Langevin Forces Generalized Transfer Fields Modelling Deep ...downloads.hindawi.com/journals/vlsi/2001/082542.pdf · nic devices is the impedance field method [1,2], a deterministic

VLSI DESIGN2001, Vol. 13, Nos. 1-4, pp. 85-90Reprints available directly from the publisherPhotocopying permitted by license only

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Langevin Forces and Generalized Transfer FieldsNoise Modelling in Deep Submicron Devices

P. SHIKTOROVa’*, E. STARIKOVa’a, V. GRUINSKISa, T. GONZ/LEZb, J. MATEOSb

D. PARDOb, L. REGGIANIc, L. VARANIa’t and J. C. VAISSIREa

aSemiconductor Physics Institute, Vilnius, Lithuania," bUniversidad de Salamanca, Salamanca, Spain;Universittt di Lecce, Lecce, Italy," a Universitk Montpellier II, 34095 Montpellier Cedex 5, France

We present a generalized transfer field method with the microscopic noise sourcesdirectly connected with the velocity and energy change during single scattering events.The advantages of this method are illustrated by hydrodynamic calculations of currentand voltage noise spectra in several two-terminal submicron structures.

Keywords: Noise; Modeling; Hydrodynamics; Submicron devices

INTRODUCTION

A standard technique to compute noise in electro-nic devices is the impedance field method [1,2], adeterministic approach able to provide analyticaland numerical solutions for the noise spectra ofelectronic devices at a hydrodynamic level [3].However, even in its advanced form [2] thismethod is not appropriate for deep submicrondevices where spatial correlations between noisesources cannot be neglected [4]. To overcome thesedifficulties, we have recently developed a newscheme [5] where the noise sources are directlyconnected with the velocity and energy changesduring single scattering events. The aim of thiswork is to implement the new scheme within ageneralized transfer field (GTF) method and

present the corresponding computational proce-dures which are then applied to noise calculationsin several submicron structures such as GaAsn+nn +, n + n-Schottky-contact and Si p+ndiodes operating under various conditions includ-ing voltage, current, and circuit-noise operationmodes.

GENERALIZED TRANSFER FIELDMETHOD

The GTF relates macroscopic fluctuations of thedevice characteristics under interest (e.g., current,voltage, etc.) to microscopic fluctuations of thedynamical variables of a single carrier (e.g.,velocity, energy and higher order moments of

* Present address: Semiconductor Physics Institute, Goshtauto 11, 2600 Vilnius, Lithuania. Tel.: (3702) 614920, Fax: (3702)627123, e-mail: [email protected]

*Corresponding author. Tel." / 33 4 67143822, Fax: / 33 4 67547134, e-mail: [email protected]

85

86 P. SHIKTOROV et al.

carrier momentum p) which are originated bysingle scattering events of carriers inside thedevice. At the kinetic level of description, thefluctuations of the carrier distribution function inmomentum space due to random scattering eventsare described by the microscopic Langevin force,(p,x,t). At the hydrodynamic (HD) level ofdescription, the fluctuations of average dynamicvariables such as velocity v(x,t), energy e(x,t),etc., are described by the HD Langevin force,

fa(p)(p, x, t)dp, where c v, e, etc. Withinthe Green-function formalism, the fluctuation ofany HD H-characteristic of a two-terminal devicecaused by the HD Langevin force can berepresented as (for simplicity let us consider aone-dimentional geometry):

f0L

f06H(x, t) dxo drG(x, xo, r)

(1)

Here H(x, t) :_ {n(x, t), v(x, t), e(x, t), E(x, t), ...)can represent various local physical quantities suchas carrier concentration, average velocity, averageenergy, conduction current density, electric field,electrical potential, etc., G(x, Xo, r) is the single-particle Green-function which describes the linearresponse of the H-characteristic at point x andtime to a local perturbation of the dynamicalvariable a of a single particle at point x0 and time

t-% and L is the device length. Owing to the f-like correlation of the Langevin forces both in timeand space, the spatio-temporal dependence of theauto-correlation (H H’) and cross-correlation(H:H’) functions of fluctuations of any twoH-characteristics takes the form:

(2)

where Sea(xo is the single particle spectral densityof the Langevin force and ns(xo) the carrier con-centration pertaining to stationary conditions.By neglecting carrier-carrier scattering S3(x)takes the form:

G (x)2

m(x) f dpf dp’[c(p)-c(P’)]

[fl(p) fl(p’)] W(p, p’)f,(p’, x)(3)

where a, 3 v, e, etc., f(p, x) is the steady-statedistribution function normalized to carrier con-centration at point x, and W(p, p’) is the collisionrate due to all scattering mechanisms.

In the general case of a local H(x, t)-character-istic such as the local conduction current,j(x, t) en(x, t)u(x, t), the local electric field,E(x,t), etc., the correlation function given byEq. (2) and its Fourier transform will depend on apair of local coordinates, (x,x’). Usually one isinterested in fluctuations of global H(t)-character-istics, such as, the total current density (H _= J(elL) f n(x, t)v(x, t)dx) or the voltage drop be-tween the terminals (H-- U- fE(x, t)dx) whichare already some integrals of the correspondinglocal characteristics over the whole device. Ac-cordingly, the response functions of these globalcharacteristics and their Fourier transform willdepend only on x0. By applying the Wiener-Khintchine theorem to the fluctuations of a globalcharacteristic one obtains the GTF formula in theform:

S(co) 2 SH(t)SH(t + s)texp(icos)ds

(4)

where

F(xo, co) ds exp(-icos)G(xo, s)

LANGEVIN FORCES 87

is the generalized transfer field determined as theFourier transform of the function G(xo,s) whichgives the linear response of the global H(t)-characteristic of the whole device (i.e., the con-duction current flowing through the structure orthe voltage drop between the structure terminals)to perturbations of velocity, energy and higherorder moments (a u, c, etc.) appeared at pointx0. In the case when fluctuations of localcharacteristics are considered, the spectral densityof fluctuations takes a .form similar to Eqs. (4) and(5), however Sm-l,(X, x’,v) will describe the cross-correlation of fluctuations in two local points xand x’, and F(x, xo, t) will describe the responseof the local H(x, t)-characteristic at point x to a

perturbation of velocity, energy, etc., at point x0.

COMPUTATIONAL PROCEDURES

The above approach ig sufficiently general and canbe applied to various HD models. Below we shallapply it in the framework of the velocity andenergy conservation equations. Accordingly,microscopic fluctuations of carrier velocity andenergy will form the basis of noise sources.

Model

The HD model includes the continuity equationfor carrier concentration n(x, t) as well as the con-servation equations for carrier average velocityv(x, t) and mean energy c(x, t) written in the form[5]:

0-- + 0 (6)

Ov 0--+ V-x + --x nQ eEm-1 + W

l(7/

OtOe 10

+ Vx +-n-x (nQe) eEv + ( eth)l,’e ---en(8)

Depending on the task to be solved, the Poissonequation for the self-consistent electric field E(x, t)and the equations describing an external circuitcan be added to Eqs. (6)-(8).

Parameters

Equations (7) and (8) contain five energy para-metric dependencies, namely, the average of thereciprocal effective mass in the field direction m-1,the velocity and energy relaxation rates, uv andu, the variance of velocity-velocity fluctuations,Qv (v)0 and the covariance of velocity-energyfluctuations, Q (v6e)o, where brackets meanaverage over the distribution function and thesubscript 0 indicates steady-state conditions per-taining to constant and homogenous appliedelectric fields. All the parameters are assumed todepend only on the local mean energy, and as suchthey can be obtained from a stationary MonteCarlo (MC) simulation of the bulk semiconductor[3].

Noise Source

The white spectral density of the Langevin force

S3 is also treated as depending on the local meanenergy only. This quantity is determined, withina MC simulation, in parallel with the calculationsof the other parameters of the HD model byaveraging over a trajectory of the single-carrierrandom walk in momentum space as [5]:

2 N

S& - il mOZ m/ (9)

where Acti, A/ are the instantaneous variations ofvelocity and energy of a single carrier during the i-th scattering event and T is the total time elapsedduring N scattering events.

Green Functions

Calculations of the Green-functions are performedin two main steps. Firstly, in the absence of


perturbations due to the Langevin forces, oneshould obtain a stationary solution of Eqs. (6)-(8)coupled with the Poisson equation and, if neces-sary, another equations describing the externalcircuit. Then, a perturbation of the steady-statevalues of velocity or energy, given by Ac(x-x0),is introduced at time 0 and point x0. Usually,the spatial profile of the perturbation is givenby some approximation of the -function whichtakes some volume in x-space, as for examplea Gaussian function. The perturbation amplitudeAs is taken sufficiently small to fulfill the require-ment of linearity of the response. Then, a directnumerical solution of the system of Eqs. (6)-(8)jointly, if necessary, with the Poisson and circuitequations is carried out to give the relaxation ofthe system to the stationary state. The Green-function corresponding to the given H-character-istic is obtained from the difference between thelocal values of H(x, xo, t) calculated during therelaxation process and the values of H(x) corre-sponding to steady-state conditions. The values ofthe difference are then normalized to the amplitudeof the initial perturbation As and to n(xo) as:

6 (x x0, t)H Acns(Xo [H(x, xo, t) Hs(x)] (10)

where c v, , respectively.

4

................................................. -i[ ill

10 100 10o0

f (OHz)

FIGURE Spectral density of voltage fluctuations calculatedwith the GTF method for a 0.21-0.30-0.39 gm n+nn + GaAsstructure with doping levels n 5 x 1015 cm-3 and n +

1017 cm -3 at T 300K for a voltage of 0.5V (curve 1). Curves2 to 4 present, respectively, contributions coming fromvelocity-velocity, velocity-energy and energy-energy micro-scopic noise sources. Curve 5 shows the result of directsimulations of voltage noise with the MC method.

those of a direct simulation performed with theMC method. We find that the main contributioncomes from the microscopic noise source related tovelocity rates (curve 2). The cross-correlationbetween velocity and energy rates is found to givea negative contribution which partially compen-sates the positive contribution belonging to theautocorrelation of energy rate.

NUMERICAL RESULTS

Below the general procedure of noise calculationsis applied to several submicron semiconductorstructures.

n+nn+ Structures

Figure shows the contributions of microscopicnoise sources connected with velocity and energyrates to the total value of the spectral density ofvoltage fluctuations calculated for a 0.21-0.30-0.39 l.tm n +nn+ GaAs structure. Results obtainedwithin the GTF method are shown together with

Schottky Diode

The results of noise calculations for a 0.350.35 lam GaAs n+ n- Schottky-contact struc-ture are presented in Figures 2 and 3. Since herecurrent noise is of most interest, calculations areperformed directly for current fluctuations, i.e.,taking H- J. In so doing, intermediate quantitiessuch as transfer field, local contributions to noise,etc., useful for a spatial analysis of the noise, areobtained in a natural way. Furthermore, by usingthe microscopic noise sources originated by scat-tering events one needs to consider just the noisesources inside the structure, thus avoiding theintroduction of bulk and surface noise sources

LANGEVIN FORCES 89

6

FIGURE 2 Spectral density of current fluctuations for a0.35-0.35 gm GaAs n + -n-Schottky-barrier structure withn + 1017cm -s, n 1016cm -s and Ud= 0.575V. Continu-ous (dashed) curves refer to GTF method (MC) calculations,respectively.

10

x (m)

FIGURE 3 Spatial profiles of local contribution to the totalcurrent noise. Calculations refer to the GTF method underconstant voltage operation mode for the Schottky-barrierstructure of Figure 2. Curves to 3 correspond, respectively,to frequencies f 0, 500, 2500 GHz.

to describe the total noise of the device. Figure 2compares the spectral densities of current fluctua-tions calculated by the GTF method and the MCprocedure at U 0.575 V (solid and dashed lines,respectively). As noise source, only the S-term isaccounted for. This term is found to be sufficientto describe the noise spectrum with a good

accuracy practically in the whole frequency range.Accordingly, the shot-noise at low frequencies, thespike in the intermediate frequency range near

f-600 GHz corresponding to returning carriers,as well as the plasma peak in the high-frequencyrange near f-2.2THz are well reproduced.To emphasize the advantage given by the GTFmethod in providing the spatial analysis of thenoise contribution, Figure 3 presents the spatialprofiles of the local contribution to the conductioncurrent noise, 6Sj(xo, ) n(xo)[Fj(xo, )lzSg(x0),calculated, respectively, at frequencies f-0, 500,2500GHz (curves to 3). At low-frequency, thelocal contribution reaches a maximum value justnear to the Schottky barrier, thus confirming thatthis space region is responsible for shot noise. Atintermediate frequency, 6Sj exhibits a clear peaknear to the center of the n-region which isassociated with the returning carrier effect. Athigh frequency, Sj exhibits a plateau in the n +

region and some peaks in the n-region which arereminiscent of the formation of standing wavesdue to perturbation reflection from the barrier.

Bipolar Diode

In this case the balance Eqs. (6) to (8) writtenseparately for holes and electrons are used jointlywith the Poisson equation. By introducing sepa-rate noise sources in the velocity and energyconservation equations, the GTF method is foundto provide natural calculations of noise. Due to theshort length of the structure, generation-recombi-nation processes are neglected and, as noisesources, only the S-term is accounted for.Figure 4 presents the spectral density of hole(curves to 3) and electron (curves 4 to 6) currentfluctuations calculated by the GTF method for a

bipolar 0.3-0.4 tm Sip+n structure. Analogouslyto the case of the Schottky-barrier diode consid-ered above, the low-frequency (f< GHz) plateaucorresponding to shot noise is well reproduced forboth hole and electron current noise. The numer-ical values well agree with the usual relationSj(O) 2eJ. In the intermediate frequency range


10-1

\ i

100 101 102 103 104

f (GHz)

FIGURE 4 Spectral density of hole (curves to 3) andelectron (curves 4 to 6) current fluctuations calculated with theGTF method at, respectively, Ua 0.6, 0.65, 0.7 V applied to abipolar 0.3-0.4jam Si p+n diode with p+=1017cm-3n=5x 1015cm-3.

(f= 10+ 100GHz), a second noise plateau isexhibited by holes while electron current noiseshow a peaking behavior probably correspondingto a returning carrier mechanism. At high-fre-quencies we have found a peak of the hole currentnoise due to plasma oscillations (f THz).

CONCLUSIONS

We have presented a GTF method based onmicroscopic noise sources related to velocity,energy, and higher moments change during singlescattering events as a unifying scheme able tointerpret noise in deep-submicron semiconductordevices. By combining the new noise sources with

the transfer fields, the spectral density of thefluctuations of any macroscopic quantity can berepresented in a form similar to that given by thestandard impedance field method, i.e., as theconvolution in real space of the noise source witha transfer field. The numerical results obtainedhere for various two-terminal devices evidence thatthe GTF method offers a natural way to includespatial correlations in noise calculations of sub-micron devices under various biasing conditions.

Acknowledgments

Authors acknowledge the support of the NATOcollaborative linkage grant PST.CLG.976340, thehigh-level grant DRB4/MDL/no 99-30 of thefrench Ministere de l’Education nationale, de larecherche et de la technologie, the frenCh lithuanianbilateral cooperation n. 5380 of french CNRS, theGalileo project n. 99055, and the project PB97-1331 from the DGICYT.

References

[1] Shockley, W., Copeland, J. A. and James, R. P., In:Quantum theory of atoms, molecules and solid state, Ed.Lowdin, P. O., Academic Press (New York, 1966), p. 537.

[2] Van Vliet, K. M., Friedman, A., Zijlstra, R. J. J., Gisolf, A.and Van der Ziel, A. (1975). J. Appl. Phys., 46, 1804; ibidem1814.

[3] Starikov, E., Shiktorov, P., Gruinskis, V., Gonzfilez, T.,Martn, M. J., Pardo, D., Reggiani, L. and Varani, L.(1996). Semicond. Si. Teehnol., 11, 865.

[4] Shiktorov, P., GruZinskis, V., Starikov, E., Gonz/tlez, T.,Mateos, J., Pardo, D., Reggiani, L. and Varani, L. (1997).Appl. Phys. Lett., 71, 3093.

[5] Shiktorov, P., Starikov, E., Gruinskis, V., Reggiani, L.,Gonzfilez, T., Mateos, J., Pardo, D. and Varani, L. (1998).Phys. Rev. B, 57, 11866.

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Langevin Forces Generalized Transfer Fields Modelling Deep ...downloads.hindawi.com/journals/vlsi/2001/082542.pdf · nic devices is the impedance field method [1,2], a deterministic