Modeling of noise by Legendre polynomial expansion of the Boltzmann equation
C. Jungemann
Institute of Electromagnetic Theory RWTH Aachen University
1
Outline
• Introduction • Theory
– Noise – Legendre polynomial expansion
• Results – Bulk – 1D NPN BJT
• Conclusions
2
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
4
Electronic noise is found in all semiconductor devices and circuits
Introduction
5
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
7
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
8
Power spectral density can be calculated by Monte Carlo method, which solves
the Boltzmann equation.
Introduction
9
MC simulation of a 1D N+NN+Si structure biased at 6V
Floating body problem
Introduction
10
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
11
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
12
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"
#$% 13
Theory
stochastic deterministic
stochastic
correlated
Poisson-type noise
Example: Injection of independent particles over a barrier with rate !, current:I0 = q!Power spectral density of a Poisson process (white noise):S(! )=2!Power spectral density of current fluctuations (shot noise):
SII (! ) = 2q2! = 2qI0
PSD of independent events that occur at a rate ! is 2!14
Theory
15
Theory
Sxx (! ) = 2qI0 Hy (! ) =1
1+ j!"; ! = RC Syy (! ) =
2qI01+! 2" 2
Syy (! ) = Hy (! )Sxx (! )Hy*(! ) = 1
1+ j!"2qI0
11! j!"
=2qI0
1+! 2" 2
Shot noise passing through a low pass filter
16
Theory How to calculate noise in the framework of the BE?(single spherical valley, non degenerate conditions, bulk system)
!f (!k ,t)!t
! q!
"ET! !k f (
!k ,t) =
!sys
(2! )3 W (!k |!k ') f (
!k ',t) !W (
!k ' |!k ) f (
!k ,t)d 3k '!
" (in) " (out) drift scattering (deterministic) (stochastic)Particles are instantaneously scattered at a rate (Poisson process):#(!k ,!k ',t) =W (
!k |!k ') f (
!k ',t)
W (!k |!k ') : Transistion rate of a particle to be scattered from
!k ' to
!k
f (!k ',t) : Probability that a particle is found in state
!k '
17
Theory
Under stationary conditions the rate is given by:!(!k ,!k ') =W (
!k |!k ') f0 (
!k ')
where f0 (!k ') is the stationary distribution function.
To calculate noise the impulse response of the distribution function hf (
!k ,!k ',t,t ') is required:
"hf"t
# q!
"ET$ !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.
18
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 ')
19
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 ''!!
20
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 ''
21
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 !
22
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.
23
Theory
Legendre Polynomial expansion
24
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,$1(!,") = 34#
sin!sin"
Y1,0 (!,") = 34#
cos!
Y1,1(!,") = 34#
sin!cos"
Yl,m (!,")Yl',m' (!,")d% = &l,l '&m,m' d% = sin!d!d"!''
ε
25
Theory Spherical harmonics expansion:Xl,m (!) = X(
!k(!,",#))Yl,m (",#)d$"%%
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!
26
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*
27
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!
28
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
29
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<#.. = $q
!ET /!jl,m/#
+ = l,m
with!jl,m = !vg#.. Yl,md< g(#,%,&, t) = gl',m' (#, t)Yl',m' (%,&)
l ',m'>0
12345
and
= l,m = q!ET
"k/Yl,m
/%!e% +
1sin%
/Yl,m
/&!e&
0
123
45gd<#..
30
Theory
! l,m (", t) = q!ET
"k(")#Yl,m
#$!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.
31
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'.
32
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
$
33
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
36
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
41
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
44
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
Top Related