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 mechanism

Atoms 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).

Materials flow (the atom) is opposite the vacancy flow

direction.

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.

Generation of Point Defects

Point defects are caused by:

1. Thermal energy

)](exp[ kTEC

nn

X defect

site

defectdefect −==

kTECX defectdefect /ln]ln[ −=

Ln[X]

1/T

Edefect/k

*

ExampleIf, 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 termsof the mass flux - mass of atoms diffusing through unit area per unit time, (e.g.,kg/m2-second).

AtMJ =

tM

AJ

δδ1

= (Kg m-2 s-1); where M is the mass of atoms diffusing through the area A during time t.

Area Ain

outSteady-State Diffusion

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

xC

δδ

•Negative sign indicates direction of gradient•It is the “driving force”•[m2/s (kg/m3)/m] = kg/(m2 s)

xCDJ

δδ

−=

Flux does not change with timeConcentration 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

xxcc

xc

−−

=δδ

Fick’s First Law of DiffusionxCCDJor

dxdCDJ

Δ−

=−= 21

Where J: the number of atom diffusing down the concentration gradient per second per

unit area, unit: atoms/cm2⋅sC: the concentration of molecules (or the number of diffused molecules per unit

volume), unit: atoms/cm3

x: atomic jump distanceD: diffusion coefficient, unit: cm2/s

Ji units[ ] =g

s ⋅ cm 2Ji units?[ ] = Dcm2

s⎡

⎣ ⎢

⎤

⎦ ⎥ ⋅

∂C∂x

⎛ ⎝

⎞ ⎠

gcm 4

⎡ ⎣ ⎢

⎤ ⎦ ⎥ , 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 atmosphere

Fe

t=0.1cmCalculate: the number of carbon atoms transport to back surface per second through an area of 1cm2

Solution:The concentration of carbon (atoms/cm3):

Density of Fe: ρ = 7.9g/cm3

D = 8.9×10-7 cm2/s at 1000K

AC

Fe NA

wtC ⋅⋅

=ρ%

scmatomscm

cmatomsscmt

CCDdxdCDJ

Ccmatoms

molatomsmolg

cmgC

⋅×=

××=

−=−=

=×=

×⋅×

=

−

215

3202721

2

320

233

1

/109.61.0

/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 Ddepends on the temperature

RTQ

o

d

eDD−

=

T RQ - D = D d

olnln

D is the Diffusivity or Diffusion Coefficient (m2 / sec )Do is the prexponential factor or Diffusion constant (m2 / sec )Qd is 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 DiffusionDiffusion 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

Clow

dx

Chigh

dA

Jin Jout

dV=dA⋅dx

dAJJdVtC

outin )( −=∂∂

Fick’s 2nd Law:

The rate of change of the number of atoms in the slice dV

The rate that atoms entering the slice –the rate that atoms leaving the slice

=

2

2

)(

xCD

xCD

x

xJ

dVdAJJ

tC

outin

∂∂

=⎟⎠⎞

⎜⎝⎛

∂∂

−∂∂

−=

∂∂

−=−=∂∂

⇒ 2

2

xCD

tC

∂∂

=∂∂

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

xCD

tC

∂∂

=∂

∂

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 - CC - C

os

ox

where CS = surface concentrationCo = initial uniform bulk concentration Cx = concentration of element at distance x from surface at time tx = distance from surfaceD = diffusivity of diffusing species in host latticet = timeerf = error function = erf (x/2 Dt) is the Gaussian error function – this is like a continuous probability density function from 0 to x/2 Dt

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

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

⎟⎠

⎞⎜⎝

⎛Dt2

x erf- 1 = C- CC- C

os

ox

Special Case

Desired to achieve some specific concentration of solute, C1 in an alloy, then

constant= C- CC- C

os

ox constant=Dt2

x

ExampleThe 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 γ – 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 = ?

RTmolJ /000,142

−

RTmolJ /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(×

−

⎟⎠

⎞⎜⎝

⎛−===−−

=−−

Dtxerf

CCCC

OS

OX

212.0

2.004.0

2.04.02.024.0

⇒ erf(z) = 0.8, where z = Dt

x2erf(z) = 0.8

12

1

12

1

)()()()(

zzzz

zerfzerfzerfzerf

−−

=−−

90.095.090.0

7970.08209.0797.08.0

−−

=−

− z

⇒ z = 0.906 ⇒ = 0.906Dt

x2

⇒ 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

=−

−=

−

−+

=−−

+=

CCCC

CC

CCC

CCCC

CCtxC

s

s

s

s

s

seff

Fick’s 2nd Law: )2

(15.00

0

Dtx

erfCCCC eff

s

−==−−

erf (0.5) ≈ 0.5 ⇒ xeff ≈ Dt

Effective penetration distance

In general, for most diffusion problems

xeff =

where γ: a geometry-dependent parameterγ = 1 for a flat plateγ = 2 for cylinders

Dtγ

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 Cuin 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.

( )( )( ) hours

smxhourssmx

DtD

t

tDtDtconsDt

4.110108.4

10.103.5

tan

1214

1213

500

600600500

600600500500

===

==

−−

−−

(C) Using the equation

⎥⎦

⎤⎢⎣

⎡−−

−

−

−

−

−−

===

==

500600

500

600

500

600

500600

500

600

500600 and

RTQ

RTQ

RTQ

RTQ

RTQ

o

RTQ

o

RTQ

oRTQ

o

dd

d

d

d

d

dd

e

e

e

eD

eDDD

eDDeDD

( ) ( )

( ) ( )( )

( ) ( ) ( )[ ]

1

1213121411

500600

600500

500600500600500600

500

600

.7.134773

1873

1.103.5ln.108.4ln..31.8

11lnln

11lnlnln

−

−−−−−−

=

⎥⎦⎤

⎢⎣⎡ −

−=

⎥⎦

⎤⎢⎣

⎡−

−=

⎥⎦

⎤⎢⎣

⎡−−=⎥

⎦

⎤⎢⎣

⎡−−=−=⎥

⎦

⎤⎢⎣

⎡

molkJQKK

smxsmxKmolJQ

TT

DDRQ

TTRQ

RTQ

RTQ

DDDD

d

d

d

ddd