Modeling of noise by Legendre polynomial expansion of the ...
Transcript of Modeling of noise by Legendre polynomial expansion of the ...
Modeling of noise by Legendre polynomial expansion of the Boltzmann equation
C. Jungemann
Institute of Electromagnetic Theory RWTH Aachen University
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Outline
• Introduction • Theory
– Noise – Legendre polynomial expansion
• Results – Bulk – 1D NPN BJT
• Conclusions
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3
Introduction
Introduction
• Noise is fundamental and cannot be avoided (E. g. Nyquist noise: SVV = 4kBTR)
• Noise degrades the performance of circuits (E. g. Noise limits the minimum signal that can be detected)
• Noise occurs in all frequency ranges
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Electronic noise is found in all semiconductor devices and circuits
Introduction
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Current fluctuates due to particle scattering (and trapping) (displacement current)!
Stochastic electron motion in a constant electric field
Introduction
6
Noise analysis for stationary processes
Current fluctuations:( ) ( )
Correlation function:( ) ( ) ( )
Power spectral density:
( ) 2 ( )
II
jII II
I t I t I
I t I t
S e dωτ
δ
ϕ τ δ τ δ
ω ϕ τ τ∞
−
−∞
= −
= +
= ∫
Electric power absorbed in a resistor by current fluctuations:( ) (2 )IIdP f S f Rdfπ=
Introduction
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Macroscopic models (often approximations): Nyquist noise:(equilibrium)
( ) 4II BS k TGω =
Shot noise:(non-equilibrium)
( ) 2IIS qIω =
The same microscopic origin!
How can we calculate noise on a microscopic basis?
Introduction
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Power spectral density can be calculated by Monte Carlo method, which solves
the Boltzmann equation.
Introduction
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MC simulation of a 1D N+NN+Si structure biased at 6V
Floating body problem
Introduction
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MC noise simulation
Small tail of the ACF determines low-frequency noise MC CPU time: 3 weeks (easy to simulate) MC too CPU intensive for device noise below 100GHz "
Introduction
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A deterministic BE solver is required and should have similar numerical properties
as the classical approaches (DD, HD)
Spherical harmonics expansion (SHE)! (Baraff, Bologna group, Maryland group, etc)
Requirements: • Self-consistent solution of BE and PE • Exact stationary solutions • ac and noise analysis directly in the frequency
domain (including zero frequency) • Large signal simulations (Harmonic balance) • Rare events (small currents, deep traps, ...) • Full bands, magnetic fields, Pauli principle, etc
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Theory
Noise
Two LTI-systems (linear, time-invariant)Time domain (convolution, hy (t): impulse response):
! y(t) = hy (t ! t ')! x(t ')dt '!"
t
!
! z(t) = hz (t ! t ')! x(t ')dt '!"
t
!Frequeny domain:!Y (! ) = Hy (! )" X (! )
"Z(! ) = Hz (! )" X (! )
Cross correlation of " y(t) and ! z(t) :
! yz (! ) = ! y(t +! )" z(t) = hy (t +! ! t ')! x(t ')dt '!"
t+!
! hz (t ! t ')! x(t ')dt '!"
t
!
= hy (t +! ! t '') ! x(t '')! x(t ') hz (t ! t ')dt ''dt '!"
t+!
!"#
t
! = hy (t +! ! t '')! xx (t ''! t ')hz (t ! t ')dt ''dt '!"
t+!
!"#
t
!Wiener-Lee theorem:
Syz (! ) = Hy (! )Sxx (! )Hz*(! ) Syy (! ) = Hy (! )
2Sxx (! ) ! 0"
#$% 13
Theory
stochastic deterministic
stochastic
correlated
Poisson-type noise
Example: Injection of independent particles over a barrier with rate !, current:I0 = q!Power spectral density of a Poisson process (white noise):S(! )=2!Power spectral density of current fluctuations (shot noise):
SII (! ) = 2q2! = 2qI0
PSD of independent events that occur at a rate ! is 2!14
Theory
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Theory
Sxx (! ) = 2qI0 Hy (! ) =1
1+ j!"; ! = RC Syy (! ) =
2qI01+! 2" 2
Syy (! ) = Hy (! )Sxx (! )Hy*(! ) = 1
1+ j!"2qI0
11! j!"
=2qI0
1+! 2" 2
Shot noise passing through a low pass filter
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Theory How to calculate noise in the framework of the BE?(single spherical valley, non degenerate conditions, bulk system)
!f (!k ,t)!t
! q!
"ET! !k f (
!k ,t) =
!sys
(2! )3 W (!k |!k ') f (
!k ',t) !W (
!k ' |!k ) f (
!k ,t)d 3k '!
" (in) " (out) drift scattering (deterministic) (stochastic)Particles are instantaneously scattered at a rate (Poisson process):#(!k ,!k ',t) =W (
!k |!k ') f (
!k ',t)
W (!k |!k ') : Transistion rate of a particle to be scattered from
!k ' to
!k
f (!k ',t) : Probability that a particle is found in state
!k '
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Theory
Under stationary conditions the rate is given by:!(!k ,!k ') =W (
!k |!k ') f0 (
!k ')
where f0 (!k ') is the stationary distribution function.
To calculate noise the impulse response of the distribution function hf (
!k ,!k ',t,t ') is required:
"hf"t
# q!
"ET$ !khf =
%sys
(2! )3W (!k |!k '')hf (
!k '',!k ',t,t ') #W (
!k '' |!k )hf (
!k ,!k ',t,t ')d 3k ''&
+ 1(2! )3
! (!k #!k ')! (t # t ')
hf (!k ,!k ',t,t ') is the probability that a single particle generated in the state
!k '
at time t ' appears in state !k at time t. Otherwise the system is empty.
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Theory Stationary system!hf (!k ,!k ',t ! t ',0) = hf (
!k ,!k ',t,t ')
Transfer function:
H f (!k ,!k ',! ) = hf (
!k ,!k ',! ,0)
!"
"
# e! j!"d!
Solving directly in the frequency domain yields:
j!H f !q!
"ET$ !k H f =
%sys
(2! )3W (!k |!k '')H f (
!k '',!k ',! ) !W (
!k '' |!k )H f (
!k ,!k ',! )d 3k ''#
+ 1(2! )3
! (!k !!k ')
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Theory Scattering consists of particle creation and annihilation (in and out scattering)Fluctuation of the distribution function by scattering:G(!k ,!k ',!k '',! ) = H f (
!k ,!k ',! ) ! H f (
!k ,!k '',! )
! ! creation annihilationThe particle vanishes out of state
!k '' and re-appears in
!k ' due to scattering!
G(!k ,!k ',!k '',! ) is the transfer function of particle scattering
PSD of the distribution function:
S ff (!k1,!k2 ,! ) =
4!sys
(2! )6 G(!k1,!k ',!k '',! )W (
!k ' |!k '') f0 (
!k '')G*(
!k2 ,!k ',!k '',! )d 3k 'd 3k ''!!
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Theory
Expected values:
x(t) = 2(2! )3
X (!k ) f (
!k ,t)d 3k!
PSD of two macroscopic quantities x and y:
Sxy (! ) =1
(2! )6X (!k1)S ff (
!k1,!k2 ,! )Y (
!k2 )d
3k1d3k2!!
This eqution is too CPU intensive (12D integral):
Sxy (! ) =4"sys
(2! )12X (!k1)! G(
!k1,!k ',!k '',! )! W (
!k ' |!k '') f0 (
!k '')Y (
!k2 )!!
G*(!k2 ,!k ',!k '',! )d 3k1d
3k2d3k 'd 3k ''
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Theory
Sxy (! ) =4!sys
(2! )61
(2! )3X (!k1)G(
!k1,!k ',!k '',! )d 3k1" W (
!k ' |!k '') f0 (
!k '')""
1(2! )3
Y (!k2 )G
*(!k2 ,!k ',!k '',! )d 3k2" d 3k 'd 3k ''
=4!sys
(2! )6GX (!k ',!k '',! )W (
!k ' |!k '') f0 (
!k '')GY
* (!k ',!k '',! )d 3k 'd 3k ''""
with
GX (!k ',!k '',! ) = 1
(2! )3X (!k1)G(
!k1,!k ',!k '',! )d 3k1"
= 1(2! )3
X (!k1) H f (
!k1,!k ',! ) # H f (
!k1,!k '',! )$% &'d
3k1" = HX (!k ',! ) # HX (
!k '',! )
with
HX (!k ',! ) = 1
(2! )3X (!k1)H f (
!k1,!k ',! )d 3k1"
HX (!k ',! ) is a direct solution of the adjoint BE, CPU time similar to solving for f0 !
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Theory
Example: PSD of velocity fluctuations !v(t) at equilibrium
Svv (! ) =4!sys
(2! )6 Hv (!k ',! ) ! Hv (
!k '',! )
2W (!k ' |!k '') f0 (
!k '')d 3k 'd 3k ''!!
colored colored white noise
=4!vx
2
1+! 2" 2 =4kT! µ(! ){ }The noise source of the BE is white (instantaneous scattering), but the transfer functions are not resulting in colored noise for all usual microscopic quantities.
lim!!"
HX (k ',! ) !1!
! lim!!"
SXX (! ) !1! 2
Noise of all observable quantities vanishes at high frequencies.
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Theory
Legendre Polynomial expansion
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Theory Spherical harmonics
k-space energy-space (angles are the same as in k-space) (kx,ky,kz) (ε,ϑ,φ) with ε = ε(k,ϑ,φ) and k = k(ε,ϑ,φ)
Dependence on angles is expanded with spherical harmonics: Complete set of orthogonal functions Yl,m(,):
Y0,0 (!,") = 1
4#
Y1,$1(!,") = 34#
sin!sin"
Y1,0 (!,") = 34#
cos!
Y1,1(!,") = 34#
sin!cos"
Yl,m (!,")Yl',m' (!,")d% = &l,l '&m,m' d% = sin!d!d"!''
ε
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Theory Spherical harmonics expansion:Xl,m (!) = X(
!k(!,",#))Yl,m (",#)d$"%%
X(!,",#) = Xl,m (!)m=& l
l
'l=0
(
' Yl,m (",#) = Xl,m (!)Yl,m (",#)l,m'
Example: group velocity (spherical band structure)
!v = v(!)sin"cos#sin"sin#cos"
)
*
+++
,
-
.
.
.= v(!) 4/
3
Y1,1(",#)
Y1,&1(",#)
Y1,0 (",#)
)
*
+++
,
-
.
.
.
Nonzero elements:
!v1,&1 = v(!) 4/3!ey , !v1,0 = v(!) 4/
3!ez , !v1,1 = v(!) 4/
3!ex
Only three nonzero elements!
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Theory Spherical harmonics expansion of the distribution function:
gl,m (!, t) = 1(2")3 # ! $ !(
!k)( )Yl,m (%,&)f (
!k, t)d3k'
= Z(!) Yl,m (%,&)f (!k(!,%,&), t)"'' d( (spherical bands)
with the (reduced) density-of-states (DOS)
Z(!) = k2
(2")3
)k)!
Expectations:
x(t) = 2(2")3 X(
!k)f ('!k, t)d3k
= 2 Xl,m (!)gl,m (!, t)d!'l,m*
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Theory ExamplesParticle density:
n(t) = 2(2!)3 f ("
!k, t)d3k 1= 4!Y0,0( ) = 2 4! g0,0 (#, t)d#"
Only the zero order component carries charge!Particle current density (spherical bands):
!j(t) = 2
(2!)3
!v(!k)f ("!k, t)d3k = 2 4!
3v(#)
g1,1(#, t)
g1,$1(#, t)
g1,0 (#, t)
%
&
''''
(
)
****
d#"
Only the first order components carry current!
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Theory Spherical harmonics expansion of the Boltzmann equation:
1(2!)3 " # $ #(
!k)( )Yl,m (%,&) BE{ }d3k'
(Balance equation for gl,m:
)gl,m
)t$ q!ET )!jl,m)#
+ * l,m = Wl,m g{ }with
1(2!)3 " # $ #(
!k)( )Yl,m (%,&)
)f)t
+,-
./0
d3k'
= ))t
1(2!)3 " # $ #(
!k)( )Yl,m (%,&)fd3k'
+,-
./0=)gl,m
)t
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Theory Drift term:
1(2!)3 " # $ #(
!k)( )Yl,m (%,&) $ q
!
"ET'"kf
()*
+,-
d3k.
= $1(2!)3
q"ET
!" # $ #(
"k)( )Yl,m (%,&)'!kfd
3k.
= $q!ET /!vgYl,m
/#$ 1"k
/Yl,m
/%!e% +
1sin%
/Yl,m
/&!e&
0
123
45g
6
788
9
:;;d<#.. = $q
!ET /!jl,m/#
+ = l,m
with!jl,m = !vg#.. Yl,md< g(#,%,&, t) = gl',m' (#, t)Yl',m' (%,&)
l ',m'>0
12345
and
= l,m = q!ET
"k/Yl,m
/%!e% +
1sin%
/Yl,m
/&!e&
0
123
45gd<#..
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Theory
! l,m (", t) = q!ET
"k(")#Yl,m
#$!e$ +
1sin$
#Yl,m
#%!e%
&
'()
*+g(",$,%, t)d,#--
= q!ET
"k(")#Yl,m
#$!e$ +
1sin$
#Yl,m
#%!e%
&
'()
*+Yl',m' d,#--
.
/00
1
233l ',m'
4 gl',m' (", t)
= q!ET
"k(")
!bl,m,l ',m'gl',m' (", t)
l ',m'4
with!bl,m,l ',m' =
#Yl,m
#$!e$ +
1sin$
#Yl,m
#%!e%
&
'()
*+Yl',m' d,#--
!bl,m,l ',m' is a constant that can be readily calculated by computer algebraic methods.
The sum over l',m' couples the balance equation for l,m with the other ones. For even l the drift term couples only with odd l' and vice versa.
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Theory !jl,m (!, t) = !
v(!,",#)Yl,m (",#)g(!,",#, t)d$"%% = v(!)
!al,m,l ',m'gl',m' (!, t)
l ',m'&
with!al,m,l ',m' = Yl,m
!e!Yl',m' d$"%%
!al,m,l ',m' has the same odd/even coupling
property as !bl,m,l ',m'.
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Theory Scattering integral (neglecting Pauli principle):
W f{ } = !s
(2")3 W#(!k,!k ')f (
!k ', t)$ %W#(
!k ',!k)f (!k, t)d3k '
#&
Transition rate of process # (constant energy transfer, dependsonly on the scattering angle):
W#(!k,!k ') = 1
!s
c# '(!k),cos"(
!k,!k ')() *+, '(
!k) % '(
!k ') % #-#( )
Expansion of the transition rate (addition theorem):
cos!("k,"k ') = cos.cos. '+ sin.sin. 'cos(/ %/ ')( )
c# '(!k),cos"(
!k,!k ')() *+ = c#l '(
!k)() *+ Yl,m (.,/)Yl,m (. ',/ ')
m=% l
l
&l=0
0
&with
c#l '(!k)() *+ = 2" Pl (u)c# '(
!k),u() *+du
%1
1
$
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Theory
Velocity randomizing scattering (e.g. phonons):c!l "(
!k)#$ %& = 4'c! "(
!k)#$ %&(l,0
Projection of the scattering integral:1
(2')3 ( " ) "(!k)( )Yl,m (*,+)W f{ }d3k, = Wl,m g{ }
Wl,m g{ } = Z(")c!l "#$ %&gl,m (" ) !-!, t) ) Z(" + !-!)c!0 " + !-!#$ %&gl,m (", t){ }
!.
The projected scattering integral is local in l,m. Only in the case of a full bandstructure or inclusion of the Pauli principle this is no longer the case. The scattering integral is nonlocal in energy.
34
Theory Additional effects included in the simulator: • Full bands for holes (bulk) • Modena model for electrons • Magnetic fields • Pauli principle (bulk) • Traps (bulk) • Large signal simulation by harmonic balance
method (bulk) • Real space with maximum entropy dissipation
stabilization (1D, 2D)
Theory
35
• Boltzmann and Poisson equations are solved with the Newton-Raphson method
• Green’s functions are calculated based on the Jacobian of the Newton-Raphson scheme by the adjoint method
• The resultant large systems of equations are solved CPU and memory efficiently with the robust ILUPACK solver
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Results
Bulk
37
Stationary bulk results
• Rare events are easily simulated by SHE
• MC requires statistical enhancement which forestalls noise simulation
• Required for simulation of floating body problems
EDF for 300kV/cm in <100> direction
38
AC bulk results
• Excellent agreement of MC and SHE
• 3rd order expansion sufficient for bulk
• SHE works at low and high frequencies
PSD of velocity for an electric field of 30kV/cm at room temperature
Only phonon scattering
39
AC bulk results
CPU time MC: 50000sec (95% CL) SHE: 173sec SHE about 300 times
faster for similar error! "
MC device simulation is many orders of magnitude more CPU intensive
Relative error of the velocity PSD for an electric field
of 30kV/cm at room temperature
40
AC bulk results
• SHE can handle GR processes with arbitrary life times
• SHE can handle zero frequency
• Even 1/f-noise models can be simulated in the framework of the full Boltzmann equation
PSD of current for a doping of 1017/cm3
and an electric field of 10kV/cm
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Cyclostationary bulk results
E(t) = 50kV/cm*sin(2pf0t), f0=500GHz
MC data: S. Perez et al., J. Appl. Phys., 88 (2), p. 800, 2000.
42
Cyclostationary Bulk results
• For 1kV/cm only upconversion at f0
• For 30kV/cm velocity saturation leads to upconversion at multiples of f0
• Impossible to simulate with MC at technically relevant frequencies
317000 /10 ,5 ,1 )],2cos(1[)( cmNnsGHzftfEtE Dl ===+= τπ
43
Degenerate bulk systems
Silicon, n=1020/cm3
Pauli exclusion principle [1-f(k)] W(k|k’) f(k’)
Scattering is only possible if the final state is empty! f(k) is often approximated in MC device simulators
Deep traps ε
ε Cε T
Vε
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Degenerate bulk systems Electrons in silicon at room temperature, zero field
Mobility
µUT = !vx2 (1" f0 )
Full:
Isotropic approximation:
µUT = !vx2
t is the same in both cases
[5] E. Ungersboeck and H. Kosina, Proc. SISPAD, p. 311, 2005
45
Degenerate bulk systems Electrons in silicon at room temperature, zero field
PSD of velocity
Svv = 4 !vx2 (1" f0 )
Full:
Isotropic approximation:
Svv = 4!vx
2
1" f0
Both approximations fail!
46
Degenerate bulk systems Electrons in silicon at room temperature, n=1021/cm3
Comparison with exact analytical solutions for zero field
Simulations with and without Pauli principle
1D NPN BJT
1D NPN BJT
VCE=0.5V
SHE can handle small currents without problems
50nm NPN BJT
Modena model for electrons in silicon with analytical band structure
1D NPN BJT VCE=0.5V
SHE can handle huge variations in the density without problems
VCE=0.5V, VBE=0.55V
1D NPN BJT
Transport in nanometric devices requires at least 5th order SHE
VCE=0.5V, VBE=0.85V
Dependence on the maximum order of SHE
1D NPN BJT
A 2nm grid spacing seems to be sufficient
VCE=0.5V, VBE=0.85V
Dependence on grid spacing
1D NPN BJT
Rapidly varying electric fields pose no problem Grid spacing varies from 1 to 10nm
VCE=3.0V, VBE=0.85V
1D NPN BJT VCE=1.0V, VBE=0.85V
1D NPN BJT Collector current noise due to electrons, VCE=0.5V, f=0Hz
Up to high injection the noise is shot-like (SCC=2qIC)
1D NPN BJT Collector current noise, VCE=0.5V, f=0Hz
Spatial origin of noise can not be determined by MC
1D NPN BJT Collector current noise due to electrons, VCE=0.5V
MC can not cover the full frequency range
57
Conclusions
58
Conclusions
• Noise can be calculated based on the Langevin Boltzmann equation
• Allows full AC analysis, arbitrary frequencies and simulation of rare events
• Enables the investigation of slow processes (e. g. 1/f noise) based on the full BE
• Calculation of cyclostationary noise for Si based on the full BE
• Device solutions of the LBE including the spatial origin of current noise