Testing Hydrodynamical Models on the Characteristics of a One-Dimensional Submicrometer Structure

bCo.Ri.M.Me.,Stradale Primo Sole 50, 95121 Catania (Italy) Recent advances in technology leads to increasing high speed performance of submicrometer electron devices by the scaling of both process and geometry. In order to aid the design of these devices it is necessary to utilize powerful numerical simulation tools. In an industrial environment the simulation codes based on the Drift-Diffusion models have been widely used. However the shrinking dimension of the devices causes the Drift-Diffusion based simulators to become less accurate. Then it is necessary to utilize more refined models (including higher order moments of the distribution function) in order to correctly predict the behaviour of these devices. Several hydrodynamical models have been considered as viable simulation tools. It is possible to discriminate among the several hydrodynamical models on the basis of their results on the output characteristics of the electron device which are measurable (I-V curves). We have analyzed two classes of hydrodynamical models: i) HFIELDS hydrodynamical models and HFIELDS drift-diffusion model; ii) self-consistent extended hydrodynamical models with relaxation times determined from Monte Carlo simulations.


INTRODUCTION
Recent advances in technology leads to increasing high speed performance of submicrometer electron devices by the scaling of both process and geometry.In order to aid the design of these devices it is neces- sary to utilize powerful numerical simulation tools.In an industrial environment the simulation codes based on the Drift-Diffusion models have been widely used.However the shrinking dimension of the devices causes the Drift-Diffusion based simulators to become less accurate.Then it is necessary to utilizb more refined models (including higher order moments of the distribution function) in order to cor- rectly predict the behaviour of these devices.Short of a direct Monte Carlo simulation (hereafter MC), which requires prohibitively large computational cost in an industrial setting, hydrodynamical models have been considered as viable simulation tools.
Several hydrodynamical models have been consid- ered in the literature with various degrees of sophisti- cation and completeness, since the pioneering work of Blotekjaer 1].For a given device structure the vari- ous hydrodynamical models can give widely different results for the velocity and energy profiles according to the various assumptions made in the model (the crucial parameter seems to be the heat conductivity of the electron gas).However the different velocity or energy profiles are hardly accessible to experimental detection.Therefore it is mandatory to discriminate among the various hydrodynamical models on the basis of their results on the output characteristics of the electron device which are measurable (I-V curves).
We have analyzed two classes of hydrodynamical models: * HFIELDS drift-diffusion model (hereafter HDD) and the HFIELDS hydrodynamical model based on the Blotekjaer, Baccarani and Wordeman model (here after BBW) [2]-[4].HFIELDS is a general purpose two-dimensional device simulator developed at Bologna University which performs a wide range of tasks like steady-state, small-sig- nal and transient analysis.It solves the semicon- ductor equations incorporating the most important physical effects using a wide range of models.Default parameters are used both in the drift-diffu- sion version and in the hydrodynamical version ('Cp, :w [5], heat conductivity).The latter is phe- nomenologically represented by the Weidemann- Franz law with c =-2.1 (this choice has no physi- cal justification and is motivated only by fitting the velocity and energy curve to MC simulations for the particular n + n n + device [6], which will be considered in sec.3 ).
the Anile, Pennisi and Muscato model [7], [8], [9] (hereafter APM): this is a self-consistent extended hydrodynamical models with relaxation times determined from inhomogeneous Monte Carlo simulations.Heat conductivity is consist- ently represented by linear Fourier law plus a (dominant) convective term.
2 DESCRIIrFION OF THE HYDRODYNAMICAL MODELS The hydrodynamical models we consider can be derived from the moment equations of the BTE (sem- iclassical Boltzmann transport equation for charged carriers in semiconductors).Throughout the deriva- tion spherical parabolic bands and the effective mass approximation have been assumed.
The models we consider can be described within the general framework of the following balance equa- tions.For the electron component (neglecting elec- tron-hole generation and recotnbination, which is inessential for the applications we shall present in the sequel) these equations comprise the particle number balance equation an a-7 + v.
the Poisson equation V (eV,) q(n U). (4) Here m* is the electron effective mass, n, u, T the electron density, average velocity and temperature, h is the heat flux, E is the electric field, E =-V0, ) the potential, q the absolute value of the electric charge, N the ionized donors density and e the lattice die- lectric constant.Finally Q, Qw. are the momentum and energy productions due to collisions.Notice that in the balance equations (1-3) we have neglected any anisotropic terms in the pressure tensor.This can be justified because these terms, for the applications we shall consider, are rather small [6].
The various hydrodynamical models we investigate differ on choices about: i. the closure problem, i.e. a constitutive law for the heat flux h, ii.modeling the production terms Q, Qw.As stated in the Introduction we have examined the BBW model and the recent APM.
2.1 The BBW model i) The closure problem Without any justification it is assumed that the heat flux vector obeys Fourier's law h -kVT, (5) and the heat conductivity k is given by the Wiede- mann-Franz law (which holds for electrons near ther- mal equilibrium) 5 nT'cp k-(-+c)k2 rn* (6) with Zp the momentum relaxation time (to be speci- fied later) and c an adjustable constant.The default value of c =-2.1 is chosen because it gives the best agreement with the corresponding MC simulations for a particular n +n n + diode [6].
ii) Modeling the production terms Both Q and Qw are assumed to be of the relaxation type Q -=-, where W m*u_ 2 + 72nkgT is the total energy den- 3 sity, W 0 2nkgTL (T L being the lattice tempera- ture)."w is the energy relaxation time.
Both "Cp and "c W are assumed to be functions of the electron temperature (and of the doping density).In the original Baccarani-Wordemann formulation they are given by phenomenological expression [4] m lanO TL :P T (9 where gn0 is the low field mobility and Vs 1. 10 7c is the saturation velocity.However it s seems that in the model implemented in ST-HFIELDS, Zp and z w are taken to be constant.

the APM model i) The closure problem
This problem is solved by using the methods of extended Thermodynamics [10], [1 1], i.e. by requir- ing that the balance equations be compatible with a supplementary conservation law to be interpreted as entropy production law (entropy principle).By linear- izing around the state of local thermal equilibrium one obtains for h 2 m* VT + 2 "Cp where qSq is the relaxation time for the energy flux.
Notice the convective term which is absent in the BBW model.According to MC simulations [12], [13] the convective term is the dom- inant contribution to the heat flux.Furthermore the presence of this term renders the extended model compatible with the Onsager reciprocity relations [9]  near thermal equilibrium (at variance with the BBW model which has not the correct limit near thermal equilibrium).
ii) Modeling the production terms For 1-D flow one can assume the same relaxation type expression eqs.( 7),( 8) as in the BBW model.In 2 or 3 dimensions however a more general expression is required for Q The relaxation times :p, Xq, zm are obtained with the DAMOCLES TM code [13] (with parabolic bands for consistency with the derivation of the hydro mod- els) for the structure which will be considered in the next section.
We remark that in the APM model there are no free parameters and adjustable constants.

THE STRUCTURE
We consider a n + n n + diode consisting of a 0. n + region followed by a 0.4t, nregion, and ending with a 0.1bt n+.In the n+region the doping density is N + 1018cm -3 while in the n region it is N= 1016cm-3; the lattice temperature is T/ 300K.The simplest boundary conditions are: n= N, T TI, (13) 4,5 vs. Voltage computed by using the HFIELDS drift-diffusion model (HDD), the HFIELDS hydrodynamical model (BBW), the APM model and the Monte Carlo code DAMOCLES at Xmin and Xmax.Also qC(Xmin) T log( n n i )' qC(Xmax) T log( i. + qVb, (14) where V/) is the applied bias and n/the intrinsic con- centration n 1.4 x 101cm-3.For the numerical solution we employ the methods developed by Gard- ner, Jerome and Rose 14] and Gardner 15], adapted to this model by Anile, Maccora and Pidatella [16] and Anile, Maccora, Muscato and Pidatella 17]

RESULTS AND CONCLUSIONS
In fig.we plot the characteristics I-V for the structure previously described.We tested this structure with: a) the HDD model (where the mobility model used is the CVT [18]- [19]),b) the BBW model as implemented in HFIELDS (with the default values for the parameters "Cp, "c w, determined from comparison with homogene- ous MC simulations and c =-2.1 obtained by tuning with MC data for a Vbias--" 1.5 g [6]), c) the APM model with the relaxation times obtained from inho- mogeneous MC simulations.
We remark that all four curves start from the same initial point, corresponding to a zero bias thermal equilibrium state.This is consistent with the physical requirement that all models must yield indistinguisha- ble results near thermal equilibrium.The curves cor- responding to the HDD and BBW models are almost indistinguishable.We notice that at high voltages the difference in current density between the HDD and APM models can be as large as 20%: we ascribe this difference to the fact that the APM model is consist- ently obtained from the Boltzmann equation.In par- ticular this entails the presence of the convective term in the heat flux constitutive equation at variance with the BBW model where the Fourier equation with the Wiedemann-Franz law is simply postulated.Furthermore the relaxation times (henceforth the mobility) are consistently obtained from inhomogeneous MC simulations.We notice that the only MC datum we have (through DAMOCLES) is much closer to the APM curve as shown in fig. 1. Therefore such a In fig. 2 we plot the velocity obtained with the APM model, the BBW model and DAMOCLES for Vbias=l V.In fig. 3 we plot the temperature obtained with the APM model, the BBW model and DAMOCLES for Vbias=l g.
We remark that care must be exercised when com- paring the theoretical curves with the experimental ones, due to the several approximations employed in deriving the theoretical models neglect of non-para- bolicity, of the hole current, 1-D geometry, etc.).In particular a rigorous assessment of the non parabo.licityeffects is required, following approaches employed by Tang et al. [20].

6 FIGURE 2 EFIGURE 3
FIGURE 2 Electron temperature vs. distance computed by using the HFIELDS hydrodynamical model (BBW), the APM model and the Monte Carlo code DAMOCLES for Vbias V