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FYS4310. DOPING. Fys4310 doping. DIFFUSION. phenomenological. Ficks laws Intrinsic diffusion - extrinsic diffusion Solutions of Fick ’ s laws ’ predeposition ’ , ’ drive-in ’ electrical field enhancement Concentration dependant diffusion ( As). Diffusion program I. - PowerPoint PPT Presentation
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DOPING
Diffusion program I
Ficks laws Intrinsic diffusion - extrinsic diffusion Solutions of Fick’s laws ’predeposition’, ’drive-in’ electrical field enhancement Concentration dependant diffusion ( As)
phenomenological
atomistic
Si-Self diffusion
Vacancy diffusion
Charged vacancy model
Impurity diffusion
Vacancy -dopant interactions
Diffusion program II
Numerical solutions of Ficks lawsComputer packages
Math & practical
experimental./ref.reading
Measurements of diffusion
SIMS, Radio tracer, RBS
Differential Hall, SRM, CV, stain etch
High concentration effectsPairs, quartets, defects
Materials science
Bolzman Matano method
Diffusion equipment, furnaces, boilers, sources
Diffusion program III
B in SiAs in SiP in Si, high concentrationEmitter pushZn in GaAs
Examples
etc.
These points/topics are woven into other headings
High doping effectsSolubility, Metastable dopingDopant interactionsDefects introduced in diffusion doping, StressesDefining doping areas, masking, Diffusion in SiO2 vs SiDiffusion of metallic impurities
Diffusion phenomenological
Ficks laws
Ficks 1st lovDefines Dphonomenologic
Continuity equation
Ficks 2nd law
J: flux
Continuity equation should be well known
Derivation
One dimentional
J: fluxC: concentrationt : time
x x+dx
JoutJin
Diffusion in general microscopically
Random walkBrownian motion
Microscopic lattice modelling
Comparison gives D Phenomenologically Ficks 1. og 2.
a: jump distancev:trial rateZ: geometry
Ea= Em+Ev :migr.+vac6: std for cubic
Analytic solution diffusion equation 1
Boundary cond. - ”predeposition” from gas or solid source
C(z,0) =0C(0,t)=Cs
C(∞,t)=0
Solution
Total amount
Analytic solution diffusion equation 2
Boundary cond. - ”drive - in” on surface
Solution
no flux predep
Drive in
Dtpre<<Dtinndr
Curve shape for idealized caseINTRINSIC Diffusion
Fig. 3-7 a b, Cambell
Pre deposition Drive in
Electric field assisted diffusion - doping
Elec field
Flux
Atomic diffusion -self diffusion
Usually requires little energy
exhange
vacancy-diffusion
1 2 12
Interstitialdiffusion
V V
Interstitials can knockout regular atoms - interstitialcy
Atomic diffusion
Interstitials can move from on interstitial position to the next-Interstitial diff. Usually requires little energy
substitutional interstitial
Si
Int O, Fe, Cu, Ni, Zn
Sub P, B, As, Al, Ga, Sb, Ge
Substitutionals may move in various ways
exchange
vacancy-diffusion
1 2 12
V V
Atomic diffusion
Fig 3.6
Fig 3.5
Various diffusion mechanisms for doping atoms
B and P may diffuse this way -also depends whether have Si(I) or V
Frank-Turnbull
’kick out’
’Interstitialcy’
Atomic diffusion
Assume DSi can be measured
Si self diffusion - guess vacancy diffusionCan be measured by labeling the Si atoms by radioactive Si*
P conc.. DSi dependant on the doping density
Why?
Finds:
Atomic diffusion Vacancy diffusion
Vacancy concentration depends on doping concentration
Probability for jump via vacancy depends on vacancy concentration
i.e. Diffusivity D depends on doping concentration
Schematic
Diy :diffusivity
of Si by vac. w. charge y in intrinsic case
atomistic diffusion consequencesFor dopant atoms
n=n(C) n: electron concentrationp=p(C) C:doping concentration
i.e.D=D(C) and C=C(x)
ie EXTRINSIC DIFFUSJON
Diffusion equation Must be solved numerically
Diffusion, atomistic model consequences
Assume D can be measured, How?
From
With measurements of D vs. n you can determine D*, D-, D=
Fit to y: charge state *,-,+,=
Measured cm2/s eV
Diffusion, typical profiles Fig 3.8 B high conc.
B acceptor B- V-
Fig 3.9 As high conc.
As donor, As+
As+ V- , As+ V+
n>>ni : h=2,
Diffusion, High doping effects
Cluster formation
Dislocations
Band gap shrinkage
Segregation
Diffusion, High doping effects Dislocations
Stress-no disloc.f.example B doping
Disadvantage for stress in membranes for MEMS, ie for pressure sensors
Stress in heavy B doping of Si tried compensated by adding Ge
Elastic energy released by creating a dislocation
Diffusion, High doping effects Band-gap narrowing
Very low doping conc. High doping conc.
e-e int.act.. Donor donor interaction.Ed
Eg variesni varies[V] varies.
Diffusion, High doping effects Band-gap narrowing, qualitatively?
Van Overstraeten, R. J. and R. P. Mertens, Solid State Electron. 30, 11 (1987) 1077- 1087.
For 1017 ≤ N ≤ 3·1017 cm-3 .∆Eg
el ~ 3.5·10-8·Nd1/3 (eV)
(Nd in cm-3)
Diffusion, High doping effects Cluster formation
Various reactions
ln(C)ln
(n) solubility
VAs2-complexSi
Si
V
As
As
V=+2As+=VAs2
kVl+mAs++ye=(VKAsm)(m+lk-y)
Complex neutral, el.neg, [V]>>[2V]Gives k=1 and m=1,2,4
Complexes/clustersMaybe precursor to segregation
One model
Diffusion, High doping effects Cluster formation
How to find out about cluster models.?
ln(C)ln
(n) solubility
V2As2-complex
2V-+2As+=V2As2
kVl+mAs++ye=(VKAsm)(m+lk-y)
Én model
As
AsSi
V
V
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.
r* r
∆G
surface
volumebulk
∆G*
Diffusion, Numerical calculation considerations
Finite differencing
Solving the diffusion equations (by simulations)
General methods
Finite elementMonte CarloSpectral
Variational
Diffusion, numeric ’finite difference’Representations of time and space differentials
example. diff.eq.
Combining a and b giver representation FTCS Forward Time Centered Space
Forward Euler differentiation, Accuracy to 1ste order ∆t
NB Unstable*
FTCS explicit so Ujn+1 can be calculated for each j from known points
Stability achieved by Stability just as much art as science
Diffusion, numeric ’finite difference’
Leapfrog
accuracy 2nd order in ∆t
FTCS:
Often stability means many time steps
Here
F: Flux
C: Flux
(explicitly)
Stab. criteria:
Diffusion equation in simple case
FTCS:
=Crank-Nicholson method
(explicit)
(implicit) Backward time
Combining the two
(explicit)(implicit) Crank-Nicholson
Allows large time-steps
Diffusion, numeric ’finite difference’
Diffusion equation w. Crank -Nicholsen
collect left and right
ie
Diffusion equation w Crank -Nicholsen
A and B tridiagonal i.e.
Boundary cond , j=1 i.e. surface
R=1 reflectionR=0 trapping
R=0.5 segregation coeff
j=2
ie.
ie.
Diffusion equation w Crank -Nicholsen
unknown
=
known
If D=D(C) i.e. we have a nonlinear set of equations
Can readily be solved when D only a function of x and t but not of C.
Diffusion, numeric examplesschematic
Implanted profileVacancy creation on surfaceVacancy annihilation in ion-damaged regionGe i Si
Implanted profileVacancy creation on surfaceVacancy annihilation in ion-damaged region Surface segregationAs i Si
Diffusion, numeric general
unknown known
If D=D(C) ie
we have a nonlinear system of equations
But assume
as 1st approx. then calculate Cn+1, and put in A for next it.
vector matrix Initial conditions
This equation is solved for example by Newton’s method in the general case
Diffusion, numeric SOFTWARE PACKAGES
SUPREM III (1D), SUPREM IV(2D) [Stanford] TSUPREM4, SSUPREM4[Silvaco],ATHENA[Silvaco]ICECREAM, DIOS, STORM Floops, Foods, ACES,
’curve-fitting' models / physical models
1D, 2D, 3D
All processing stages/ all thermal/ just diffusion
Integrated, -masks, device, system-simulation (icecream silvaco)
Required for ULSI
Diffusion, Measurements of diffusion profiles, D(C)
SIMS, TOFSIMSNeutron activationRBS, ESCA, EDAXIR abs.
So measurement of C, and n el. P is needed
In many situations we have C=C(x), We can find D(C) by measuring C(x) at different T
Can find D*, D+, D-, D= from the dependency D(n)
Example P in simple caseMethods for C Methods for n
Spreading ResistanceDifferential Hall i.e.. +stripCapacitance, C-V + stripSCM, SRM
Diffusion, Measurement of diffusion, process surveillance
sphere
sphere
Top view
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
Diffusion, Measurements of diffusion profiles
van der Peuw method
2
1
4
3
More common geometry
12
3
4
Diffusion, Measurement of electric profileHall measurements
Measurement-procedure
Measure s, RHS gives µH
Strip a layerRepeat to d
Calc individual n and µ
Diffusjon, Measurement of electric profile
Schottky
p-n junction
electrolyte
Diffusion, Measurement of diffusion profile
ScanningProbeMicroscopy
SSRMSCM
Diffusion, Measurement of diffusion profile
Diffusion, Measurement of diffusion profile
Diffusion, Measurement of diffusion profileSpreading resistancemeasurements
beveling
needle
Diffusion, Measurement of diffusion profileScanning capacitance microscopy
Electric
Diffusion, Measurement of diffusion profileScanning capacitance microscopy
Elektriske
Diffusion, Measurement of diffusion profileScanning capacitance microscopy
Elektriske
Diffusion, Measurement of diffusion profileSIMS,
Physical ElectronicsUniversity of Oslo
Equipment at MRL-UiO:
Installed April-Aug. 2004
principle
SIMS Cameca 6F
UiO SIMS
Characteristics of SIMS:
SIMS karakteristiske trekk
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
Well suited for measurements of diffusion profiles in high tech 1/2 conductors:
Diffusion, Measurement of diffusion profile
RBS
Absolute measurement
Depth resolution 10-30 nm
Sensitivity: Matrix effects. Heavy in light matrix 1018 cm-3
Detection limits
Analysis depths
Diffusion, doping sources
Bolzman Matano method
Point: Extract D(C) from C(x)
Introduce a new coordinate system x’ by
C=C(x’,t) is a function only of i.e. C=C()
We may write Ficks 2 law by
LHS
RHS
Setting LHS=RHS
Bolzman Matano method
Here we have the same differential on both sides, so we integrate
Integration between C=0 and C=C1 and assume
C1 is one particular conc. we wish to investigate
Slight rewriting:
I
Put in to [I ] and choose a particular experimentally measured time for diffusion t1:
A
B
1. x’ origo from B2. dC/dx’ from curve3. Int(x’,C=C1..C0)
1+2+3 gives D from A
Method
Diffusjon, example P
P donor, P+
P+ V- , P+ V+
?observations
Vacancies are created at the surfaceConc of vac depends on n at surfaceV= + P+ -> VP-
EC-EF depends on depth when dC/dx <0VP- breaks up -> exess vac’Conc. of vacancy controlled by surface and not by local doping
Fair -Tsai model
Diffusion, ”Emitter-push”
Many models
Fair-Tsai diff. Model for P +Band gap narrowing
Describes results adequately
Diffusion, Example: Zn i GaAs
Zn acceptor in GaAs , Substitutes for Ga
VGa+ , VGa
* ..Zni+, Zns
-
Confinement of waveguides, etc..