Chapter3 Diffusion 2013-1

7/27/2019 Chapter3 Diffusion 2013-1

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ECE 6450, Chapter 3 1

Chapter 3: Diffusion

3.1 Ficks Diffusion Equation & Solutions

3.2 Diffusion Coefficients

3.3 Analysis of Diffusion Profiles

3.4 Simulation of Diffusion Profiles

Literature: Campbell, Chapter 3, pages 43-73

September 6, 2013 ECE 6450: Chapter 3 Diffusion O. Brand, 1 of 32

What is Diffusion?

Diffusion refers to movementof material in response to a

concentration gradient

In semiconductor devices,diffusion either refers to (i) the

movement of doping atoms

or (ii) charge carriers

(electrons or holes) fromareas of high concentration to

low concentration

Diffusion processes carriedout at elevated temperatureestablish the doping profiles of

e.g. pn junctions or control thematerial resistivity


Campbell, Fig. 3.1


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Application of Diffusion Processes

Diffusions are most commonly used for Bases, emitters, and resistors in bipolar technology Form wells, source/drain regions and dope polysilicon gate/

interconnect lines in MOS technology.

When to use it and when not to use it Use when damage from Ion Implantation leads to unacceptable

decreases in minority carrier lifetime, when electrical junctions need

to be very deep, or a cheap easy solution is needed

Do not use it for ultra-shallow junctions, majority carrier devices(use ion implantation instead) or when total impurity dose is

critical (ex. MOSFET channel)


September 6, 2013 ECE 6450: Chapter 3 Di ffus ion O. Brand, page 4 of 32

3.1 Ficks Diffusion Equation

Ficks 1stLawstates that the net flux of material J is proportional tothe negative gradient of the concentration gradient dC/dx with the

diffusion coefficient D as proportionality factor:

Assuming only diffusion, the change of material concentration dC/dtin a length element dx equals the negative gradient of the material

flux across dx

Combining both equation yields Ficks 2ndLaw

J = "D#C(x,t)

#x !J = "D grad C(x,y,z,t)

"C(x,t)

"t= #

"J

"x "C(x,y,z,t)

"t= # div

!J(x,y,z,t)

"C(x,t)

"t=

"

"xD

"C

"x

#

$%

&

'(

"C(x,y,z,t)

"t= div D gradC[ ]


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September 6, 2013 ECE 6450: Chapter 3 Di ffus ion O. Brand, page 5 of 32

Ficks Diffusion Equation (cont.)

Assuming D is independent of the position x (isotropic materialproperties), Ficks 2nd Law can be expressed by

The above DE can generally only be solved numerically;however, analytical solutions exist for 2 important cases:

(1) Constant surface concentration: C(0,t) = CS(2) Constant total dopant concentration (i.e. dose):

Case (1) approximates a so-calledpre-deposition,Case (2) a drive-in process

"C(x,t)

"t=D

"2C(x,t)

"x2

"C(x,y,z,t)

"t=D div grad C[ ] =D #2C

C(x,t)dx = QT0

"

#

1-D 3-D

Pre-Deposition

Constant-Surface-Concentration Diffusion Boundary conditions for DE: C(x,0) = 0; C(0,t) = CS; C(,t) = 0 Solution for t>0 involves complementary error function

with the surface concentration CS and the characteristic diffusionlengthDt; the erfc function is tabulated in Campbell, Appendix V

The dopant dose QTdeposited by the pre-deposition step isobtained by integrating over the doping profile

With a fixed surface concentration, the dose increases with thesquare-root of the pre-deposition time


C(x,t) = CS erfcx

2 Dt

"

#$

%

&'

QT(t) = C(x,t)dx0

"

#=

2

$

CS Dt


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Some Error Function Algebra


erf(x) =2

"e#y

2

dy0

x

$

erfc(x) = 1# erf(x)

erf(0) = 0

erf(%) = 1

erf(x) &2

"x for x 0 involves a Gaussian centered at x = 0

Note that in this case the surface concentration C(0,t) decreaseswith time as more and more dopant diffuses into the wafer (awayfrom the surface)

Typical problems involve first a pre-deposition step from agaseous, liquid or solid dopant source, followed by a drive-in

process to achieve the desired doping profile; the aboveequations provide a good approximation for this case as long as


C(x,t) =

QT

"Dte#x

2 / 4Dt

C(x,t)dx = QT

0

"

#

D tpredep


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Pre-Deposition Sources @ GT

The IEN cleanrooms offer solid sources for B and P doping(Supplier: Saint-Gobain, http://www.bn.saint-gobain.com;Techneglas, http://www.techneglas.com)

The solid sources are ceramic wafers of BN (for p-type) andP2O5 (for n-type), which are loaded together with the siliconwafers in an alternating fashion in a quartz boat

Note: the ceramic wafers must be initially oxidized, resulting in asurface glass layer of B2O3 in case of BN wafers

During the pre-deposition step, the glass slowly evaporates fromthe BN wafer and condenses on the silicon wafer; silicon doping

occurs via the following reaction2 B2O3 + 3 Si 4 B + 3 SiO2

To insure a reproducible pre-deposition, the solid solubility limitfor the dopant should be reached at the wafer surface

The pre-deposition dose and the resulting sheet resistance arecontrolled by the pre-deposition temperature and time


SiO2 Diffusion Barriers

The common silicon dopantsdiffuse slower in thermally grown

SiO2 than in Si; as a result, oxidelayers can be used as a diffusionmask during pre-deposition

In order to prevent dopant diffusionthrough the mask, a minimal oxidethickness is required, which

depends on dopant and processtemperature

Because ion implantation is widelyused today for pre-deposition,

oxide diffusion masks are of lessimportance

9/6/13 ECE 6450: Chapter 3 Diffusion O. Brand, 12 of 32

Sze (1sted.), Fig. 10.14


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3.2 Diffusion Coefficients

Impurities (e.g. doping atoms) introduced in a semiconductorcrystal either sit between lattice sites in an interstitial position

(interstitial impurity) or replace atoms at a lattice site(substitutional impurity)

For silicon Substitutional impurities: P, B, As, Al, Ga, Sb, Ge Interstitial impurities: O, Au, Fe, Cu, Ni, Zn, Mg

Note: Doping atoms generally have to occupy lattice sites to beelectrically active!

Depending on the impurity type, we have different diffusionmechanisms: interstitial impurities diffuse by interstitial

diffusion and are typically characterized by large diffusionconstants; substitutional impurities often diffuse by vacancydiffusion, described by Fairs vacancy model


Fairs Model for Vacancy Diffusion

Depending on the chargeassociated with the vacancy

(-4,0,+4), different diffusioncoefficients apply

The overall diffusion coefficientcan then be estimated bysumming up the individual

diffusivities weighted by theappearance probability of the

respective vacancy charge,yielding

Thereby, each D follows an Arrhenius behavior:September 6, 2013 ECE 6450: Chapter 3 Diffusion O. Brand, 14 of 32

Energetically Favored Process

Campbell, Fig. 3.3

D = D0+

n

ni

"

#$

%

&'D

(+

n

ni

"

#$

%

&'

2

D2(

+

n

ni

"

#$

%

&'

3

D3(

+

n

ni

"

#$

%

&'

4

D4(

+

p

ni

"

#$

%

&'D

+

+

p

ni

"

#$

%

&'

2

D2+

+

p

ni

"

#$

%

&'

3

D3+

+

p

ni

"

#$

%

&'

4

D4+

D = D0 e"E

a/ kT


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Fairs Vacancy Model (cont.)

Note: the intrinsic carrier concentration is evaluated at thediffusion temperature, i.e. it is orders of magnitude larger than

the room-T ni!

For p-type dopants, the n terms can be neglected and viceversa

If the doping concentration is small compared to ni, we haven = p = ni and the overall diffusivity is not concentrationdependent and has an Arrhenius form

This is often considered the intrinsic diffusion range

At higher doping concentrations ( ni), the diffusion coefficientsbecome concentration dependent


D = D0 e"Ea / kT

Diffusion Coefficients


Campbell, Tab. 3.2

Note: all other coefficients can be neglected


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Intrinsic(Low Doping)

DiffusionCoefficients


Sze, Fig. 13.4

D = D0e"Ea / kT

Si GaAs

Model for Diffusion Coefficients Besides vacancy diffusion, other diffusion mechanisms exist

(see Campbell, page 48-49, for details)

Interstitialcy diffusion Frank-Turnball diffusion Kick-out diffusion

However, experimental data for diffusion coefficients can berather well described by Fairs model even though otherdiffusion mechanisms might play a role



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Intrinsic Carrier Concentration

From semiconductor devicephysics, we recall

with ni0 = 7.3x1015 cm-3 for Si

and ni0 = 4.2x1014 cm-3 for GaAs

At the diffusion temperature, theT-dependence of Eg must beaccounted for

with Eg0, , and being 1.17eV,

0.000473eV/K and 636K for Si,and 1.52eV, 0.000541eV/K and

204K for GaAs; T is in Kelvin


ni = ni0 T3 /2 e

"Eg /2kT

Eg = Eg0 "

# T2

$ + TCampbell, Fig. 3.4

How do you calculate n and p?

Diffusivities for Common Dopants

Boron: for doping concentrations below 1020 cm-3, borondiffusion follows Fairs model with neutral and first positivevacancy term; for higher concentrations, some B atoms are nolonger substitutional and the resulting diffusivity decreases

sharply in crystalline Si (but increases in amorphous Si)

Arsenic: diffuses via neutral and single negatively chargedvacancies; relatively low diffusivity in Si makes As very suitable

for shallow diffusion profiles (minimal redistribution); for dopingconcentrations above 1020 cm-3, As tends to form interstitial

clusters and resists thermal activation, flattening the top of highconcentration carrierprofiles

Phosphorus: diffuses more rapidly than As in silicon, and isused primarily for well and isolation diffusions; high

concentration doping profiles have characteristic kink region



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More on Diffusion Coefficients


Diffusion profiles in case ofconcentration-dependent D

Phosphorus diffusion profiles

Sze, Fig. 13.9

Sze, Fig. 13.8

As, B in Si

Au, Pt in Si

Impurity Redistribution during Oxidation

Thermal oxidation will cause dopant redistribution in siliconsubstrate

Similar to the dopant segregation at the liquid/solid interfaceduring crystal growth, dopant atoms segregate at the interface

between Si and SiO2, with k denoting the impurity concentrationratio between Si and SiO2

Depending on k (0.3 for B, 10 for P) and the diffusivity of thedopant in SiO2, different cases of impurity redistribution

(enrichment and depletion) in the silicon substrate can occurduring a thermal oxidation

As a result, the doping profile can change considerably,affecting device characteristics



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Impurity Redistribution during Oxidation


Sze, Fig. 13.12

3.3 Analysis of Diffusion Profiles Sheet Resistance Measurement

Four-Point Probe Van der Pauw Method

Carrier Concentration Measurement Hall Effect

Profiling Techniques C-V Measurement

Scanning Capacitance Microscopy

Spreading Resistance Profilometry Secondary Ion Mass Spectroscopy (SIMS)


RS =1

q (C(z))C(z)dz"

Information

on integralof

profile only

Information

on profile


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Four-Point Probe

The resistivity of a semiconductorcan be determined by a current-

voltage measurement using afour-point probe

W is the substrate thickness,CF a correction factor (see graph)

The four-point probe can also beused to measure the sheetresistance of a doping profile

where CF = /ln(2) if the probe-spacing of a co-linear

probe is much larger than the junction depth

! =V

IW CF

RS

=

V

I CF

/ln2 = 4.532

Van der Pauw Method

Sheet resistance measurement usingintegrated test structure: diffused

region with 4 point contacts

Correction factor F depends onprobe geometry and becomes F = 1

for an ideal square geometry

Rotating the voltage and currentcontacts by 90 and averaging the

measurements improves accuracy


RS=

"

ln2

F R

R =1

4

V12

I34

+

V23

I41

+

V34

I12

+

V41

I23

#

$%

&

'(

12

3

4

Campbell, Fig. 3.13


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Measurement of Carrier Concentration Hall Effect

The carrier concentration n or pcan be measured using the Halleffect

Assume: p-type semiconductorplate with L > W t (thin plate)with current I applied in x-directionand a magnetic induction Bzapplied along z-direction

Holes experience a Lorentz forcein y-direction:

Hole accumulation creates electric field in y-direction (electrostatic force =Lorentz force, because no current flow in y-direction), resulting in a Hallvoltage VH:

x

y

zBz

+

+

VH

VI

L

t

Wx

y

P-Type Semiconductor

F = q (!

v !!

B) = (0,"qvxB

z,0)

VH=!

yW = v

xB

zW =

Jx

qpB

zW =

1

qp

"RH

!

IBz

t

Hall Coefficient

Measurement of Junction Depth

Using Stain Solution

Wafer is either beveled or grooveis formed and then immersed in

staining solution

Staining solutions 1:30:10 HF:HNO3:CH3COOH

will turn p-type Si dark

1:1:10 HF:H2O2:H2O withbright light exposure is usedfor GaAs

From size of stained region andgeometry of bevel/groove, the

junction depth can be estimated


Campbell, Fig. 3.14


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C-V Measurement

Applied to pn-junction and MOS capacitor to measure dopingconcentration vs. depth; in case of pn-junction the depletioncapacitance under reverse bias is measured

Depletion capacitance of one-sided step junction is function ofthe depletion region width (extending into the lower doped side

of the junction with doping Nsub(z)); varying V changes W andthus the depth z at which Nsub(z) is probed:

Pros: easy to perform, non-destructive measurementCons: applicable in Si only for Nsub < 10

18 cm-3; depletion edgesare not abrupt, thus step junctions are not well resolved


C ="A

W(V)=

q"A2Nsub(z)

2(V + Vbi)

Nsub(z) =8(V + Vbi)

3

q"A2dC(z)

dV

#

$%

&

'(

2

Spreading Resistance Profilometry

Sample is beveled at shallow angle Pair of probes are placed in contact with the

surface at predetermined force

Measurement of resistancealong the beveled sampleand comparison of measurementto calibration standard

Due to current crowding at thepoint contacts, the resistivity inthe vicinity of the contacts is

dominating the measured resistance

Pros: able to measure profiles ranging from 1013

-1021

cm-3

Cons: tip conditioning and frequent calibration are essential; difficultto measure shallow junctions; material must be similar to standard

See e.g. Solecon Laboratories (http://www.solecon.com)September 6, 2013 ECE 6450: Chapter 3 Diffusion O. Brand, 30 of 32

http://www.wikipedia.org


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Chemical Profiling Secondary Ion Mass Spectroscopy

Sample (in vacuum) is bombarded with 1-5 keV ion beam Energetic atoms strike surface and eject material (sputtering);

ionized secondary atoms are accelerated towards massspectrometer and then analyzed with respect to their mass

Result is the materialcomposition as afunction of the sputteringtime, which can be

translated into depthinformation

Detection limit for dopingatoms in Si in 1014-1015cm-3 range

Cons: expensive, time consumingSeptember 6, 2013 ECE 6450: Chapter 3 Diffusion O. Brand, 31 of 32

http://www.ifw-dresden.de

3.4 Simulation of Diffusion Profiles Numerical methods are used

for calculating 1, 2, and 3-Ddiffusion profile

Stanford University PRocessEngineering Module(SUPREM) is widely usedsimulator

Details via homeworkassignments


Campbell, Fig. 3.18

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Chapter3 Diffusion 2013-1