Q

RT

Q

RT

QDD

D

D

d

d

d

ddd