part 8 semiconductor diffusion

8/8/2019 Part 8 Semiconductor Diffusion

1/17

Fall 2008 EE 410/510:

Microfabrication and Semiconductor ProcessesM W 12:45 PM 2:20 PMEB 239 Engineering Bldg.

Instructor: John D. Williams, Ph.D.Assistant Professor of Electrical and Computer Engineering

Associate Director of the Nano and Micro Devices CenterUniversity of Alabama in Huntsville

406 Optics BuildingHuntsville, AL 35899Phone: (256) 824-2898

Fax: (256) 824-2898email: [email protected]

Tables and Charts taken from Cambell, Science and Engineering of Microelectronic Fabrication, Oxford 2001And Wolf and Tauber, Introduction to Silicon Processing for the VLSI Era, Vol. II


2/17

Ficks Laws of Diffusion

Ficks 1st law

Accurately describes diffusion

No convenient measure of current density

Ficks 2nd law

Combines first law with continuity equation

Yields concentration over time as a function of second derivative of theconcentration gradient through the diffusion constant

Solution requires knowledge of at least two boundary conditions

t

CAdx

x

JAdxJJA

=

= )( 12

CDt

C 2=

x

txCDJ

=

),(


3/17

Understanding Atomistic Diffusion

Physical Mechanisms of Diffusion

To use Ficks second law, we must assume that the crystal is isotropic

Assumption breaks down when the concentration of the dopant is large

At large concentrations, diffusivity becomes a function of concentrationand therefore depth.

Interstitial and substitution diffusion

Assume atoms are correctly represented as minima in parabolicpotential wells .

These atoms are oscillate slightly due to thermal excitation

An inserted impurity atom may then sit between lattice sites interstitially.


4/17

Understanding Atomistic Diffusion Interstitial and substitution diffusion

These impurities diffuse rapidly due to the sharp localized changes inpotential energy and do not contribute to doping

Diffusion, however allows the impurity to move into an empty lattice site,thereby substituting for its potential into the lattice in place of the matrixmaterial

Vacancies filled by substitution remain within the lattice site until sufficient

energy is provided for the impurity to move to another empty lattice site.This is achieved by charge redistribution to minimize the free energy of thelattice

Vacancies are very dilute in semiconductors at typical process conditions

Each of the possible sites can be treated as independent entities. The diffusion coefficient then becomes the probability of all possible

diffusion coefficients, weighted by the probability of existence

=+

+

+=

1a

a

a

i

a

a

i

i DnpD

nnDD


5/17

Intrinsic Carrier Concentrations The intrinsic carrier concentration is

where nio =7.3*1015cm-3 for Siand 4.2*1014cm-3 for GaAs

The bandgap can be determined by

where Eg0, , and are 1.17 eV,0.000473 eV/K and 636 K forSi and 1.52 eV, 0.000541eV/K and 204K for GaAs

)2/(2/33)()(

TkE

ioibgeKTncmn

=

)(

)( 2

0KT

KTEE gg +

=


6/17

Understanding Atomistic Diffusion In heavily doped silicon, the bandgap is further reduced by the bandgap

narrowing effect

For heavily doped diffusions (C>>ni) the electron or hole concentration is justthe impurity concentration

For low concentration diffusions (C


7/17

Understanding Atomistic Diffusion If very dilute impurity profiles are measured before and after diffusion, then

diffusion coefficients can be determined.

Repetition of the procedure over several temperatures provides

Where Eia is the activation energy of the intrinsic diffusivity

Dio is a nearly temperature independent term that depends on vibrational energy and geometryof the lattice

)/( TkEoii

biaeDD =

Donors (D) Acceptors (A)

Dio Eia Do+ Ea

+

0.066 3.443.9 3.66

0.21 3.650.037 3.46 0.41 3.361.39 4.41 2480 4.20.37 3.39 28.5 3.920.019 2.63000 4.167E-6 1.20.1 3.20.7 5.6

Dio Eia Do+ Ea

+

12 4.0544 4.37 4.4 4

15 4.08

As in Si DP in Si D

Sb in Si DB in Si AAl in Si AGa in Si AS in GaAs DSe in GaAs D

Be in GaAs AGa in GaAs IAs in GaAs I I is interstitial


8/17

Analytic Solution of Ficks 2nd Law:

(Constant Source) In practice, dopant profiles area sufficiently complex and the assumption

that the coefficient of diffusion is constant is questionable, thus numerical

solutions are generally required However rough approximations can be made using analytic solutions

Solutions are provided for two theoretical conditions

1st : Predeposition Diffusion: source concentration (Cs) is fixed for all

times, t > 0

=

=

==

=

DtzefrcCtzC

tC

CtoCzC

CDt

C

s

s

2),(

0),(

),(0)0,(

2

Boundary Conditions

Solution, t > 0

Ficks 2nd Law in 1-D


9/17

Estimation of Diffusion profiles

Dose of predeposition profiles varies with the time of diffusion

Dose can be obtained using

measured in impurities per unity area (cm-2)

The depth of the profile is typically less than 1 m Dose of 1015 cm-2 will produce a large volume concentration (>1019 cm-3)

Since the surface concentration (Cs) is fixed for a predeposition diffusion,the total dose increases as the square root of time

DttCdztzCtQT ),0(2

),()(

0

==


10/17


(Constant Dose) 2st approach: Drive Diffusion: Initial amount of impurity (QT) is introduced

into the lattice

)4/(

0

2

2

),(

),(

0),(

0),(

0)0,(

DtzT

T

eDt

QtzC

QdztzC

tC

z

toC

zC

CDt

C

=

=

=

=

=

=

Boundary Conditions

Solution

z K 0

Ficks 2nd Law in 1-D

QT = constant

t > 0


11/17


(Constant Dose) With dose is constant, surface concentration must decrease with time:

At x = 0, dC/dz is zero for all t K 0.

One classic real world example of these two solutions is a predeposition surfacefollowed by drive in diffusion

Recall that the boundary condition for drive in was that the initial impurity concentrationwas zero everywhere except at the surface

Thus drive in is a good approximation for diffusion provided that

Boron (B) is diffusing into Si that as a uniform composition of phosphorus (P), CB. Also assume that CS>>CB A depth will exist at which CS = CB Since B is p-type and P is n-type, a p-n junction will exist at this depth, xj:

=

DtC

QDtx

B

Tj

ln4

driveinpredep DtDt


12/17

Diffusion of Various Dopants in Si

Online Thermal Diffusion Calculator:http://www.ece.gatech.edu/research/labs/vc/c

alculators/DiffCalc.html
http://www.ece.gatech.edu/research/labs/vc/calculators/DiffCalc.htmlhttp://www.ece.gatech.edu/research/labs/vc/calculators/DiffCalc.htmlhttp://www.ece.gatech.edu/research/labs/vc/calculators/DiffCalc.htmlhttp://www.ece.gatech.edu/research/labs/vc/calculators/DiffCalc.html


13/17

Corrections to Simple Theory

Substitutional impurities are almost completely ionized at room temperature

Thus an electric field always exist within the substrate

Total current due to the field effects both drift and diffusion components

Recalling Ohms Law:

Where is the mobility, E is the electric field, and the Einstein relationship between mobility

and diffusivity as been invoked.

Assuming that the carrier concentration is completely determined by the dopingprofile, then the field can be calculated directly

Where is the screening factor varying from 0 to 1.

dZ

dC

Cq

Tk

E

B 1

=

xCEz

CDJ +

=

dZ

dCDJ )1( +=


14/17

Corrections for Doping under

Oxides and Nitrides For (Cdoping >> ni, CSub), the profiles own electric filed will enhance movement of the impurity

Note that the equation is identical to Ficks first law with the slight modification of thescreening factor multiplier

Comparison of inert, oxidizing, and nitridizing dopant diffusion experiments has provided thefollowing conclusions: Diffusion of impurities depends directly on the concentration of impurities

Oxidized semiconductors produce a high concentration of excess interstitials at the oxidesemiconductor interface

Interstitial concentration decays with depth due to recombination

Near surface, these interstitials increase the diffusivity of B and P Therefore it is believed that B and P impurities diffuse primarily interstitially

Arsenic is diffusivity is found to decrease under oxidized conditions Excess interstitial concentration is expected to decrease local vacancy concentration, therefore, arsenic is primarily

believed to diffuse through vacancy mechanisms (at least in oxidized systems)

These results have been confirmed by using nitride silicon surfaces which are dominated primarily byvacancies and NOT interstitials.

Dopant diffusivities under oxidizing conditions

Where the exponent n has been found experimentally to range from 0.3-0.6 and the term is (+) foroxidation and (-) for nitridation

n

ox

dt

dtDiDDiD

+=+=


15/17

High Concentration Doping

At high concentrations, field enhancement is evident

This leads to maximum carrier concentrations of

Arsenic Phosphorous

+= iitail DDD

( )Asi

i

As Dn

nD 2 ( )

+ i

i

iPh Dn

nDD

2


16/17

4 Probe Analysis

of Diffused Profiles

4 probe Resistance measurement

Sheet carrier concentration canalso be combined with ameasurement of junction depth toprovide a complete description of

the diffused profile Ce(z) is the carrier concentration

(C) is the concentrationdependent mobility

[ ] 1)(

23

41

12

34

41

23

34

12

)(

1

)2ln(

4

1

=

=

+++=

dzCCqRsqR

Rsq

I

V

I

V

I

V

I

VR

ze

DttCdztzCtQT ),0(2

),()(

0

==

Recall from before:


17/17

Hall effect Analsysis

of Diffused Profiles

wBvV

BvE

BvqF

zxh

zxy

==

=r

r

r

sej

h

xxje

X

e

x

e

j

e

ej

xx

RCqx

qV

BIxCDxC

DxCx

C

Cqwx

Iv

j

j

1

1

0

0

=

==

=

=

Hall Voltage

integrated carrier concentration

Lorentz Force

Hall mobility (for epitaxy considerations)

part 8 semiconductor diffusion

Documents