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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.
Professor N Cheung, U.C. Berkeley
Lecture 10EE143 F2010
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
Professor N Cheung, U.C. Berkeley
Lecture 10EE143 F2010
3
(d) Liquid Source.
Professor N Cheung, U.C. Berkeley
Lecture 10EE143 F2010
4
Solid Solubility of Common Impurities in Si
oC
C0 (cm-3)
Professor N Cheung, U.C. Berkeley
Lecture 10EE143 F2010
5
Diffusion Coefficients of Impurities in Si
10-14
10-13
10-12
B,P
As
10-6
Au
Cu
kTE
O
A
eDD
Professor N Cheung, U.C. Berkeley
Lecture 10EE143 F2010
6
Temperature Dependence of D
.,/106.8
0
5
0
tabulatedareEDkelvineV
constantBoltzmankeVinenergyactivationE
eDD
A
A
kTAE
Arrhenius Relationship
Professor N Cheung, U.C. Berkeley
Lecture 10EE143 F2010
7
Mathematics of Diffusion
Fick’s First Law:
sec][:
,,
2cmDconstantdiffusionD
xtxCDtxJ
J
C(x)
x
Professor N Cheung, U.C. Berkeley
Lecture 10EE143 F2010
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”
Professor N Cheung, U.C. Berkeley
Lecture 10EE143 F2010
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.)
Professor N Cheung, U.C. Berkeley
Lecture 10EE143 F2010
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
Professor N Cheung, U.C. Berkeley
Lecture 10EE143 F2010
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
Professor N Cheung, U.C. Berkeley
Lecture 10EE143 F2010
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)
Professor N Cheung, U.C. Berkeley
Lecture 10EE143 F2010
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
Professor N Cheung, U.C. Berkeley
Lecture 10EE143 F2010
14
Dtx
eDt
CoxC
tDtC
dxtxCtQ
42
2
,
0
0
[1] Predeposition dose
[2] Concentration gradient
Professor N Cheung, U.C. Berkeley
Lecture 10EE143 F2010
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
Professor N Cheung, U.C. Berkeley
Lecture 10EE143 F2010
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
Professor N Cheung, U.C. Berkeley
Lecture 10EE143 F2010
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
Professor N Cheung, U.C. Berkeley
Lecture 10EE143 F2010
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.
Professor N Cheung, U.C. Berkeley
Lecture 10EE143 F2010
19
PredepositionPredeposition Drive-inDrive-in
Semilog Plots of normalized Concentration versus depth
Professor N Cheung, U.C. Berkeley
Lecture 10EE143 F2010
20
Note: is the implantation doseNote: is the implantation dose
Diffusion of Gaussian Implantation Profile
Professor N Cheung, U.C. Berkeley
Lecture 10EE143 F2010
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)
Professor N Cheung, U.C. Berkeley
Lecture 10EE143 F2010
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
Professor N Cheung, U.C. Berkeley
Lecture 10EE143 F2010
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.
Professor N Cheung, U.C. Berkeley
Lecture 10EE143 F2010
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
Professor N Cheung, U.C. Berkeley
Lecture 10EE143 F2010
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
Professor N Cheung, U.C. Berkeley
Lecture 10EE143 F2010
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
Professor N Cheung, U.C. Berkeley
Lecture 10EE143 F2010
27
Illustrating the relationship of No, NB, xj, and RS
Professor N Cheung, U.C. Berkeley
Lecture 10EE143 F2010
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
Professor N Cheung, U.C. Berkeley
Lecture 10EE143 F2010
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