Modeling of noise by Legendre polynomial expansion of the Boltzmann equation

Transcription

1 Modeling of noise by Legendre polynomial expansion of the Boltzmann equation C. Jungemann Institute of Electromagnetic Theory RWTH Aachen University 1

2 Outline Introduction Theory Noise Legendre polynomial expansion Results Bulk 1D NPN BJT Conclusions 2

3 Introduction 3

4 Introduction Electronic noise is found in all semiconductor devices and circuits Noise is fundamental and cannot be avoided (E. g. Nyquist noise: S VV = 4k B TR) Noise degrades the performance of circuits (E. g. Noise limits the minimum signal that can be detected) Noise occurs in all frequency ranges 4

5 Introduction Stochastic electron motion in a constant electric field Current fluctuates due to particle scattering (and trapping) (displacement current) 5

6 Introduction Noise analysis for stationary processes Current fluctuations: δ It () = It () I Correlation function: ϕ ( τ) = δit ( + τ) δit ( ) II Power spectral density: jωτ SII ( ω) = 2 ϕii ( τ) e dτ Electric power absorbed in a resistor by current fluctuations: dp( f ) = S (2 π f ) Rdf II 6

7 Introduction Macroscopic models (often approximations): Nyquist noise: (equilibrium) S II ( ω ) = 4k TG B Shot noise: (non-equilibrium) SII ( ω ) = 2qI The same microscopic origin How can we calculate noise on a microscopic basis? 7

8 Introduction Power spectral density can be calculated by Monte Carlo method, which solves the Boltzmann equation. 8

9 Introduction MC simulation of a 1D N + NN + Si structure biased at 6V Floating body problem 9

10 Introduction 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 " 10

11 Introduction 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 11

12 Theory Noise 12

13 Theory Two LTI-systems (linear, time-invariant) Time domain (convolution, h y (t): impulse response): t y(t) = h y (t t ') x(t ')dt ' " t z(t) = h z (t t ') x(t ')dt ' " Frequeny domain: Y ( ) = H y ( )" X ( ) " Z( ) = H z ( )" X ( ) Cross correlation of " y(t) and z(t) : t+ yz ( ) = y(t + )" z(t) = h y (t + t ') x(t ')dt ' h z (t t ') x(t ')dt ' t t+ " = h y (t + t '') x(t '') x(t ') h z (t t ')dt ''dt ' = h y (t + t '') xx (t '' t ')h z (t t ')dt ''dt ' "# " Wiener-Lee theorem: t t " t+ "# " S yz ( ) = H y ( )S xx ( )H * z ( ) " S yy ( ) = H y ( ) 2 S xx ( ) 0 $ # stochastic stochastic deterministic correlated % 13

14 Theory Poisson-type noise Example: Injection of independent particles over a barrier with rate, current: I 0 = q Power spectral density of a Poisson process (white noise): S( )=2 Power spectral density of current fluctuations (shot noise): S II ( ) = 2q 2 = 2qI 0 PSD of independent events that occur at a rate is 2 14

15 Theory Shot noise passing through a low pass filter S xx ( ) = 2qI 0 H y ( ) = 1 1+ j" ; = RC S yy ( ) = 2qI " 2 S yy ( ) = H y ( )S xx ( )H y * ( ) = 1 1+ j" 2qI j" = 2qI " 2 15

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 " E T k f ( k,t) = sys (2 ) W ( k k ') f ( k ',t) W ( k ' k ) f ( k,t)d 3 k ' 3 " (in) " (out) drift (deterministic) scattering (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 ' 16

17 Theory Under stationary conditions the rate is given by: ( k, k ') = W ( k k ') f 0 ( k ') where f 0 ( k ') is the stationary distribution function. To calculate noise the impulse response of the distribution function h f ( k, k ',t,t ') is required: "h f "t # q " E T $ k h f = % sys (2 ) & W ( k k '')h 3 f ( k '', k ',t,t ') #W ( k '' k )h f ( k, k ',t,t ')d 3 k '' + 1 (2 ) 3 ( k # k ') (t # t ') h f ( 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. 17

18 Theory Stationary system h f ( k, k ',t t ',0) = h f ( k, k ',t,t ') Transfer function: " H f ( k, k ', ) = # h f ( k, k ',,0) e j" d " Solving directly in the frequency domain yields: j H f q " E T $ k H f = % sys (2 ) # W ( k k '')H 3 f ( k '', k ', ) W ( k '' k )H f ( k, k ', )d 3 k '' + 1 (2 ) 3 ( k k ') 18

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 annihilation The 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 ( k 1, k 2, ) = 4 sys (2 ) G( k 6 1, k ', k '', )W ( k ' k '') f 0 ( k '')G * ( k 2, k ', k '', )d 3 k 'd 3 k '' 19

20 Theory Expected values: x(t) = 2 X ( k ) f ( k,t)d 3 k (2 ) 3 PSD of two macroscopic quantities x and y: S xy ( ) = 1 (2 ) 6 X ( k 1 )S ff ( k 1, k 2, )Y ( k 2 )d 3 k 1 d 3 k 2 This eqution is too CPU intensive (12D integral): S xy ( ) = 4" sys X ( k (2 ) 12 1 ) G( k 1, k ', k '', ) W ( k ' k '') f 0 ( k '')Y ( k 2 ) G * ( k 2, k ', k '', )d 3 k 1 d 3 k 2 d 3 k 'd 3 k '' 20

21 Theory 1 3 S xy ( ) = X ( k1 )G( k1, k ', k '', )d k1w ( k ' k '') f 0 ( k '') 6 "" 3 " (2 ) (2 ) * Y ( k )G ( k, k ', k '', )d k d k ' d k '' " (2 ) 4 sys * 3 3 = G ( k ', k '', )W ( k ' k '') f ( k '')G ( k ', k '', )d k ' d k '' X 0 Y 6 "" (2 ) 4 sys with G X ( k ', k '', ) = 1 3 X ( k )G( k, k ', k '', )d k " (2 ) 1 3 = X ( k1 ) $% H f ( k1, k ', ) # H f ( k1, k '', ) &' d k1 = H X ( k ', ) # H X ( k '', ) 3 " (2 ) with 1 H X ( k ', ) = X ( k1 )H f ( k1, k ', )d 3k1 3 " (2 ) H X ( k ', ) is a direct solution of the adjoint BE, CPU time similar to solving for f 0 21

22 Theory Example: PSD of velocity fluctuations v(t) at equilibrium S vv ( ) = 4 sys (2 ) H 6 v ( k ', ) H v ( k '', ) 2 W ( k ' k '') f 0 ( k '')d 3 k 'd 3 k '' colored colored white noise =4 v x 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 " H X (k ', ) 1 lim " S XX ( ) 1 2 Noise of all observable quantities vanishes at high frequencies. 22

23 Theory Legendre Polynomial expansion 23

24 Spherical harmonics Theory k-space energy-space (angles are the same as in k-space) (k x,k y,k z ) (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 Y l,m (, ): Y 0,0 (,") = 1 4# Y 1,$1 (,") = Y 1,0 (,") = Y 1,1 (,") = 3 4# sin sin" 3 4# cos 3 4# sin cos" ε '' Y l,m (,")Y l',m' (,")d% = & l,l' & m,m' d% = sin dd" 24

25 Theory Spherical harmonics expansion: X l,m () = "%% X( k(,",#))y l,m (",#)d$ ( l ' m=&l X(,",#) = ' X l,m () Y l,m (",#) = X l,m ()Y l,m (",#) l=0 ' l,m Example: group velocity (spherical band structure) ) sin "cos#, v = v() + sin "sin #. * + cos" -. Nonzero elements: = v() 4/ v 1,&1 = v() e 3 y, v 1,0 = v() Only three nonzero elements 4/ 3 ) Y 1,1 (",#), +. + Y 1,&1 (",#). + * Y 1,0 (",#). - 4/ 3 e z, v 1,1 = v() 4/ 3 e x 25

26 Theory Spherical harmonics expansion of the distribution function: g l,m (,t) = 1 #( $ ( k) )Y (2") 3 l,m (%,&)f ( k,t)d ' 3 k = Z() Y l,m (%,&)f ( k(,%,&),t) d( (spherical bands) "'' with the (reduced) density-of-states (DOS) Z() = k2 )k (2") 3 ) Expectations: 2 x(t) = X( ' k)f ( k,t)d 3 k (2") 3 = 2* ' X l,m ()g l,m (,t)d l,m 26

27 Examples Theory Particle density: 2 n(t) = " f ( k,t)d 3 k (2) 3 1 = 4Y 0,0 ( ) = 2 4 " g (#,t)d# 0,0 Only the zero order component carries charge Particle current density (spherical bands): 2 j(t) = " v( k)f ( k,t)d 3 k = 2 4 (2) 3 3 % ' v(#) ' ' & ' Only the first order components carry current " g 1,1 (#,t) g 1,$1 (#,t) g 1,0 (#,t) ( * * d# * ) * 27

28 Theory Spherical harmonics expansion of the Boltzmann equation: 1 " # $ #( ' ( k) )Y (2) 3 l,m (%,&){ BE}d 3 k ( Balance equation for g l,m : )g l,m )t with $ q E ) j T l,m )# + * = { l,m Ŵl,m g } 1 " # $ #( ' ( k) )Y (2) 3 l,m (%,&) = ) )t +, - )f )t. / d 3 k "(# $ #( k)., ' )Y (2) 3 l,m (%,&)fd 3 k/ - 0 = )g l,m )t 28

29 Theory Drift term: 1 "(# $ #( ( k) )Y (2). 3 l,m (%,&)) $ q * " E T = $1 q (2) 3. " E T ' " k f "(# $ #( k) " )Y l,m (%,&)' k fd3 k +, d 3 k - = $q E 6/ vgy T l,m $ 1 0 /Y l,m /# "k /% e + 1 /Y l,m 3 2 e % sin % /& & g 9 8 ; 78 :; d< with jl,m = 0 3 #.. vg Y l,m d< g(#,%,&,t) = > g l',m' (#,t)y l',m' (%,&) and = l,m = q E T #.. = $q E T / "k # /Y l,m /% e % + 1 sin % /Y l,m /& l',m' e & gd< jl,m /# + = l,m 29

30 Theory l,m (",t) = q E T = q E T "k(") = q E T "k(") with "k(") #-- & ( ' #Y l,m #$ e $ + 1 sin $ #Y l,m #% e % ) + * g(",$,%,t)d,. & #Y l,m e #$ $ + 1 #Y l,m ) #-- ( e sin $ #% % + ' * Y d, l',m' 3g l',m' (",t) l',m' / 0 23 b l,m,l',m' g l',m' (",t) 4 l',m' & #Y b l,m,l',m' = l,m e #$ $ + 1 #Y l,m ) #-- ( e sin $ #% % + ' * Y d, l',m' b l,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. 30

31 Theory jl,m (,t) = "%% v(,",#)yl,m (",#)g(,",#,t)d$ = v() a l,m,l',m' g l',m' (,t) & l',m' with a l,m,l',m' = Y l,m e Y l',m' d$ "%% a l,m,l',m' has the same odd/even coupling property as b l,m,l',m'. 31

32 Theory Scattering integral (neglecting Pauli principle): Ŵ{ f } = s & (2") $ W 3 # ( k, k')f ( k',t) % W # ( k', k)f ( k,t)d 3 k' # Transition rate of process # (constant energy transfer, depends only 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 ('( k) * #l ) + with 1 $ 0 & l=0 c #l ('( k) * ) + = 2" P (u)c ( '( k),u* l # ) + du %1 l & m=%l Y l,m (.,/)Y l,m (.',/') 32

33 Theory Velocity randomizing scattering (e.g. phonons): c l #"( k) % $ & = 4'c # "( k) % $ & ( l,0 Projection of the scattering integral: 1 (2'), ((" ) "( k) )Y 3 l,m (*,+)Ŵ { f }d 3 k = Ŵl,m g { } Ŵ l,m { g} =. Z(")c l # $ " % & g (" ) -,t) ) Z(" + - )c #" + - l,m 0 $ % & g (",t) l,m { } The projected scattering integral is local in l,m. Only in the case of a full band structure or inclusion of the Pauli principle this is no longer the case. The scattering integral is nonlocal in energy. 33

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) 34

35 Theory 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 35

36 Results Bulk 36

37 Stationary bulk results EDF for 300kV/cm in <100> direction Rare events are easily simulated by SHE MC requires statistical enhancement which forestalls noise simulation Required for simulation of floating body problems 37

38 AC bulk results PSD of velocity for an electric field of 30kV/cm at room temperature Only phonon scattering Excellent agreement of MC and SHE 3rd order expansion sufficient for bulk SHE works at low and high frequencies 38

39 " AC bulk results Relative error of the velocity PSD for an electric field of 30kV/cm at room temperature 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 39

40 AC bulk results PSD of current for a doping of /cm 3 and an electric field of 10kV/cm 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 40

41 Cyclostationary bulk results E(t) = 50kV/cm*sin(2 f 0 t), f 0 =500GHz MC data: S. Perez et al., J. Appl. Phys., 88 (2), p. 800,

42 Cyclostationary Bulk results E( t) = E τ cm 17 0 [1 + cos(2πf 0t)], f0 = 1GHz, l = 5ns, N D = 10 / 3 For 1kV/cm only upconversion at f 0 For 30kV/cm velocity saturation leads to upconversion at multiples of f 0 Impossible to simulate with MC at technically relevant frequencies 42

43 Degenerate bulk systems Silicon, n=10 20 /cm 3 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 C ε T ε V 43

44 Degenerate bulk systems Electrons in silicon at room temperature, zero field Mobility Full: µu T = v 2 x (1" f 0 ) Isotropic approximation: µu T = v x 2 [5] E. Ungersboeck and H. Kosina, Proc. SISPAD, p. 311, 2005 is the same in both cases 44

45 Degenerate bulk systems Electrons in silicon at room temperature, zero field PSD of velocity Full: S vv = 4 v 2 x (1" f 0 ) Isotropic approximation: S vv = 4 v 2 x 1" f 0 Both approximations fail 45

46 Degenerate bulk systems Electrons in silicon at room temperature, n=10 21 /cm 3 Comparison with exact analytical solutions for zero field Simulations with and without Pauli principle 46

47 1D NPN BJT

48 1D NPN BJT Modena model for electrons in silicon with analytical band structure 50nm NPN BJT V CE =0.5V SHE can handle small currents without problems

49 1D NPN BJT V CE =0.5V, V BE =0.55V V CE =0.5V SHE can handle huge variations in the density without problems

50 1D NPN BJT Dependence on the maximum order of SHE V CE =0.5V, V BE =0.85V Transport in nanometric devices requires at least 5th order SHE

51 1D NPN BJT Dependence on grid spacing V CE =0.5V, V BE =0.85V A 2nm grid spacing seems to be sufficient

52 1D NPN BJT V CE =3.0V, V BE =0.85V Rapidly varying electric fields pose no problem Grid spacing varies from 1 to 10nm

53 1D NPN BJT V CE =1.0V, V BE =0.85V

54 1D NPN BJT Collector current noise due to electrons, V CE =0.5V, f=0hz Up to high injection the noise is shot-like (S CC =2qI C )

55 1D NPN BJT Collector current noise, V CE =0.5V, f=0hz Spatial origin of noise can not be determined by MC

56 1D NPN BJT Collector current noise due to electrons, V CE =0.5V MC can not cover the full frequency range

57 Conclusions 57

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 58