Diffusion & OxidationDoping

Diffusion

1st Fick’s law

Fick's First Law is used in steady state diffusion, i.e., when the concentration within the diffusion volume does not change with respect to time.

( )

dxdnDJ

dxdnv

nnv

SdtdNJ

−=

−=−−−==3

)()(6

λλλ

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

TkE

DDB

diffexp0 Ediff: activation energy for diffusion

Diffusion

2nd Fick’s lawFick's Second Law is used in non-steady state diffusion,i.e., when the concentration within the diffusion volume changes in time.

[ ]

2

2

)()(

:Continuity

xnD

dtdn

dxdJ

dtdn

dtdVdxdJSdtxxJxJdN

dxdnDJ

∂∂

=⇒−=

−=∆+−=

−=

DiffusionConstant source diffusion

“pre-deposition”Limited source diffusion

“drive-in”

( ) ∫∞

−=

⎟⎠

⎞⎜⎝

⎛=

x

x dxex

Dtxntxn

22erfc

;2

erfc),( 0

π

2

2exp),( ⎟

⎠

⎞⎜⎝

⎛=Dtx

DtQtxnπ

DtndxtxnSNtQ 0

0 32),()( Dose === ∫

∞

(complimentary error function)

DiffusionEnhanced diffusion:• Stress, electric field• Ionization• Grain boundaries (defects)

Parameters that affect diffusion:• Temperature• Pressure• Crystal direction (channeling)

Probability to escape in thermalactivation process:

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−⋅⋅=

TkeaEWanPB

2/expω

w- attempt frequency

DiffusionIn gases:

D[cm2/s]

H2 in O2 0.7

CO2 in air 0.14

O2 in O2 0.18

sugar in water 3e-6

σσ PTv

nvvlD Bk~~~

scm][

2

=DxcDj

dd

−=

Mobility B, force F, drift velocity u of ionsBFu =

DLt

2

~Time of equalization

BTD Bk= Einstein's equation

L

Doping

Mechanism of Diffusion

a) substitutional diffusionb) impurity atom replaces Si atomc) impurity atom does not replace Si atom

Doping

O3HOB3OHB6BrO2B3O4BBr

3SiO4B3SiO2B

232262

23223

232

+→++→+

+→+

Uniformity on Wafers

O3HOP4O2PH6ClO2P3O4POCl

5SiO4P5SiO2P

25223

25223

252

+→++→+

+→+

⎟⎟⎠

⎞⎜⎜⎝

⎛∆

⎟⎟⎠

⎞⎜⎜⎝

⎛∆

==

DkrI

rr

DkrI

CrC

drrd

00

0

00

0

0

2

2)(

)1()/(

φ

Doping by thermal diffusion

Tool #433Rapid Thermal Processor (RTP) AG Heatpulse 610

a PC-controlled halogen lamp furnace. It accommodates 2", 3", and 4" wafers, or small pieces placed on a wafer.

It is typically used for alloying of ohmic contacts on III-V semiconductors. No other substrates than III-V are allowed.

Specifications:Protecting gas: N2Wafer size: pieces, 2", 3", 4“Max temperature:1000°CHeating rate: 120°C/secTemperature sensor: Thermocouple / Pyrometer

Diffusion

http://www.oki.com/

diffusion of dopant atoms in the direction parallel to the surface of semiconductor; undesired in device manufacturing as it causes distortion of the device geometry.

Thermal Oxidation of Si

222

22

2HSiOO2HSiSiOOSi

+→+→+

1. Oxygen is transported from the bulk of the gas phase to the gas-oxide interface2. Oxygen diffuses through the growing solid oxide film3. When oxygen reaches the Si/SiO2 interface, it chemically reacts with Si and forms SiO2

Interface Reaction Kinetics

)( sgggs CChJ −= sss CkJ =

sgsgs

gs JJ

hkC

C =+

= for 1

⎟⎠⎞

⎜⎝⎛−∝

RTEks exp

ks>>hg: mass transfer

ks<<hg: surface reaction

Interface Reaction Kinetics( )( )

iS

i

gg

CkJd

CCDJ

CChJ

=

−=

−=

3

0

02

01

constant ratereaction chemical SiOin oxygen of coeff. diff.

tcoefficien transportmassinterface Si/SiO at theoxygen ofion concentrat interface gas/SiO at theoxygen ofion concentrat

gas in theoxygen ofion concentrat

2

20

2

−−

−−−

−

S

g

i

g

kD

hCC

C

Ddk

hk

DdkC

CS

g

S

Sg

0

0

0

1

1

++

⎟⎠⎞

⎜⎝⎛ +

=

Ddk

hk

CC

S

g

S

gi

01 ++=

growth) state-(steady 321 JJJ ==

Interface Reaction Kinetics

Ddk

hk

DdkC

CS

g

S

Sg

0

0

0

1

1

++

⎟⎠⎞

⎜⎝⎛ +

=

Ddk

hk

CC

S

g

S

gi

01 ++=

( )00 limitedreaction 1

dkD

hk

CCC S

G

S

gi >>

+=≈

( )00 limiteddiffusion :0 ; dkDCCC Sig <<≈≈

)(

)(

020

00

τ+=+

=

tBAdd

NCktd

dtd iSthe actual growth

depends on J3:

Interface Reaction Kinetics

)(

)(

020

00

τ+=+

=

tBAdd

NCktd

dtd iS

the actual growthdepends on J3:

[ ]( )

any) (if thicknessoxide initial

/;/2);(2

cm (wet) 104.4 (dry) 102.22

011

322220

−

+==+=

××=−−

−

i

iigSg

d

BAddNDCBkhDA

N

τ

BAttABtd

BAtBttdBA

tAtd

4/),()(

4/,)(

14/

12

)(

20

20

20

<<++≅

>>≅

−+

+=

ττ

τ

Oxide ExpansionCracks – lower breakdown voltage (5-10 MV/cmLeakageTrapped charges

Mechanical Relaxation Effectsduring oxide growth (expansion)

Stress relaxation: Strain relaxation:

( )

212

2110

21

21

21

s/cmdyn 108.2dyn/cm 106.6for

/expstrain)(constant

;

⋅⋅=

⋅=

−=−=

==

+=

η

ησσεε

ησεσε

εεε

Y

Yt

Y

x

xx

T

&&

& ( )[ ]ησε /exp10 YtY

−−=

Si substrate

SiO2

Usually a mixtureof strain and stressrelaxations

τ ~ 4 sec

A volume change of ~200%!

Diffusion Along Grain BoundariesStressDefects

easier diffusion along grain boundaries

http://shasta.mpi-stuttgart.mpg.de/research/thinfilmcnt/thinfilms.htm

Interdiffusion & Transformationsin Thin Films

Electro-migration

Interdiffusion is crucialin multilayers

MoSimultilayers

•EUV lithography •EUV metrology •EUV microscopy •synchrotron optics •x-ray astronomy •soft x-ray lasers •element analysis •plasma physics

M. Ohring: Materials Science of thin filmshttp://www.iof.fraunhofer.de/departments/optical-coatings/

Si Crystal Structurediamond structure

Two interleaving FCC cells offset by 1/4 of the cube diagonal

Isotropic vs Anisotropic Wet Etching

Chemicals IsotropicHF:HNO3:CH3COOH:H20HFHF:NH4F Masking Materials

Photoresist (Acids Only)Si3N4SiO2Anisotropic

KOHEDP (Ethylenediamine Pyrocatechol)CsOHNaOHN2H4-H20 (Hydrazine)

Chemicals KOH EtchingEtch rate

(110) > (100) > (111)(111) > (110) > (111) w/ IPA

Varies with T and concentration

MasksSi3N4: is the best, very slow etch rate, selectivity > 1000SiO2 :selectivity >> 100

Application of Anisotropic Etch

Pits

Steps

Holes

Cantilevers

Bridges

MicromachiningM. Madou “Fundamentals of microfabrication”

Etch Stop Layers

Boron Stops Etching~1020 cm-3 reduces etch rate 1000 times

Boron-doped Layer

masking

Laser-Assisted Wet Etching

M. Madou, Fundamentals of Microfabrication

Macroporous Si http://www.macroporous-silicon.com/

Photogenerated minority carriers (in the case of n-type Si this means "holes") diffuse from the back side of the sample to the pore (etch) pits and promote dissolution there, because of the enhanced electrical field in the space charge layer (SCL). www-tkm.physik.uni-karlsruhe.de/