A 3D Monte Carlo seiconductor device siulator for subicron silicon transistors* MOS K. Tarnay,^ p Masszi,& A. Poppe,* R Verhas," T. Kocsis," Zs. Kohari* " Technical University of Budapest, Departent of Electron Devices, H-1521 Budapest, Hungary & Departent of Electronics, Institute of Technology, Uppsala ABSTRACT A new 3D Monte Carlo siulator has been developed for analyzing the properties of subicron Si MOS transistors using the olecular dynaics ethod. This progra, called MiCroMOS uses the possible deepest first principles instead of any abstractions or siplifications. In this way, probles arising fro the usage of a continuu view, drift-diffusion equations or effective obility concept are inherently avoided. One of the ost iportant new features of the progra is that instead of solving Poisson's equation, the exact potential and electric field caused by the charged particles inside the siulated structure is analytically calculated, resulting in an exact odelling of all Coulob scattering echaniss and in real particle trajectories. 'This research has been sponsored by the Digital Equipent Co. EERP HG-001 and SW-003 projects, and by the Swedish and Hungarian governents' scientific research funds.
346 Software Applications in Electrical Engineering NOTATION $(r) potential at point r ^tp(r) potential at point r arising fro charged ipurities $n(r) potential at point r arising fro electrons #p(r) potential at point r arising fro holes ^t(r) potential arising fro interface charges #g(r) potential at point r caused by applied biases and boundary conditions Eip(r) Aeld at point r caused by charged ipurities En(r) field at point r caused by electrons Ep(r) field at point r caused by holes E,n((r) field at point r caused by interface charges ~J reciprocal effective ass tensor t transversal effective ass / longitudinal effective ass k wave vector INTRODUCTION Existing Monte Carlo siulators use abstractions and siulate usually only 2D structures. Assuing a silicon MOS structure having a gate area of 1x1 p* and a doping density of 10^3 ~3, the nuber of ionized ipurities under the gate in the depletion layer is about 10^. The nuber of inversion carriers is in the sae order oi agnitude. The relatively sall nuber of carriers and the increasing CPU power of the up-to-date coputers suggest the developent of a Monte Carlo siulator, where the trajectories of each carrier are individually followed both in real space and in k-space. Encouraged by the above facts we developed a 3D Monte Carlo siulation progra, called MiCroMOS. This progra applies the Monte Carlo ethod for the active region only (see Fig. 1). Outside the active region (deep in the source and drain) classical approxiations are used. More precisely, MiCroMOS is a quasi-3d siulator, because in the y-direction artificial boundary conditions are used. Still, the carrier otion is followed in all (even in the y) directions. The device structure The progra MiCroMOS is a hybrid Monte Carlo siulation in the sense, that a certain part of the source and drain together with the deeper part of the bulk is treated by classical ethods, while in the depleted regions (including the inversion channel) foring ost of the active region all carriers are exained individually. Since we have extended the active region siulated by the Monte Carlo ethod soewhat into the source and drain regions, the coputational need of the progra has increased. However, this price in CPU tie consuption is acceptable in case of shallow junctions (0.1-0.3 /z for typical subicron devices). The latest test version of our progra assues graded source and drain junction profiles, arbitrary source and drain doping concentrations, * hoogeneous channel and bulk doping concentrations.
Software Applications in Electrical Engineering 347 Source neutrality region Drain neutrality region Az Ax Figure 1: The MOS structure siulated by the MiCroMOS progra. We have to note, that siilarly to the way the junction profiles are introduced, inhoogeneous bulk doping ay easily be ipleented. The diensions of the exained device (see Fig. 1) are the following: On the x-axis: The total length of the calculated structure is the source region plus the channel length (between the source-channel and the channel-drain etallurgical junctions) plus the drain region, where the length of the source region is the su of a sall Ax region and the classically calculated n-side source region. Charge neutrality is forced in that sall Ax region. In a siilar way, the length of the drain region is coposed of the classically calculated n-side drain depletion region and another sall Ax region, in which charge neutrality is forced. The total length along the x-axis can be expressed as follows: Ltotal = Ax + dsdepl + Lchannel + dodepl + Ax The assued depth is calculated as follows: The p-side depletion width of the drain-bulk junction is deterined by classical approxiation. The total depth is given by the su of the distance of the Si-Si02 interface to the drain-bulk etallurgical junction, the p-side drain-bulk depletion width and a sall Az region, where charge neutrality is forced. The total depth considered is the following: l = Ddrain + i + A general assuption is the presence of strong inversion in the channel region. Outside the active region classical approxiations are used. The progra has an open structure, thus it is possible to write a ore sophisticated control structure (eg. for calculating ID - Ucs characteristics, etc.), to ipleent new features and/or new physical odels, or to replace the present odels with new ones.
348 Software Applications in Electrical Engineering SIMULATION PRINCIPLES In addition to the practical considerations already entioned in the pervious section the following considerations have lead us to apply the so-called olecular dynaics Monte Carlo ethod for the siulation of subicron seiconductor structures: The classical drift-diffusion ethod or the hydrodynaic ethod - both utilizing a continuu view of the transport processes in seiconductor structures (charge transport in the drift-diffusion approach, charge and energy transport in the hydrodynaic approach) - are no ore applicable, since in a subicron device structure the nuber of carriers is in the order of agnitude of a few thousands. In the widely used Monte Carlo ethods (based on charge clouds, superparticles, etc.) the basic physical phenoena of the siulated syste are less pronounced due to the applied sophisticated approxiations (eg. distribution functions, scattering rates, Boltzann transport equation), which can lead to siulated carrier behaviour being incorrect fro a physical point of view. Potential calculation In our icroscopic view, where we consider point charges the "life" of which is followed individually, using Poisson's equation would cause ajor probles. Probles with Poisson's equation. Solving Poisson's equation requires that a 2D or 3D grid is generated. An effective charge density is assigned to each eleent of this grid. In conventional device siulators (using either the drift-diffusion or the hydrodynaic ethod) the charge density is usually a generic quantity and the charge assignent proble does not exist. However, in our case, when we have point charges, defining effective charge densities for a Poisson solver iplies physical probles: If a carrier is assigned to a grid eleent with a given weight function (W), the original (q) and the assigned charge distributions (6p) are different, thus the potential and the electric field acting on carriers will be different, too. This causes a self-accelerating electric field as one can see in Fig. 2 [1]. Even if we have just a single electron, there will be a given electric field due to the effective charge density assigned to the esh eleent where the electron resides. Since the electric field accelerates all charged particles, our single electron would accelerate itself. Carriers very close to each other are affected by the sae electric field and cannot feel each other's attracting or repelling field. If the speed value or direction of these carriers are only slightly different they can travel together in the structure unless one of the is scattered. * Another proble arises fro the relatively sall nuber of point charges: the assigned charge density can be very rough. This leads to nuerical instabilities in the Poisson solver slowing down its convergence. Line charges vs. point charges. To reduce CPU tie consuption, the siulation of physical effects in 3D space can be soeties siplified by a 2D approach (by taking a unit-width 3-diensional syste, assuing perfect unifority in the neglected direction).
Software Applications in Electrical Engineering 349 P' d(r;ppi) = -d(rp,;r) Vc is the volue of the doain Figure 2: Charge assignent for the solution of Poisson's equation This siplification causes that the charges in this odel are considered as line charges for which thefieldhas an 1/r character and the potential has a logarithic character, whereas for point charges these are l/r% and 1/r, respectively (see the corresponding potentials in Fig. 3). The effect of this difference can be neglected over large distances, but it is questionable if such a siplification can be applied for a subicron MOSFET where all diensions are in the order of agnitude of a half icron. Since the force acting on carriers is proportional to the field, the calculated trajectories in the 2-diensional case are essentially different fro the real 3-diensional ones. Our approach for potential calculation. To eliinate the previously entioned probles, the so-called olecular dynaics ethod (in which all charges - including dopant ions - are treated as point charges without charge assignent) sees to be the best suitable technique. However, it is ore tie consuing and requires ore coputing power. A unique feature of the MiCroMOS progra is that the E(r) field and the #(r) potential distributions are calculated without solving Poisson's equation. We apply another ethod which is detailed below. Both the electric field and potential are divided into two parts, one originating fro the charges inside the Monte Carlo siulated region (and fro the Si - Si02 interface), and the other (#g(r)) caused by the external voltages (boundary conditions): The field and potential arising fro charges in the active region- and at the Si - SiO? interface are analytically deterined by using the Gaussian law: * n,p,ip,int \ = -grad $n,p,tp,t(r) The potential caused by the charged particles can be seen in Fig. 4. (2) (3)
350 Software Applications in Electrical Engineering 40 60 80 100 120 140 160 180 200 distance in n Figure 3: Potential of point and line charges Potential [V] Figure 4: The potential coponent caused by the charged particles
Potential [V] Software Applications in Electrical Engineering 351 Figure 5: The potential coponent arising fro the solution of the Laplace equation The field and potential due to external voltages (ie. the boundary conditions at the Monte Carlo siulated region) are deterined by solving the Laplace equation div grad ##(r) = 0. (4) Satisfying the boundary conditions is assured by the solution of the Laplace equation for odified boundary conditions. The physical boundary condition (basically deterined by the external voltages) is given by $ph(b)- Thus the odified boundary condition for surfaces with constant potentials at boundary B is obtained by subtracting the potential values given by the electrons, holes, ionized ipurities and interface states fro For the Si - SiO? interface the boundary condition is the following (8)). (5) E,e ' grad (Vcharge, + *Laplace) ' ^, =,', grad (Vcharge. + * Lap/ace) ' %L, (6) The potential coponent arising fro the solution of the Laplace equation is shown in Fig. 5 The total potential (the su of the potential caused by the charged particles and the potential given by the solution of the Laplace equation) can be seen in Fig. 6. These results were obtained for a device structure corresponding to Fig. 1, under the conditions detailed in Table 1. The advantage of this ethod is self-explanatory: without solving Poisson's equation directly - which would result in self-accelerating carriers, as shown earlier - the exact field caused by point charges is calculated analytically for all carrier positions. Only the Laplace equation ust be solved nuerically. This enables exact siulation of carrier trajectories and thus the exact evaluation of all Coulob scattering processes.
352 Software Applications in Electrical Engineering Paraeter L D W D.drain clox NA ND Nss Gate Ud. v,. (Channel length) (Channel depth) (Channel width) (Depth of the drain) (Oxide thickness) (Substrate doping) (Source/drain doping) (Surface state density) (Drain-source voltage) (Gate-source voltage) Value: 0. 25E-06 0. 30E-06 0. 25E-06 0. 12E-06 4. 50E-09 1. OOE+23 1. OOE+24-1.OOE+15 Al 0. 50E+00 2. 50E+00-3 -3-2 V V Table 1: Paraeters of the structure used in the potential calculations (see Figures 4-6). Carrier dynaics The carrier otion is described by the classical Newtonian law of otion where (8) The effect of the agnetic field is neglected. The second integral of Eq. (7) deterines the carrier path. Applying a tie increent Af sall enough to assue a constant force during this period, the integration for deterining the carrier trajectories can be carried out by afirstorder nuerical quadrature forula: (7) or in a slightly different for p(t + At) = p(t) + ~ (9) (10) The particle oentu at the tie instant t + Ai is For various kinds of carriers different reciprocal effective rrtass tensors are used. It is assued that: 1. The reciprocal effective ass tensors are diagonal. (H) 2. The diagonal eleents of the reciprocal effective ass tensors are independent of the k- vector (p = h k, h is Planck's constant) and their values correspond to the zero energy surface.
Software Applications in Electrical Engineering 353 Figure 6: The total potential in the structure. See Table 1. for the structure paraeters and the applied voltages. For electrons, there exist six ellipsoid-shaped constant energy surfaces. For a <100> oriented crystal, the principal axe of these ellipsoids lies on the positive or negative coordinate axes in the k-space and the centers of constant energy ellipsoids are located at 0.85 k \ax- For holes, the diagonal eleents are equal, because of the spherical syetry of the constant energy surfaces. It eans, that a scalar effective ass can be considered. Different effective asses are used for the light and heavy holes. The centers of the constant energy surfaces are located at the origin of the k-space, thus the hole velocity vector is directly proportional to the k-vector. Scattering processes The advantage of our approach to the potential and field calculation is apparent: since for each charged particle the exact electric field is analytically deterined, the charged particle interactions are inherently accounted for. This results in the exact siulation of ionized ipurity scattering, scattering on charged interface states and carrier-carrier scatterings. Consequently, the real carriers trajectories are autoatically followed. In order to derive scattering rates in the post-processing phase of our progra, an epirical factor ust be used to ake a decision whether the change of the direction in the oveent of a particle is large enough to consider it as a Coulob scattering or^not. In the present state of the progra the phonon scatterings are odelled quite siply. In the future, ore sophisticated odels should be applied to take into account real phonon spectra. For the intervalley scattering of the electrons a therodynaic approach is applied. For surface scattering,,the following phenoena are taken into consideration: the Coulob scatterings caused by charged interface states (as entioned earlier),
354 Software Applications in Electrical Engineering elastic or specular surface scatterings occur if a carrier reaches the Si - SiOi interface. The ratio between the elastic and specular surface scatterings are controlled by the so-called Fuchs paraeter. This is the only epirical paraeter used in our ethod so far. The entire siulation algorith is suarized by the flowchart presented in Fig. 7. CONSEQUENCES OF OUR SIMULATION METHOD Since we have point charges, the conventional ters such as charge density or dopant profile cannot be interpreted in the usual way. This eans that a continuu view cannot be applied any ore, eg. in a 1 p^ volue of Si assuing a doping density of 10^ ~^ there are only 10^ dopants resulting in 100 particles over a 1 /zra length. However, it is possible to define ethods which would atch acroscopic paraeters used by process engineers (eg. doping profile) to our icroscopic view. Due to our approach the conventional technological input data need to be pre-processed and the acroscopic output paraeters (currents, obilities) should be identified by eans of statistical post-processing of our calculated results. DISCUSSION OF RESULTS With the ethod applied in MiCroMOS all the carriers are individually traced both in the real space and the k-space at each siulation tie instant. The carrier-carrier and carrierionized ipurity interactions, together with soe other scattering processes are siulated with no approxiation. During the developent of our ethod we ainly concentrated on applying the deepest possible first physical principles in the active device region. The siulation is based on the effective ass concept. As long as this concept is valid, the results can be considered as an exact description of the real physical processes. Statistical evaluation Special postprocessor progras are used to evaluate the results (to convert our detailed results into "huan readable" forat for people used to classical device siulators). There are tools to calculate the drain current coponents and the RMS noise of the drain current; regression analysis of nuber of carriers, electrons entering to drain, the average tie spent in the active region; the cross correlation between the initial and final positions during a single siulation step, and the averaged velocity coponents; distribution of electrons between different ellipsoids vs. tie spent in structure; distribution of the electron teperature vs. tie spent in structure; statistics of scattering rates; estiation about obility coponents; etc. Test run results With thefirstworking version MiCroMOS [3], we perfored test runs on DEC's new, JOOOseries achines built around the 64-bit Alpha chips. Based on the detailed results obtained fro these runs we calculated the net drain current and the electron obility in the inversion channel. The applied voltages were as follows: Uds = 0.5 V and Ugg = 2.5 V. The runs were perfored on the test structure shown in Fig. 8. The structure paraeters are suarized in Table 2 (see Fig. 8 for the notations).
Software Applications in Electrical Engineering 355 Structure definition Setting siulation paraeters Initialization Deterination of potential and Reid distributions caused by ionized ipuritiei free carriers, interface charges and external voltages (see Eq. 1-6) Evaluation of carrier-carrier interactions (otion of carriers in the real space and in the k-space, see Eq. 7-11) Evaluation of carrier-particle (quasi particle) interactions (scattering, generation-recobination, etc.) END 1 PCstprocesso rs for the evaluation ai id visualisation of the results. Figure 7: The flow chart of the MiCroMOS progra
356 Software Applications in Electrical Engineering MC (iul&ted regie. Figure 8: The structure used for testing thefirstversion of the MiCroMOS progra. Paraeter L D W d_ox NA ND Nss Gate (Channel length) (Channel depth) (Channel width) (Oxide thickness) (Substrate doping) (Source/drain doping) (Surface state density) Value: 2. 33E-07 3. OOE-07 5. OOE-07 4. 50E-09 1. OOE+23 1. OOE+24 -:.OOE+15 Al _3-3 -2 Table 2: Paraeters of the structure used for our test runs.
Software Applications in Electrical Engineering 357 Val iey < 1 0, 0 > o, 0 > < 0 1, 0 > < 0' -1, 0 > < 0 0, 1 > < 0 o, -1 > E 1 4 18 20 6 24 75 Id 8065 1291 4951 0435 9679 3447 7868 6/(,/<% [/J A/3] 0 0047-0 0116 0 0276-0 0050-0 0136 00 0113 0133 inoiae 0.2937 00.4091.8725 0.8900 01.5522.0348 1.9454 Table 3: Results concerning the Id drain current derived torn the detailed siulation results of the MiCroMOS progra. Scattering type Lattice scattering Ipurity scattering Interface state scattering Electron - electron scatt. Electron - hole scatt. Elastic surface scattering Specular surface scattering Resulting obility (Mathesian rule) (ultiple scatterings accounted once) Scattering rate b3s~1] 6 5924 1 6333 0 0000 1 3363 0 0000 2 0193 1 9599 Mobility l?/vs] 0.0831 0.3355 0.4101 0.2714 0.2796 0.0406 0.0487 Table 4: The scattering rates and obility values resulting for the statistical evaluation of the carrier trajectories. An averaging was perfored for the entire device structure. In Table 3 portions of detailed siulation reports concerning the drain current Id are presented. Note that the different current coponents represented by electrons belonging to different valleys are distinguished. In addition to the drain current, obility values*can also be obtained. In Table 4 we suarized derived scattering rates and the resulting obility values.
358 Software Applications in Electrical Engineering References [1] R.W. Hockney and J.W. Eastwood: Coputer Siulation Using Particles Ada Hilger, Bristol and Philadelphia, 1988 [2] C. Jacoboni and P. Lugli: The Monte Carlo Method for Seiconductor Device Siulation Springer Verlag, Wien New York, 1989 [3] Tarn ay, K. - Masszi, F. - Kocsis, T. - Poppe, A. - Verhas, P.: Quantu Mechanical Approach for the Exaination of the Properties of MOS Inversion Layers in p-type Silicon Proceedings of the 15th NORDIC SEMICON (Haenlinna, Finnland), pp. 187-190., June, 1992.