a benchmark study of the wigner monte-carlo method
TRANSCRIPT
A benchmark study of the Wigner
Monte-Carlo
method
J.M. Sellier *, M. Nedjalkov **, I.Dimov *, S. Selberherr **
* Bulgarian Academy of Sciences, IICT, Sofia** TU Wien, Inst. Microelec., Vienna
LSSC 2013
Topics
• Motivations
• The Wigner equation and its semi-discrete form
• A Monte Carlo approach based on particles sign
• The annihilation technique
• Wigner vs. Schroedinger benchmark test
Motivations
Quantum Effects – First set of
challenges• When device dimensions are reduced, quantum effects start
to appear.
• Particles can tunnel through barriers, energies are discretized.
• The behavior of an electron is more similar to a wave than to• The behavior of an electron is more similar to a wave than toa particle, the dynamics is described by Schroedingerequation.
• Boltzmann equation does not predict quantum effects.
• A new approach must be taken.
Quantum effects – Second set of
challenges
• Scattering effects are relevant even in nanodevices
• Full-Quantum approach
Time-dependent phenomena• Time-dependent phenomena
• Self-Consistent Calculations
• A simulator that possibly runs on a common machine in reasonable run-times
The Wigner Equation and its
Semi-Discrete FormSemi-Discrete Form
The Wigner equation
• The Wigner equation reads
• where
( ) ( ) ( ) ( )tkrQftkrfkt
tkrfWWrk
W ,,,,1,, rrrrr
h
rr
rr =∇∇+∂
∂ ε
• where
( ) ( ) ( )',',', krfkkrVkdkrQf WWW
rrrrrrrr −= ∫
( ) ( ) ∫
−−
+= ⋅−
2
'
2
''
2
1, ' r
rVr
rVerdi
krV rkidW
rr
rrr
h
rr rr
π
Wigner equation in semi-discrete form
• Taking into account the fact that a semiconductor device
has limited dimensions, it is possible to re-formulate the
Wigner equation in a semi-discrete form.
( ) ( ) ( ) ( )tnMrfnrVtMrfkMmt
tMrfWWWr
W ,,,,,,,
*−=∇∆+
∂∂
∑+∞ rrrrh
r
r
• The phase space is discretized w.r.t. the pseudo-wave
vector coordinates.
( ) ( ) ( )tnMrfnrVtMrfkMmt W
nWWr ,,,,,
*−=∇∆+
∂ ∑−∞=
( ) ( ) ( )( )∫ −−+= ⋅∆−2/
0
211,
Lskm
W srVsrVesdLi
nrVrrrrr
h
r rr
Comment
• The Wigner equation is a difficult task in a
finite difference framework.
• The distribution is known to be rapidly varying • The distribution is known to be rapidly varying
and the diffusion term cannot be calculated
correctly.
( )tkrfWr ,,rr
r∇
A Monte-Carlo Approach Based on
Particles SignParticles Sign
Wigner equation in integral form
• The semi-discrete Wigner equation can be re-
formulated in terms of an integro-differential
equation.
( ) ( )( ) ( )( )−=∫− ,0,, 0 mxfetmxf i
dyyx
W
trr
rγ
• where
( ) ( )( )
( )( ) ( ) ( )( ) ( ) ( )( ) ( )∫ ∑∞ ∞+
−∞=
−−−∫Γ
=−
0'
''''',,'',',''
,0,,
' xtxxttemmxtmtxfdt
mxfetmxf
Dm
dyyx
W
iW
t
trrrrr
r
θδθγ
( ) ( ) ( )∑∑+∞
−∞=
−+∞
−∞=
+ ==m
Wm
W mxVmxVx ,,rrrγ
( )( ) ( )( ) ( )( ) ( )( ) ','','','',,' mmWW txmmtxVmmtxVmmtx δγ rrrr +−−−=Γ ++
( ) ( )''*
ttm
kmxtx −∆−=
rhr
Mean value of a function
• Finally, using the fact that the adjoint equation of the
integro-differential equation is a Fredholm integral
equation of second type, one can show that:
( ) ( ) ( ) ( )∑ ∑∫∫+∞ +∞∞
=−= W AtmxAtmxfxddtA ,,, τδτ rrr
• where (for example)
( ) ( ) ( ) ( )∑ ∑∫∫−∞= =
=−=m i
iW AtmxAtmxfxddtA0
0,,, τδτ
( ) ( ) ( )( ) ( )( )∑∫+∞
−∞=
−∫='
0',',' 0
mi
dyyx
ii mxAemxfxdAi ττ
τγrr
( ) ( ) ( )( ) ( )
( ) ( )( ) ( )( ) ( )τδ
θτ
γ
γ
−∫Γ⋅
⋅∫=
−∞+
−∞=
∞
+∞
−∞=
−∞
∑∫
∑∫∫
ttmtxAemmxdt
xemxfdxdtA
t
t
t
i
dyyx
mt
mD
dyyx
iii
,,',,
,'
11'
'101
'1
'
0r
Physical interpretation of the terms
• A particle starts at xi with momentum m’∆k at time 0. The exponent gives the probability that the particle remains on the trajectory, provided that the scattering rate isϒ.
• Consider as a particle generation rate.
The Wigner potential gives rise to the creation of two particles, one positive and one negative, and the sign carries the quantum information.
( ) ( ) ( )∑∑+∞
−∞=
−+∞
−∞=
+ ==m
Wm
W mxVmxVx ,,rrrγ
Similarities in the 2 approaches
• Boltzmann • Wigner
( ) ( )∑+∞
−∞=
+=m
W mxVx ,rrγ
dtdttkWtkWdttPt
]'))'((exp[))(()(0∫−=
rr
∫= )',(')( kkSdkkWrr
dtdttxtxdttPt
]'))'((exp[))(()(0∫−= rr γγ
The Annihilation Technique
Some comments
• The only unknown of the problem is the quasi
distribution function
• Particles with opposite signs in the same • Particles with opposite signs in the same
phase-space cell do not contribute to the
calculation of the unknown
• Created particles can annihilate (i.e. removed)
Wigner vs. Schroedinger
Benchmark TestBenchmark Test
The Schroedinger Equation
and its Discretization
The Schroedinger Equation
and its Discretization
