kinetic lattice monte carlo simulations of dopant diffusion/clustering in silicon
DESCRIPTION
Kinetic Lattice Monte Carlo Simulations of Dopant Diffusion/Clustering in Silicon. Zudian Qin and Scott T. Dunham Department of Electrical Engineering University of Washington SRC Review February 25-26,2002. Outline. Introduction to KLMC Simulations High Concentration Arsenic Diffusion - PowerPoint PPT PresentationTRANSCRIPT
Kinetic Lattice Monte Carlo Simulations of Dopant Diffusion/Clustering in Silicon
Zudian Qin and Scott T. Dunham
Department of Electrical Engineering
University of Washington
SRC Review
February 25-26,2002
Outline
Introduction to KLMC Simulations High Concentration Arsenic Diffusion Acceleration of KLMC Simulations Fermi Energy Level Modeling in Atomistic Scale Random dopant fluctuations (initial results) Summary
Set up a silicon lattice structure (10-50nm)3
Defects (dopant and point defects) initialized
- based on equilibrium value
- or imported from implant simulation
- or user-defined
Kinetic Lattice Monte Carlo SimulationsKinetic Lattice Monte Carlo Simulations
V
Vacancy Mechanism Interstitial Mechanism
Si
Dopant
Fundamental processes are point defect hop/exchanges.
Kinetic Lattice Monte Carlo SimulationsKinetic Lattice Monte Carlo Simulations
Vacancy must move to at least 3NN distance from the dopant to complete one step of dopant diffusion in a diamond structure.
Tk
EE
B
fi
2exp0
Simulations include B, As, I, V, Bi, Asi and interactions between them.
Hop/exchange rate determined by change of system energy due to the event.
Energy depends on configuration and interactions between defects with numbers from ab-initio calculation (interactions up to 9NN).
Kinetic Lattice Monte Carlo SimulationsKinetic Lattice Monte Carlo Simulations
Kinetic Lattice Monte Carlo SimulationsKinetic Lattice Monte Carlo Simulations
1
1
4
1
N
m jmjt
Calculate rates of all possible processes.
At each step, Choose a process at random, weighted by relative rates.
Increment time by the inverse sum of the rates.
Perform the chosen process and recalculate rates if necessary.
Repeat until conditions satisfied.
Experiments found strong enhancement of diffusivity above 1020 cm-3.
High Concentration Arsenic DiffusionHigh Concentration Arsenic Diffusion
Dunham/Wu found strong D increase using KLMC simulations.
t
xD
6
2
High Concentration Arsenic DiffusionHigh Concentration Arsenic Diffusion
List et al. found reduced D in long term of simulations with fixed number of Vs in system.
The reason for the discrepancy is the formation of AsnV clusters during the simulation---number of free V drops.
Si
As V
Dunham/Wu did a relatively short simulation before clusters can form. ---Possible transient effects.
Solution: Long term simulations tracking free V concentration.
Problem: Computationally demanding for good statistics.
Once a cluster is formed, the system can spend a long time just making transitions within a small group of states.
state0
state1
state2
state3
~ eV
r01
r10
r23
r12
r21
r32
Acceleration of KLMC SimulationsAcceleration of KLMC Simulations
state0
state1
state2
state3
~eV
r0k
rk0
rk3
r12
r21
r3k
state K
The solution is to consider the group of states as a single effective state.
States inside the group are near local equilibrium.
kBkB
kBk
kk
TkETkE
jkTkEEjkpj
/exp[
)(/][exp)()(
Acceleration of KLMC SimulationsAcceleration of KLMC Simulations
Comparison of time that a vacancy is free as a function of doping concentration via simulations and analytic function
Both simulations with/without acceleration mechanism agree with the analytic prediction, but acceleration saves orders of magnitude in CPU time.
Acceleration of KLMC SimulationsAcceleration of KLMC Simulations
Equilibrium vacancy concentration increased significantly since the formation energy is lowered due to presence of multiple arsenic atoms.
At high concentration, vacancy likely interacts with multiple dopant atoms. The barrier is lowered due to attraction of nearby dopant atoms.
1
10
100
1000
10000
100000
1E+16 1E+17 1E+18 1E+19
Ar seni c Concent r at i on ( cm- 3)
Norm
aliz
ed V
acan
cy C
once
ntra
tion
(Cv
/Cv0
)
High Concentration Arsenic DiffusionHigh Concentration Arsenic Diffusion
High Concentration Arsenic Diffusion --- KLMC Results
High Concentration Arsenic Diffusion --- KLMC Results
As seen experimentally, simulations show arsenic diffusivity has strong increase with doping level: polynomial or exponential form.
Effect stronger at lower T, critical for As modeling (Pavel Fastenko).
Fermi Energy Level ModelingFermi Energy Level Modeling
Tk
EE
n
nC
B
FF
ix
iexp
Dopant atoms are ionized (e.g. As+, B-) and exposed to the field.
n
n
q
TkE B
The concentration of charged point defect is a function of Fermi level.
Continuous models derive Fermi level from dopant profiles.
22 4)()(5.0)( iADAD nNNNNxn
At atomistic scale, dopant atoms are discrete. Each donor (acceptor) contributes an electron (hole) cloud around itself.
110 -7 210 -7 310 -7 410 -7
210 19
41019
610 19
810 19
11020
1.2 10 20
5.0
0
3
)(
)/( Length, Debye
8
)/exp()(
pnq
qTkkl
l
lxxn
BSiD
D
D
Fermi Energy Level ModelingFermi Energy Level Modeling
Contributions of all charged dopants and defects add to give the total electron density.
Simulation of in a nonuniform background.
Residence time follows electron density, as predicted by continuous model.
0
1E+20
2E+20
3E+20
4E+20
5E+20
6E+20
7E+20
8E+20
9E+20
1E+21
Depth
Elec
tron
Den
sity
(cm
-3)
0
10000
20000
30000
40000
50000
60000
70000
Resi
denc
e Ti
me(r
elat
ive
valu
es)
El ect ron Densi ty Resi dent Ti me
Fermi Energy Level ModelingFermi Energy Level Modeling
i
totBF
Njjtot n
xn
q
TkxExnxn
i
D
)(ln)( ,)()(
-V
Fermi Energy Level ModelingFermi Energy Level Modeling
Example of KLMC simulations with incorporated field effect.
Fi el d Eff ect on Dopant Di ff usi on ( KLMC Si mul at i ons)
1E+19
1E+20
1E+21
1E+22
0 10 20 30 40 50 60 70 80 90
Depth ( Uni t=5. 43A)
Dopant Conc. (cm-3)
As i ni t i al
B i ni t i al B af t er KLMC
As af t er KLMC
Random Dopant FluctuationsRandom Dopant Fluctuations
Initial simulations show like dopant atoms tend to repel each other, resulting in a more uniform potential.
Dopant Fl uctuati on Si mul ati ons
0
10000
20000
30000
40000
50000
60000
0. 5 0. 51 0. 52 0. 53 0. 54 0. 55 0. 56 0. 57 0. 58
Fermi Energy Level (eV)
Sampling Points Random Af t er KLMC
Summary
• As diffusion at high concentrations shows a strong increase with doping level that is consistent with experimental measurements.
• Acceleration mechanism improves simulation efficiency, significantly reducing CPU time.
• Developed a Fermi level model and incorporated into LAMOCA KLMC simulation code.
• Initial dopant fluctuation simulations give more uniform Fermi level than random distribution (dopant/dopant repulsion).