ee143 f2010 lecture 10 dopant diffusion (1) predepositionee143/fa10/lectures/lec_10.pdf ·...

Professor N Cheung, U.C. Berkeley

Lecture 10EE143 F2010

1

Dopant Diffusion

(1) Predeposition dopant gas

SiO2SiO2

Si

dose control

(2) Drive-in Turn off dopant gasor seal surface with oxide

SiO2SiO2

Si

SiO2

Doped Si region

profile control(junction depth;concentration)

Note: Predeposition by diffusion can also bereplaced by a shallow implantation step.Note: Predeposition by diffusion can also bereplaced by a shallow implantation step.



2

Dopant Diffusion Sources

(a) Gas Source: AsH3, PH3, B2H6

(b) Solid SourceBN Si BN Si

B oxide +SiO2

(c) Spin-on-glass SiO2+dopant oxide



3

(d) Liquid Source.



4

Solid Solubility of Common Impurities in Si

oC

C0 (cm-3)



5

Diffusion Coefficients of Impurities in Si

10-14

10-13

10-12

B,P

As

10-6

Au

Cu

kTE

O

A

eDD



6

Temperature Dependence of D

.,/106.8

0

5

0

tabulatedareEDkelvineV

constantBoltzmankeVinenergyactivationE

eDD

A

A

kTAE

Arrhenius Relationship



7

Mathematics of Diffusion

Fick’s First Law:

sec][:

,,

2cmDconstantdiffusionD

xtxCDtxJ

J

C(x)

x



8

From the Continuity Equation

x)t,x(CD

xt)t,x(C

x)t,x(CD

xx)t,x(J

t)t,x(C

0t,xJt

t,xC

“Diffusion Equation”



9

If D is independent of C(i.e., D is independent of x).

C x tt

D C x tx

, ,

2

2

Concentration Independent Diffusion EquationConcentration Independent Diffusion Equation

Concentration independence of D

State of the art devices use fairly high concentrations,causing variable diffusivity and other significant side-effects (transient-enhanced diffusion, for example.)

State of the art devices use fairly high concentrations,causing variable diffusivity and other significant side-effects (transient-enhanced diffusion, for example.)



10

Boundary Conditions

Initial Condition

C x t C solid solubility of thedopantC x t

C x t

:

:

,,

,

00

0 0

0

A. Predeposition Diffusion Profile

Justification:Si wafers are ~500um thick, dopingdepths of interest are typically < several um

At time =0, there is no diffused dopantin substrate



11

0

0

200

2

2

21,2

CDt

DtxerfcC

dyeCtxC Dtx

y

C0

t3>t2t2>t1

x=0x

t1

Characteristic distance for diffusion.

Surface Concentration (solid solubility limit)

Diffusion under constant surface concentration



12

erf (z) =2

0

z

e-y2 dy erfc (z) 1 - erf (z)

erf (0) = 0 erf( ) = 1 erf(- ) = - 1

erf (z) 2

z for z <<1 erfc (z) 1

e-z2

z for z >>1

d erf(z)dz = -

d erfc(z)dz =

2

e2-z

d2 erf(z)dz2 = -

4

z e2-z

0

z

erfc(y)dy = z erfc(z) +1

(1-e-z2 )

0

erfc(z)dz =1

Properties of Error Function erf(z)and Complementary Error Function erfc(z)



13

The value of erf(z) can be found in mathematical tables, as build-in functions in calculators andspread sheets. If you have a programmable calculator, this approximation is accurate to 1 part in107: erf(z) = 1 - (a1T + a2T2 +a3T 3 +a 4T4 +a5T 5) e-z2

where T = 11+P z and P = 0.3275911

a1 = 0.254829592 a2 = -0.284496736 a3 = 1.421413741 a4 = -1.453152027 a5 = 1.061405429

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6

erfc(z)exp(-z^2)

Practical Approximations of erf and erfc



14

Dtx

eDt

CoxC

tDtC

dxtxCtQ

42

2

,

0

0

[1] Predeposition dose

[2] Concentration gradient



15

B. Drive-in Profile

DtxerfcCotxC

ConditionsInitial

xC

txCConditionsBoundary

x

20,

:

0

0,:

0

x

C(x)

x=0

Predep’s (Dt)

Physical meaning of C/x =0:No diffusion flux in/out of the Sisurface. Therefore, dopant dose isconserved



16

xQtxC 0,

C(x,t=0)

x

Solution of Drive-in Profilewith Shallow Predeposition Approximation:

C x t QDt

edrive in

xDt drive in,

2

4C(x,t)

x

t1 t2

Q

C Dt predep0 2

Approximate predep profileas a delta function at x=0



17

indrive

predep

DtDtRLet

x

Approximation over-estimates conc. here

Approximationunder-estimatesconcentration here.

Goodagreement

C(x)/C0

R=1

R=0.25

Exact solutionDelta functionApproximation

How good is the (x) approximation ?

For reference onlyFor reference only



18

Summary of Predeposition + Drive-in

22

2

42

1

22

110

2

2

1

1

2 tDx

etDtDCxC

tDtD

Diffusivity at Predeposition temperature

Predeposition time

Diffusivity at Drive-in temperature

Drive-in time

*This will be the overall diffusion profile after a “shallow” predepositiondiffusion step, followed by a drive-in diffusion step.



19

PredepositionPredeposition Drive-inDrive-in

Semilog Plots of normalized Concentration versus depth



20

Note: is the implantation doseNote: is the implantation dose

Diffusion of Gaussian Implantation Profile



21

T h e e x a c t s o lu tio n s w ithCx = 0 a t x = 0 ( . i.e . n o d o p a n t lo s s th ro u g h s u r fa c e )

c a n b e c o n s tru c te d b y a d d in g a n o th e r fu ll g a u s s ia n p la c e d a t -R p [M e th o d o fIm a g e s ].

C (x , t) =

2 (R 2p + 2 D t)1 /2

[e-

(x - R p )2

2 (R 2p + 2 D t) + e

-(x + R p )2

2 (R 2p + 2 D t) ]

W e c a n s e e th a t in th e lim it (D t) 1 /2 > > R p a n d R p ,

C (x ,t) e - x 2 /4 D t

(D t)1 /2 ( th e h a lf -g a u s s ia n d r iv e - in s o lu tio n )

For reference onlyFor reference only

Diffusion of Gaussian Implantation Profile (arbitrary Rp)



22

Dopants will redistribute when subjected to various thermal cycles ofIC processing steps. If the diffusion constants at each step areindependent of dopant concentration, the diffusion equation can bewritten as:

Ct = D(t)

2Cx2

Let (t)

0

t D(t’)dt’

D(t) =t

UsingCt =

C

t

The diffusion equation becomes:C

t =

t

2Cx2 or

C =

2Cx2

The Thermal Budget



23

When we compare that to a standard diffusion equation with D being

time-independent:C

Dt =2Cx2, we can see that replacing the (Dt)

product in the standard solution by will also satisfy the time-dependent D diffusion equation.

ExampleConsider a series of high-temperature processing cycles at{temperature T1, time duration t1} ,{ temperature T2, time duration t2}, etc. The corresponding diffusion constants will be D1, D2,... . Then, = D1t1+D2t2+..... = (Dt)effective

** The sum of Dt products is sometimes referred to as the “thermalbudget” of the process. For small dimension IC devices, dopantredistribution has to be minimized and we need low thermal budgetprocesses.



24

istep

ieffective )Dt()Dt(BudgetThermal

Example

Dttotal of :

Well drive-in

and

S/D annealing

Temp (t)

time

welldrive-in

stepS/D

Annealstep

Temp (t)

time

welldrive-in

stepS/D

Annealstep

For a complete process flow, only those steps with high Dtvalues are importantFor a complete process flow, only those steps with high Dtvalues are important



25

Explicit relationship between:

No

(surface concentration) ,

xj(junction depth),

NB

(background concentration),

RS

(sheet resistance),

Once any three parameters are known, the fourth one can be determined.Once any three parameters are known, the fourth one can be determined.

p-type erfcn-type erfcp-type half-gaussiann-type half-gaussian

Irvin’s Curves



26

Approach

1) The dopant profile (erfc or half-gaussian ) can be uniquely determined if oneknows the concentration values at two depth positions.

2) We will use the concentration values No at x=0 and NB at x=xj to determine theprofile C(x). (i.e., we can determine the Dt value)

3) Once the profile C(x) is known, the sheet resistance RS can be integratednumerically from:

4) Irvin’s Curves are plots of No versus ( Rs xj ) for various NB.

jx

B dxNxCxqRs

0

1

Both NB(4-point-probe), RS (4-point probe) and xj (junction staining) can beconveniently measured experimentally but not No (requires secondary ion massspectrometry). However, these four parameters are related.

Motivation to generate the Irvin’s Curves



27

Illustrating the relationship of No, NB, xj, and RS



28

Example 1 Drive-in from line source with s atoms/cm

D2Cr2

+DrCr =

Ct

C(r, t) =s

2Dt e-r2/4Dt

Diffusion Equation in cylindrical coordinates

2-Dimensional Diffusion with constant D



29

Diffusion mask

x

y

equal-conc. contours

yj (at x=0) < xj (at y )

xj

yj

Rule of Thumb : yj ~0.7-0.8 xjRule of Thumb : yj ~0.7-0.8 xj

Example 2 : Semi-Infinite Plane Source

Two-Dimensional Drive-in Profile (cont.)

C(x,y,t) =Q

2 Dt e -x2/4Dt [ 1 + erf (

y2 Dt

) ]

where Q = predep dose in #/cm2

ee143 f2010 lecture 10 dopant diffusion (1) predepositionee143/fa10/lectures/lec_10.pdf ·...

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