Fys4310%20doping

1
Fys4310 doping
FYS4310
DOPING
DIFFUSION
2
Diffusion program I
phenomenological
Ficks laws Intrinsic diffusion - extrinsic
diffusion Solutions of Ficks laws
predeposition, drive-in electrical field
enhancement Concentration dependant
diffusion ( As)
atomistic
Si-Self diffusion Vacancy diffusion Charged
vacancy model Impurity diffusion Vacancy
-dopant interactions
3
Diffusion program II
Math practical
Numerical solutions of Ficks laws Computer
packages
Materials science
High concentration effects Pairs, quartets,
defects
experimental./ref.reading
Measurements of diffusion SIMS, Radio tracer,
RBS Differential Hall, SRM, CV, stain etch
Bolzman Matano method
Diffusion equipment, furnaces, boilers, sources
4
Diffusion program III
Examples
B in Si As in Si P in Si, high concentration Emitt
er push Zn in GaAs
etc.
These points/topics are woven into other headings
High doping effects Solubility, Metastable
doping Dopant interactions Defects introduced in
diffusion doping, Stresses Defining doping areas,
masking, Diffusion in SiO2 vs Si Diffusion of
metallic impurities
5
Diffusion phenomenological
Ficks laws
Defines D phonomenologic
Ficks 1st lov
J flux
Continuity equation
Ficks 2nd law
6
Continuity equation should be well known
J flux C concentration t time
One dimentional
Derivation
7
Diffusion in general microscopically
Random walk Brownian motion
Microscopic lattice modelling
Phenomenologically Ficks 1. og 2.
Comparison gives D
a jump distance vtrial rate Z geometry
Ea EmEv migr.vac 6 std for cubic
8
Analytic solution diffusion equation 1
Boundary cond. - predeposition from gas or
solid source
C(z,0) 0 C(0,t)Cs C(8,t)0
Solution
Total amount
9
Analytic solution diffusion equation 2
Boundary cond. - drive - in
d on surface
predep
no flux
DtpreltltDtinndr
Drive in
Solution
10
Curve shape for idealized case
INTRINSIC Diffusion
Pre deposition
Drive in
Fig. 3-7 a b, Cambell
11
Electric field assisted diffusion - doping
Elec field
Flux
12
Atomic diffusion -self diffusion
exhange
1
2
1
2
vacancy- diffusion
V
V
Interstitial diffusion
Usually requires little energy
Interstitials can knock out regular atoms -
interstitialcy
13
Atomic diffusion
Si
Int O, Fe, Cu, Ni, Zn
Sub P, B, As, Al, Ga, Sb, Ge
Interstitials can move from on interstitial
position to the next- Interstitial diff. Usually
requires little energy
interstitial
substitutional
Substitutionals may move in various ways
exchange
1
2
1
2
V
V
vacancy- diffusion
14
Atomic diffusion
Various diffusion mechanisms for doping atoms
Interstitialcy
Fig 3.5
B and P may diffuse this way -also depends
whether have Si(I) or V
kick out
Fig 3.6
Frank-Turnbull
15
Atomic diffusion
Si self diffusion - guess vacancy diffusion Can
be measured by labeling the Si atoms by
radioactive Si
Assume DSi can be measured
Finds
P conc.. DSi dependant on the
doping density Why?
16
Atomic diffusion Vacancy diffusion
Schematic
Vacancy concentration depends on doping
concentration
Probability for jump via vacancy depends on
vacancy concentration
i.e. Diffusivity D depends on doping
concentration
Diy diffusivity of Si by vac. w.
charge y in intrinsic case
17
atomistic diffusion consequences
For dopant atoms
nn(C) n electron concentration
pp(C) Cdoping concentration
i.e.
DD(C) and CC(x)
ie EXTRINSIC DIFFUSJON
Diffusion equation
Must be solved numerically
18
Diffusion, atomistic model consequences
Assume can measure D
How?
From
Measure D vs. n the we can determine D, D- and
D
y charge ,-,,
Fit to
19
Diffusion, atomistic model consequences
How?
Assume D can be measured,
From
With measurements of D vs. n you can determine
D, D-, D
Fit to
y charge state ,-,,
Measured
cm2/s
eV
20
Diffusion, typical profiles
Fig 3.8 B high conc.
Fig 3.9 As high conc.
B acceptor B- V-
As donor, As As V- , As V
ngtgtni h2,
21
Diffusion, High doping effects
Dislocations
Band gap shrinkage
Cluster formation
Segregation
22
Diffusion, High doping effects
Dislocations
f.example B doping
Stress-no disloc.
Disadvantage for stress in membranes for MEMS, ie
for pressure sensors
Elastic energy released by creating a dislocation
Stress in heavy B doping of Si tried compensated
by adding Ge
23
Diffusion, High doping effects
Band-gap narrowing
Ed
e-e int.act..
Donor donor interaction.
High doping conc.
Very low doping conc.
Eg varies ni varies V varies.
24
Diffusion, High doping effects
Band-gap narrowing, qualitatively?
For 1017 N 31017 cm-3       .?Egel
3.510-8Nd1/3 (eV) (Nd in cm-3)
Van Overstraeten, R. J. and R. P. Mertens, Solid
State Electron. 30, 11 (1987) 1077- 1087.
25
Diffusion, High doping effects
Cluster formation
solubility
ln(n)
Complexes/clusters Maybe precursor to segregation
ln(C)
VAs2-complex
Si
V2AsVAs2
Various reactions
As
One model
V
kVlmAsye(VKAsm)(mlk-y)
As
Si
Complex neutral, el.neg, Vgtgt2V Gives k1 and
m1,2,4
26
Diffusion, High doping effects
Cluster formation
solubility
ln(n)
ln(C)
V2As2-complex
kVlmAsye(VKAsm)(mlk-y)
V
How to find out about cluster models.?
2V-2AsV2As2
Én model
As
Si
As
V
27
Diffusion, High doping effects
Segregation
Requires nucleation of new phase, i.e. the
concentration must correspond to super saturation
before precipitation occurs, The density of
precipitates depends upon nucleation rate and
upon diffusion. It means measurements of
solubility can require patience particularly at
low temperatures.
surface
?G
?G
r
r
volume bulk
28
Diffusion, Numerical calculation considerations
Solving the diffusion equations (by simulations)
General methods
Finite differencing
Finite element
Monte Carlo
Spectral
Variational
29
Diffusion, numeric finite difference
Representations of time and space differentials
Forward Euler differentiation, Accuracy to 1ste
order ?t
a
b
Combining a and b giver representation FTCS
Forward Time Centered Space
example. diff.eq.
NB Unstable
FTCS
FTCS explicit so Ujn1 can be calculated for
each j from known points
Stability achieved by
Stability just as much art as science
30
Diffusion, numeric finite difference
F Flux
C Flux
Leapfrog
Here
accuracy 2nd order in ?t
Diffusion equation in simple case
FTCS
(explicitly)
Stab. criteria
Often stability means many time steps
31
Diffusion, numeric finite difference
(explicit)
FTCS
(implicit) Backward time
Combining the two
Crank-Nicholson method
Allows large time-steps
(explicit)
(implicit)
Crank-Nicholson
32
Diffusion equation w. Crank -Nicholsen
collect
left and right
ie
33
Diffusion equation w Crank -Nicholsen
A and B tridiagonal i.e.
Boundary cond , j1 i.e. surface
R1 reflection
R0 trapping
R0.5 segregation coeff
ie.
j2
ie.
34
Diffusion equation w Crank -Nicholsen
known
unknown

Can readily be solved when D only a function of
x and t but not of C.
If DD(C) i.e. we have
a nonlinear set of equations
35
Diffusion, numeric examples
schematic
Implanted profile Vacancy creation on
surface Vacancy annihilation in ion-damaged
region Surface segregation As i Si
Implanted profile Vacancy creation on
surface Vacancy annihilation in ion-damaged
region Ge i Si
36
Diffusion, numeric general
If DD(C) ie we have a
nonlinear system of equations
known
unknown
But assume
as 1st approx. then calculate Cn1, and put in A
for next it.
vector
matrix
Initial conditions
This equation is solved for example by Newtons
method in the general case
37
Diffusion, numeric SOFTWARE PACKAGES
curve-fitting' models / physical models
All processing stages/ all thermal/ just diffusion
Integrated, -masks, device, system-simulation
(icecream silvaco)
1D, 2D, 3D
Required for ULSI
SUPREM III (1D), SUPREM IV(2D) Stanford
TSUPREM4, SSUPREM4Silvaco, ATHENASilvaco ICECR
EAM, DIOS, STORM Floops, Foods, ACES,
38
Diffusion, Measurements of diffusion profiles,
D(C)
In many situations we have CC(x), We can find
D(C) by measuring C(x) at different T
Can find D, D, D-, D from the dependency
D(n)
So measurement of C, and n el. P is needed
Example P in simple case
Methods for C
Methods for n
SIMS, TOFSIMS Neutron activation RBS, ESCA,
EDAX IR abs.
Spreading Resistance Differential Hall
i.e.. strip Capacitance, C-V
strip SCM, SRM
39
Diffusion, Measurement of diffusion, process
surveillance
Top view
sphere
sphere
40
Diffusion, Measurement of diffusion, process
surveillance
Four-point probe
Measures so-called sheet resistance
Can find resistivity if the depth distribution of
n and µ is found
Mostly for process surveillance-reproducibility
tests
41
Diffusion, Measurements of diffusion profiles
3
2
More common geometry
4
4
3
1
1
2
van der Peuw method
42
Diffusion, Measurement of electric profile
Hall measurements
Measurement-procedure
Measure rs, RHS gives µH Strip a layer Repeat to d
Calc individual n and µ
43
Diffusjon, Measurement of electric profile
Schottky
p-n junction
electrolyte
44
Diffusion, Measurement of diffusion profile
Scanning Probe Microscopy SSRM SCM
45
Diffusion, Measurement of diffusion profile
46
Diffusion, Measurement of diffusion profile
47
Diffusion, Measurement of diffusion profile
Spreading resistance measurements
needle
beveling
48
Diffusion, Measurement of diffusion profile
Scanning capacitance microscopy
Electric
49
Diffusion, Measurement of diffusion profile
Scanning capacitance microscopy
Elektriske
50
Diffusion, Measurement of diffusion profile
Scanning capacitance microscopy
Elektriske
51
Diffusion, Measurement of diffusion profile
SIMS,
52
UiO SIMS
Physical Electronics University of Oslo
Equipment at MRL-UiO
SIMS Cameca 6F
Installed April-Aug. 2004
principle
53
SIMS karakteristiske trekk
Characteristics of SIMS
Well suited for measurements of diffusion
profiles in high tech 1/2 conductors
Sensitivity All elements can be detected
Sensitivity depends on element. in Si B, P
1014cm-3
Accuracy Through standards ( not absolute
spectro-scopy/-metry) Good reproducibility
Matrix-effects Ionization coeff. depends on
surface, oxidation, sputter w. oxygen yields
good reproducibility Interface-effects
Depth resolution Limited by sputter process,
2 nm - 50 nm ion mixing, segregation, run
away erosion
54
Diffusion, Measurement of diffusion profile
RBS
Absolute measurement
Depth resolution 10-30 nm
Sensitivity Matrix effects. Heavy in
light matrix 1018 cm-3
55
Detection limits
56
Analysis depths
57
Diffusion, doping sources
58
Bolzman Matano method
Point Extract D(C) from C(x)
Introduce a new coordinate system x by
CC(x,t) is a function only of
i.e. CC(l)
We may write Ficks 2 law by l
LHS
RHS
Setting LHSRHS
59
Bolzman Matano method
Here we have the same differential on both sides,
so we integrate
Integration between C0 and CC1 and assume
C1 is one particular conc. we wish to investigate
I
Slight rewriting
Put in to I and choose a particular
experimentally measured time for diffusion t1
Method
1. x origo from B 2. dC/dx from curve 3.
Int(x,CC1..C0) 123 gives D from A
A
B
60
Diffusjon, example P
observations
?
P donor, P P V- , P V
Fair -Tsai model
Vacancies are created at the surface Conc of vac
depends on n at surface V P -gt VP- EC-EF
depends on depth when dC/dx lt0 VP- breaks up -gt
exess vac Conc. of vacancy controlled by surface
and not by local doping
61
Diffusion, Emitter-push
Many models
Fair-Tsai diff. Model for P Band gap
narrowing Describes results adequately
62
Diffusion, Example Zn i GaAs
Zn acceptor in GaAs , Substitutes for Ga
VGa , VGa ..Zni, Zns-
Confinement of waveguides, etc..