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