Time-dependent Monte Carlo Simulation

Transcription

1 Computational Electronics Group University of Illinois Time-dependent Monte Carlo Simulation Umberto Ravaioli Beckman Institute and Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign Urbana, IL 61801, USA

2 Monte Carlo simulation is time-dependent but to obtain true time-dependent results, the transients need to be resolved self-consistently. Noise is a difficult issue. Often, techniques to reduce noise apply averaging in a way that assumes steady-state, making the transient not physical. In steady-state simulation, we usually apply bias assuming ideal voltage sources. This eliminates the displacement current. In a transient simulation where the external circuit is present, displacement current must also be included. Is the electrostatic Poisson equation sufficient to describe a transient situation?

3 1-D simulation Consider a 1-D structure of length L, cross-sectional area A o and applied bias V D. The displacement current may be averaged and lumped in a single circuit parameter (we consider only electron current for simplicity) It () = I( xt,) + ε A total current e electron current L 1 E( x, t) It () = Ie( xt,) + ε Ao dx= L t 0 L 1 ε Ao V = Ie ( x, t) dx + L L t 0 o E( x, t) t displacement current D

4 The term ε A L Cold Capacitance o V t D contains all the information about the displacement current. It can be represented by an external capacitance, included in the circuit connected to the device. ε A L o = C c = cold capacitance I e I total C c I d V D

5 Example: Gunn diode circuit The non-linear part of the diode is simulated by using Monte Carlo (3-valleys non-parabolic model). Displacement current is accounted for by a time-dependent voltage boundary condition at the terminals of the cold capacitance.

6 Kirchhoff s equations di() t VD() t = VB L dt External circuit equations dv () () () D t V It = I t+ C + D C = C+ C dt R d tot tot cold Finite difference discretization in time t It ( + t) = [ VB VD( t) ] + I() t L t VD( t + t) = [ I() t Id() t ] + VD()1 t C tot t RC

7 Gunn diode test structures Homojunction Notch oscillator Heterojunction Heterojunction + doping spike

8 Band structure valleys in GaAs and AlGaAs

9 Structure 1: accumulation domain cycle Dead zone no leading depletion, dipole not formed Γ valley L valley X valley

10 Structure 2: well-formed domain cycle Domain forming, no dead zone Leading depletion (dipole) Γ valley L valley X valley

11 Electric potential evolution - comparison signature of dipole domain no inflection - accumulation domain Notch oscillator - accumulation domain Hetrojunction cathode oscillator - well-formed domain

12 Structure 3: accumulation domain cycle, again Γ valley L valley X valley

13 Comparative Results

14 Multi-dimensional device Current through a contact over a time-step t qz d E I() t = ( N ) a N i + εsε 0 Z dx z t dt Z = particles particles absorbed in t injected in t actual lateral width of device z = simulated lateral width displacement current over contact length qz Qt () = ( Na Ni) + εε s 0 Z E ( xtdx,) z cumulative terminal charge ( Na, Ni integrated up to time t) (Hockney and Eastwood, 1981)

15 Example: GaAs MESFET (Patil and Ravaioli, Solid-State Electronics,1991) The cumulative charge contains a steady-state contribution that can be extrapolated and subtracted, to obtain a transient contribution.

16 Multi-dimensional device

17 The fitting coefficients are found from imposing Cumulative charge decomposition Q () t = a t+ a t < t < t ss 1, ss 0, ss ss total N Q () t = a t tr n= 0 n, tr dq di() t d Q It () = and = dt dt 2 dt n tr ss ; tr ss 2 2 dq dq d Q d Q = = t = dt dt dt dt t ss (Patil and Ravaioli, Solid-State Electronics,1991)

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21 Time dependent noise analysis Prerequisite of noise analysis is to obtain accurate values of the instantaneous currents at the terminals, including the displacement contributions. For a simple quasi-2-d MESFET contact configuration (side contacts), it is possible to formulate an accurate current evaluation (Gruzinskis, Kersulis, Reklaitis, 1991). 0 x g1 x g2 x d Gate Source n+ n n+ h M y meshes Drain V D Semi-insulating substrate

22 V s Terminal Currents Source { 0Æ x } g1 Â È 1 Is() t = ÍQ vxi() t xg1 Í i ÍÎ = M y e - Â D yj xg yj t - xg yj t-dt D t j= 1 0 (ground) ( j( ) 1,, ) j( 1,, ) Q represents the linear charge density associated to a simulated particle (a rod of charge in the direction perpendicular to the simulation domain).

23 Terminal Currents Drain j= 1 { x } g2 Æ xd Â È 1 Id() t = ÍQ vxi() t ( xd - xg2) Í i ÍÎ M y e -  D yj xg yj t - xg yj t-dt D t e h V t V t t D t ( j( ) 2,, ) j( 2,, ) ( ( ) ( )) - d - d -D Terminal Currents Gate I () t = I () t - I () t g s d

24 Current-Noise operation The gate and drain voltages remain constant in time, and the fluctuations of the short-circuit currents and their correlations are investigated. The current-noise sources are represented as two correlated current generators at input and output of the device. Since the drain voltage is constant, the last two terms in the drain current equation, cancel each other. i g Noiseless MESFET i d C

25 Voltage-Noise operation The gate voltage and the drain current remain constant in time and the fluctuations of the short-circuit gate current and open circuit drain voltage and their correlation are investigated. The noise sources are represented as correlated current generator in parallel at input and voltage generator in series at the output. V d i g Noiseless MESFET C

26 Voltage-Noise operation If we consider a constant drain current I do the instantaneous drain voltage is j= 1 { x } g2 Æ xd Â Ê ˆ D t Vd() t = Vd( t-dt) - Á Id0( xd - xg2) -Q vxi() t e há i Ë My 1 +  D yj xg yj t - xg yj t-dt h ( j( ) 2,, ) j( 2,, ) This voltage is evaluated by applying an iterative technique. Poisson equation is repeated at the end of each step: first with the old value of V d and the new carrier spacedistribution, and then with the value of V d update as above by using the values of ϕ from the preceding solution.

27 Transient electromagnetic effects - Maxwell s equations B( t) Dt ( ) = ρ E() t = t Bt () = 0 Dt () H () t = + J t Dt ( ) = ε Et ( ) Bt = µ Ht () () A φ B = A E = φ A= µε t t 2 2 A A= A A= µ J = εµ ε 2 t A t

28 Wave equations 2 2 A A εµ = µ J 2 t 2 ρ φ A = t ε 2 φ ρ φ εµ = 2 t ε Instead of Poisson equation, one should solve the retarded potential wave equation, with time-dependent voltage boundary conditions. 2

29 When is full wave analysis necessary? Example: GaAs p 6 3 ε = 12.9 ε µ = µ n ( T = 300 K) = cm o µ = 8500 cm / V s µ = 400 cm / V s f n = 1.0 THz ωε = S/m i Perfect dielectric σ = 0 λ = c 83.5 µ m ε f r Intrinsic GaAs σ = σ ωε λ 83.5 µ m ( µ µ ) qn i n p S/m imperfect dielectric

30 n-type GaAs σ qµ n n When is full wave analysis necessary? 2 σ λ = f µε ωε o 11 Worst case analysis: assume in all cases the mobility for undoped material. Actual wavelength will be somewhat longer than below. 2 n σ λ cm S/m 83.1 µm cm S/m µm cm S/m µm cm S/m 8.55 µm cm S/m 2.7 µm

31 Conclusions: In the active regions of microwave and highspeeddevices with moderate dopings, the wavelength is expected to be much larger than the space scale and the electrostatic Poisson equation is adequate. Modern MESFET structures may have channel concentrations of up to cm -3. Devices with conduction channels at heterointerfaces may exhibit high concentrations and very high mobilities. Because of the large conductivity, the wavelength may be comparable with the space scale of active regions in certain regimes. For reference, the walength in copper 7 When is full wave analysis necessary? σ = S/m ε= ε µ = µ f = 1.0 THz λ 0.5 µm o o

32 Real Space Transfer Devices The real space transfer effect applies to carriers crossing a heterojunction, after gaining sufficient energy from the field along the interface.

33 Collector Up Real Space Transfer Transistor (RSTT) This is similar to a standard MODFET device, with gate biased to attract electrons.

34 Simulated RSTT Structure For scaling studies, the length of the gate L is kept equal to its distance from source and heater.

35 Details of the Hetero-interface

36 Experimental curves for a sample RSTT Source current saturates; heater current has slight Negative Differential Resistance (NDR) region; collector current shows large leakage at V H = 0.

37 Monte Carlo simulation results for RSTT

38 Monte Carlo simulation results for RSTT

39 Monte Carlo simulation results for RSTT

40 Channel behavior with varying heater voltage ( L= 1.0µm, W = 0.21 µm, V = 1.8 V) C

41 Net Real Space Transfer rate ( L= 1.0µm, W = 0.21 µm, V = 1.8 V) C

42 Actual Real Space Transfer from and into channel ( L= 1.0µm, W = 0.21 µm, V = 1.05 V, V = 1.8 V) H C

43 Total RST scross heterojunction ( L= 1.0µm, W = 0.21 µm, V = 1.8 V) C

44 Scaling with L ( W = 0.21µm, V = 1.8 V) C

45 Scaling with W ( L= 0.4µm, V = 1.8 V) C

46 Collector current transient Video animation: switching of RSTT simulated with Monte Carlo