Scattered Simulation Spatio-temporal Evolution of Local

VLSI DESIGN2001, Vol. 13, Nos. 1-4, pp. 205-209Reprints available directly from the publisherPhotocopying permitted by license only

(C) 2001 OPA (Overseas Publishers Association) N.V.Published by license under

the Gordon and Breach Science Publishers imprint,member of the Taylor & Francis Group.

Scattered Packet Method for the Simulationof the Spatio-temporal Evolution of Local Perturbations

P. GAUBERTa, L. VARANIa’*, J. C. VAISSIlREa, J. P. NOUGIERa, E. STARIKOVa’b,P. SHIKTOROVb and V. GRUZHINSKISb

aCentre d’Electronique et de Micro-Optoklectronique de Montpellier (CNRS UMR 5507),Universitk Montpellier II, Place Eugkne Bataillon, 34095 Montpellier cedex 5, France,"

bSemiconductor Physics Institute, Goshtauto 11, 2600 Vilnius, Lithuania

To calculate the noise in submicron structures we need the knowledge of the local noisesources and the generalised impedance fields. The scattered packet method has beenused to obtain both quantities at the same microscopic level. Numerical procedures usedfor the calculation of velocity and energy noise sources and the simulation of the spatio-temporal evolution of local perturbations introduced on velocity or energy are describedand some results obtained for p-type silicon are given.

Keywords: Noise; Impedance field; Scattered Packet Method

INTRODUCTION

The classical impedance field method [1] is one ofthe most powerful techniques to calculate electro-nic noise in semiconductor structures. This methodfails when applied to deep submicron devices dueto the presence of spatial correlations betweennoise sources [2]. To overcome this drawback, anew technique (generalised impedance field) hasbeen developed in the frame of a hydrodynamicsimulator [3]. Analogously to the classical method,the calculation of electronic noise requires theknowledge of two quantities: the local noise sourceand the generalised impedance fields. Instead ofusing a hydrodynamic approach to obtain thegeneralised impedance fields and a Monte Carlo

simulation for the noise sources, we have usedthe Scattered Packet Method [4-6] to computeboth quantities at the same microscopic level ofdescription.

NUMERICAL PROCEDURE

We have developed the Scattered Packet Method(SPM) a few years ago in order to solve theBoltzmann equation for carriers in semiconductorsusing only the output term in order to increase thestability of the resolution. It follows the spatio-temporal evolution of the carrier population n indifferent cells of the phase space using an evolutionoperator [B] that gives n(t+At) when applied to

*Corresponding author. Tel.: (+ 33) 467143822, Fax: (+ 33) 467547134, e-mail: [email protected]

205

206 P. GAUBERT et al.

n(t). Written in a matrix form:

this operator can be represented using threematrices corresponding to the displacement in realspace [Br], displacement in k-space [Be] and to thecollisions [Bcold:

[B] [Br] + [Be] + [Bcou]- 2[I]

[/] is the identity matrix.We choose a time step At sufficiently small (be-

tween 0.1 and fs) in order to follow the transientevolution of the Boltzmann equation and to verifythat during the time interval a carrier can only gofrom one cell to its neighbours in k and r spaces.

In that case the matrix [Br] is tridiagonal for 1Dsimulations in real space. The matrix elementsdepend only of the value k of the wave vector ofthe initial mesh (we can neglect the acceleration ofthe electric field) and are calculated using theoverlap of the initial cell and the others cells (inr-space) after the time interval At.The matrix elements of [Be] are given by the

overlap of the initial cell and its neighbours(between and 9) in k-space after the time intervalAt. Values are tabulated for different electric fields.

Using a small sub-mesh we calculate theprobability for a carrier to go from one cell tothe others cells of the k-space due to impurityscattering and interactions with acoustic andoptical phonons. The corresponding matrix [Boou]is sparse and tabulated.

Finally, the matrix element BML of [B] is thetransition probability from the cell number L tothe cell number M during the time interval At.

In the frame of the generalised impedance fieldmethod [7], we have calculated the noise sourcesand the generalised impedance fields using theSPM.

Noise Sources

The matrix [Bcon] is employed to obtain the localnoise sources corresponding to the acceleration

fluctuations of velocity and energy [8] which are

given by:

NAtZNL M

N is the total number of carriers used in thesimulation, NL the population of the cell numberL, a, aM, /3C and /3M are the hydrodynamicvelocity (v) and energy (e) corresponding to thisdifferent cells in the k space. For p-type silicon,results are presented, versus the mean en-

ergy (different static electric fields applied to thematerial) on Figure for velocity (a) and en-

ergy (b) and different doping levels between2.5 1014cm -3 and 1018cm -3. Full and open

10

5o

10 kV/em

20 kV/em

2.5 1014cm-3

1016cm-1017cm_3

108crn--60 70 80 90 100

Energy (meV)110

11

10-

9

7

,. 6

5

4

350

(b) +o

qq q 2.5 104cm-3

o + 1016cm-3

’t?? o 1018cm_3

60 70 80 90 100

Energy (meV)

110

FIGURE Local noise sources versus energy for p-type Siand different doping levels associated with velocity (a) andenergy (b) accelerations.

SCATTERED PACKET METHOD 207

triangles on (a) correspond to different sizes usedfor the k-space mesh in the simulation. Thevelocity noise source increases with energy anddoping concentration while the local noise sourceassociated with the energy shows a quasi-linearbehaviour independently of the doping. The cross-correlated noise source term between velocity andenergy decreases, starting from zero, and does not

depend on the doping. This source is not shownhere because its contribution to the total noise is

generally negligible.k=0 0

Generalised Impedance Fields

To calculate the generalised impedance field, it isnecessary to follow the spatio-temporal evolutionof a local perturbation of velocity or energy.Starting from a stationary state, the perturbationis introduced by modifying the local carrierdistribution function. A special procedure hasbeen developed in order to perturb only one

parameter (for instance velocity) without changingthe other (energy). If Ns(k, O,x) is the stationarypopulation of the cell (k, 0) in x, we transfer asmall number AN of carriers from the cell a to thecell b as shown on Figure 2. The perturbation mustbe small in order to preserve the linearity of theresponse of the system and obtain the small signalgeneralised impedance field:

AN Ns(k, O, x)F(x)Cv

with Cv-Av(x)/(2Vp); Vp is the absolute value ofthe sum of the velocities of carriers located at xwith a direction 0, between x and the wave vectork, greater than 7r/2; Av(x) is the value of the localperturbation we want to introduce at x. TypicallyAv(x)- O. v(x) and 103 m/s for low values of v(x).F(x) with 0 < F(x) < is a function of x adjustedto obtain a narrow gaussian shape along x. Thenusing the evolution operator coupled to a

Poisson solver, we follow the temporal evolutionof the perturbation of the distribution functionalong the electric field (0 =0 and 0-7r) presented

k Oj

FIGURE 2 Introduction of a local perturbation of thevelocity (without perturbing the energy) in the k space inspherical coordinate at point x.

-6

2 t=0.05psl3 t=0.1ps

t=0.41PS \5 t=lps \

6 1"9 P__S ’3\,

-3 -2 -1 0 2 3

k (109 m-1)

FIGURE 3 Temporal evolution of the perturbation of thedistribution function along the field direction at point x00.6gm where the initial perturbation on velocity was intro-duced.

on Figure 3 for x0-0.61am where the initialperturbation (curve 1) of velocity was intro-

duced. Starting from this intitial conditions, the

208 P. GAUBERT et al.

0.05

0.00

-0.05

-0.10

-0.15

-0.20

/:’

[/ t=Ops2 0.02 ps3 O.05ps4 t=O.1 ps5 t= 0.18 ps6 0.41 ps

17 t=lps

7

’".i,’-6"//.

0.60.5 0.7

FIGURE 4 Response of electric field for different times whena velocity perturbation is applied at x0 =0.6 gm at t- 0 ps in a

17p-type Si resistance. NA-- 10 cm- 3, E 10 kV/cm.

spatio-temporal evolution of the electric field iscalculated and presented in Figure 4 for an appliedelectric field E 10 kV/cm. At time t- 0, we haveintroduced a small perturbation of the localvelocity (and distribution function), but the localenergy, carrier concentration n(x) and electric field(see curve 1) are not modified. Then the localelectric field is perturbed and a maximum isobserved at about 0.2 ps. The spatial integrationof this perturbation gives the voltage responseGj(x0, t) at the device contacts plotted on Figure 5

for three different electric fields:

oh(x0, t) t)- U (x0)]

AvNs(xo) is the total variation of the velocityintroduced at x0. This sum must be extended to allthe cells of the x mesh if the initial perturbation isnot exactly located at x-x0. By Fourier trans-forming the response function we calculate thegeneralised impedance fields associated with velo-city. Figure 6 gives the real part, imaginary partand modulus of the generalised impedance field

vZv associated with velocity for three electricfields. A similar procedure can be used [8] tocalculate the generalised impedance field asso-ciated with energy. In that case the local energyperturbation is introduced by moving in the k-space part of the carriers to higher energy withoutchanging the direction of their velocity. Using thisprocedure, the local energy is increased and thelocal velocity Av(x) is modified. To suppress thisperturbation of the velocity we use the sameprocedure as described before to introduce avelocity perturbation without changing the energy.Then the voltage response function and thegeneralised impedance fields associated with en-ergy are calculated. Using the different noisesources and generalised impedance fields the

10

0 2 5

l̂i E=I0 kV/cm

[M_ E=25 kV/cm

4

Time (ps)

FIGURE 5 Voltage response function for a perturbation ofvelocity applied at x0 0.6 gm for three electric fields; in a gmp-type Si resistance, NA 1017 cm-3

(VZv) o0" I--- I.m(VZv) 10 kV/cm I1

0 2 40 10 10 10 10 10

frequency (GHz)

FIGURE 6 Real part, imaginary part and modulus of thegeneralised impedance field associated with velocitY17for _th3reeelectric fields in a gm p-type Si resistance, NA-- 10 cm

SCATTERED PACKET METHOD 209

1.4

1.2

0.4

0.2

0.0

10 101 102 10 10 10

frequency (GHz)

FIGURE 7 Spectral density of voltage fluctuation and itsdifferent contributions associated with velocity-velocity (vv),energy-energy (ee) and velocity-energy (re) terms for a gmp-type Si resistance, NA 1017 cm-3, E-- 10 kV/cm.

spectral density of voltage and its differentcontributions are obtained"

uncorrelated in time and space; they are obtainedusing the collision matrix of the evolution opera-tor. The small signal generalised impedance fieldsare calculated by studying the evolution in k, rand t, of perturbations of the velocity or energyintroduced on the local distribution function of thecarriers. Using these two quantities the spectraldensity of voltage fluctuation is obtained easily bya spatial integration.

Acknowledgements

This work have been performed with the supportof the high-level grant DRB4/MDL/no 99-30 ofthe French Ministate de l’Education Nationale, dela Recherche et de l’Industrie and the French-Lithuanian bilateral Cooperation no 5380 ofCNRS.

"L

Su(f 7Zv(x,f)Sg(X)ns(X)7Z(x,f)dx

+ VZ(x,f)ii(x)n(x)VZ(x,f)dx

+ VZ(x,f)i(x)n(x)VZ2(x,f)dx

+ VZs(x,f)Si(X)ns(x)VZ(x,f)dx

We can see on Figure 7 that the main contribu-tion to the total noise comes from the first term ofthe sum associated with velocity and that thecontribution of sum of the two last terms isnegative and practically negligible.

CONCLUSION

Using the SPM, we have calculated at the samemicroscopic level the noise sources associated withvelocity and energy accelerations and the general-ised impedance fields. These sources are bydefinition independent of the frequency and

References

[1] Shockley, W., Copeland, J. A. and James, R. P., "QuantumTheory of atoms, molecules and the Solid State", Ed. P.O.Lowdin, Academic, New York, 1966.

[2] Nougier, J. P., Vaissiere, J. C. and Gontrand, C. (1983)."Two-poing correlation of diffusion noise sources of hotcarriers in semiconductors", Phys. Rev. Lett., p. 513.

[3] Shiktorov, P., Starikov, E., Gruzhinskis, V., Reggiani, L.,Gonzalez, T., Mateos, J., Pardo, D. and Varani, L. (1998)."Acceleration fluctuation scheme for diffusion noise sourcewithin a generalised impedance field method", Phys. Rev.B, p. 11866.

[4] Vaissire, J. C., Nougier, J. P., Varani, L., Houlet, P., Hlou,L., Reggiani, L., Starikov, E. and Shiktorov, P., "Numer-ical solution of the pertubed Boltzmann equation infrequency and time domains", 3rd IWCE, (Portland, 1994),pp. 53-56, Ed. Goodnick, M., Oregon State University.

[5] Aboubacar, M., Houlet, P., Nougier, J. P., Varani, L. andVaissibre, J. C. (1996). "Transport and noise calculationsin semiconductor structures with the scattered packetmethod", Lithuanian Journal of Physics, 36(6), 583-587.

[6] Vaissire, J. C. (1997). "The scattered packet method as anew tool for microscopic investigation of noise", 14thICNF, Leuven, Belgium, Eds. Claeys, C. and Simoen, E.(World Scientific), pp. 130-135.

[7] Shiktorov, P., Starikov, E., Gruzhinskis, V., Gonzalez, T.,Mateos, J., Pardo, D., Reggiani, L., Varani, L. andVaissire, J. C., "Langevin Forces and GeneralizedTransfer Fields for Noise Modeling in Deep SubmicronDevices", 7th IWCE, (Glasgow, 2000).

[8] Gaubert, P., "Nouvelle m6thode de simulation du bruit61ectronique dans les composants submicroniques", Phdreport (Montpellier, December, 1999).

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Scattered Simulation Spatio-temporal Evolution of Local