Impact of scattering in ÔatomisticÕ device simulations

Transcription

1 Solid-State Electronics 49 (2005) Impact of scattering in ÔatomisticÕ device simulations C. Alexander *, A.R. Brown, J.R. Watling, A. Asenov Device Modelling Group, Department of Electronics and Electrical Engineering, University of Glasgow, Glasgow G12 8LT, Scotland, UK Received 10 June 2004; accepted 16 October 2004 The review of this paper was arranged by Prof. S. Cristoloveanu Abstract In this paper, by comparison of Drift-Diffusion and Monte Carlo simulation results, we investigate the impact of Coulombic scattering in atomistic device simulations and its contribution to the random dopant induced intrinsic parameter fluctuations in nano-cmos devices. By introducing ionised impurity scattering directly into the Monte Carlo simulations through the full impurity potential, we resolve the contribution of the variation in scattering on the random dopant induced current variation. In comparison, Drift-Diffusion simulations are only able to capture the corresponding electrostatic effects. This approach is first demonstrated for the simple case of a single scattering centre in the channel of a MOSFET and then used to compare current variations in a set of devices with atomistic doping. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Trapped charge; Atomistic; Monte Carlo; Drift-diffusion 1. Introduction * Corresponding author. Tel.: ; fax: address: c.alexander@elec.gla.ac.uk (C. Alexander). Scaling of conventional MOSFETs to channel lengths of 20 nm and below requires channel doping in the range of cm 3 [1]. At such sizes and doping concentrations there are, on average, only a few tens of dopant atoms within the channel, resulting in significant statistical parameter variations. This has prompted the development of simulation techniques that are able to take into account the discreteness of the doping, allowing the study of random dopant induced fluctuations on the device parameters. Previously such simulation studies have been predominantly performed using 3D Drift- Diffusion (DD) simulators [2 6]. The DD simulations only capture the electrostatic effects associated with random discrete dopant distributions, providing an estimate for the variations in the threshold voltage and the drive current. The DD simulations however, cannot capture the complex effects associated with variations in carrier transport, from device to device, due to the different numbers and configuration of ionised dopants within the channel that act as Coulombic scattering centres. For the short nano-scale devices considered here, such variations will be non-self-averaging. Therefore, results for the variation in the drive current obtained from DD simulation are likely to under estimate the real magnitude of the intrinsic parameter variations. A simulation approach that can capture, after suitable modifications, the transport variations introduced by random discrete dopants is the Ensemble Monte Carlo (EMC) technique. In this paper we use threedimensional EMC simulations utilising the Particle Particle Particle Mesh (P 3 M) methodology [7] to treat ionised impurity and carrier carrier scattering directly through the full Coulombic interaction [7], rather than through the usual scattering rate approach [8]. As a /$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi: /j.sse

2 734 C. Alexander et al. / Solid-State Electronics 49 (2005) result we are able to fully capture both the electrostatic effects and the transport fluctuations, which both contribute to the on-current variations in nano-scale CMOS devices. Although ab initio Coulombic scattering from ionised impurities through real space carrier trajectories have been included in 3D EMC simulations previously [9 11], little statistical analysis has been performed with such simulators. Gross et al. [12] have investigated threshold voltage and velocity fluctuations, and the correlation of such fluctuations to the number of dopant atoms within the channel. However, only a small sample of devices was considered. Nevertheless Gross showed the importance of the dopant position under the gate on the variations in the parameters of the studied devices. However, to date no results have been published comparing current fluctuations due to variations in scattering arising from the varying numbers and positions of discrete dopants within the channel with their associated electrostatic impact, which leads to potential fluctuations and current percolation paths within the device. In the next section of this paper we describe the ab initio Ensemble Monte Carlo approach developed and used here to accurately treat the Coulombic scattering between electrons and dopants, as well as the electron electron interactions in the device. In a careful comparison with DD simulations, this approach has been used to investigate and highlight the relative importance of the electrostatic and scattering effects when studying random dopant induced intrinsic parameter variations in nano-scale MOSFETs. Our approach is first illustrated in Section 3 by simulating the impact of a single negative trapped charge in the centre of the channel of a wellscaled MOSFET with otherwise continuous doping. In Section 4 we turn our attention to the simulation and comparison of a set of macroscopically identical MOSFETs with microscopically different random dopant distributions to illustrate the increased variation in drive current obtained from MC simulation. 2. Simulation methodology The treatment of ionised impurity scattering through direct interaction with the discrete dopant atoms instead of the traditional impurity scattering rates can be introduced into Monte Carlo device simulations of nanoscale devices using a variety of methods [9 14]. In our approach, similar to others, scattering is treated through the propagation of the MC carriers in the 3D real-space potential landscape defined by the unique configuration of ionised impurities in the device structure, as illustrated in Fig. 1. Self-consistent simulation of an ensemble of carriers allows the same treatment to be applied to carrier carrier scattering. In all cases the long-range component of the Coulombic interaction is accounted Fig. 1. Potential distribution in a 3-D volume with random discrete dopants. The carriers are scattered by Coulombic potential of the dopants.

3 C. Alexander et al. / Solid-State Electronics 49 (2005) for through the mesh-resolved solution of PoissonÕs equation. However, at short range, interpolating from the mesh force alone significantly underestimates the magnitude of the Coulombic interaction and diminishes the ability of the charge to act as a scattering centre. As a result, the simulated mobility is overestimated. One commonly used method to resolve this is to apply a short-range force correction to the mesh force in order to account for the aliasing in the mesh resolved potential [7,9,10,13,14]. This corrected force can be used in the integration of the equations of motion allowing for the full Coulombic interaction to be included in the carrier dynamics. Note that the mesh force still has a finite magnitude at short range, also containing the externally applied field (potential). Thus a direct, molecular dynamics-like evaluation of the force within some predefined radius would contain an erroneous contribution and would miss the external field. In order to avoid this double counting of the mesh force, it necessary to estimate the magnitude of the mesh resolved Coulombic potential so that it can be removed in the correction process. This task could be achieved using a variety of methods: (i) An analytical approximation to the mesh force [7] allows the use of non-uniform meshes, giving the approach a high degree of versatility [10]. (ii) Alternatively, the mesh force can be evaluated directly through the mesh solution of a single charge [9]. Here, the simulation is limited to a uniform mesh but benefits from a very efficient implementation. This approach could take into account the symmetry associated with the mesh solution, which differs from the spherical symmetry of Coulombic interactions [13,14]. (iii) A third approach introduces the short-range interaction via a scattering rate that is applied only in the region surrounding the ionised impurity [11,15]. Although this technique does not require a description of the mesh force at short-range it does restrict the mesh choice, requiring the mesh spacing to be approximately onetenth of the average inter-impurity separation in order to adequately resolve individual dopants. This limits the description of the doping to one dopant impurity per mesh-cell; such restrictions also make the approach unsuitable for a direct description of carrier-carrier scattering, which may be readily included in the same manner as the carrier-impurity interaction in methods (i) and (ii) above. In this work we use the P 3 M approach to resolve the short-range force component. Correcting to match the pure inverse square Coulomb force of a point charge at very short range is an unphysical representation of the nature of the impurity potential and leads to artificial heating of the system. Therefore, modification of the short-range force is necessary to restrict the magnitude of the force and minimise the error in particle propagation. It is desirable that such modifications should be based upon the physical nature of the problem. We assume that the electric field associated with a single donor reaches a maximum at a cut-off radius taken to be the Bohr radius of the ground state in the Hydrogenic donor model. This follows from the discussion by Gross et al. [12], and in silicon is approximately 2 nm. We similarly assume that at separations smaller than the cut-off radius the field drops to zero as we approach the centre of the donor. We also ensure that the short-range correction for the model field reproduces the full Coulomb field at distances larger than the cut-off radius, while unlike Gross we turn to an analytical form to remove the abrupt transition in the field at the cut-off and improve Fig. 2. Force interpolated from the Poisson mesh plotted alongside the corrected short-range force as described in the text. Fig. 3. Concentration dependent mobility results using ab initio and impurity scattering rates Monte Carlo simulations compared with experimental measurements.

4 736 C. Alexander et al. / Solid-State Electronics 49 (2005) the numerical integration. A suitable analytical form describing such a short-range force is given by: Qr F sr ðrþ ¼ 4pe r 2 þ 1 3=2 ð1þ 2 r2 c F sr has its maximum at a cut-off radius, r c, and tends to the Coulomb field at large distances and is zero at r =0, as illustrated in Fig. 2. This approach has been used in ÔbulkÕ simulations to reproduce the low-field concentration dependant mobility in silicon, verifying its applicability. Fig. 3 shows that the results from the simulated doping concentration dependence of the mobility are in close agreement with experimental measurements [16] over a range of doping concentrations relevant to devices. 3. Scattering from a single charge In order to highlight the variation in transport due to discrete charges, while not shrouding the result in a fully atomistic simulation where variation in number and position must also be considered, we first apply the ab initio Monte Carlo technique to investigate the impact of a single negative charge within the channel of an n-channel MOSFET. In this case, the single discrete charge is included in an otherwise continuously doped device. This can be regarded as an accurately resolved trapped electron within the traditional MC procedure. It is still debated in the literature which is the dominant effect in determining the reduction in the current in response to a single trapped electron: either the electrostatic exclusion of inversion layer charge around the trapped electron or the mobility reduction associated with the increased Coulombic scattering [17,18]. Using the described EMC simulation technique the impact of the increased scattering due to the trapped charge can be included and separated from the reduction in current density due to the local reduction in the carrier concentration observed in the DD simulations. Here we simulate a simple, yet well scaled, MOSFET with a nm channel and uniform channel doping. Doping within the source and drain regions are similarly considered as continuous. Details of the device dimensions, doping, oxide thickness, threshold voltage and sub-threshold slope are presented in Table 1. Ionised impurity scattering from the uniform doping in the device is treated by the Brooks Herring formulation incorporating degeneracy in the screening-length [19], otherwise the simulation obeys non-degenerate statistics. Surface roughness scattering is not included within the MC, while concentration dependent mobility and lateral field dependent mobility are included within the DD having been calibrated to match the MC results. This calibration of the mobility model within the DD addresses the disparity between the two techniques, as it is well known that the Brooks Herring scattering rate overestimates mobility at high concentrations. A single trapped charge is placed in the centre of the channel at the semiconductor/oxide interface where its effect on the current at low drain voltage is most pronounced [20]. A low bias of 50 mv is applied between source and drain. Under these conditions non-equilibrium carrier transport is negligible. This allows us to employ a frozen-field approximation in our MC simulations, reducing the simulation time. This also allows a more direct comparison, as in this regime we expect the results obtained from DD and MC to be in better agreement than at high drain bias. The electric field employed in the EMC simulation has been imported directly from the Drift-Diffusion simulations. The current at a series of applied gate voltages is calculated with and without the trapped charge present and the percentage difference obtained from both DD and MC simulations is plotted in Fig. 4. The current was calculated from a MC simulation time of 1 ns for gate voltages of 1.0, 0.8 and 0.6 V. For 0.4 V, the MC simulation time was increased to 3 ns in order to obtain reliable statistics. In the subthreshold region Monte Carlo simulation requires excessively long simulation times in order to gather reliable statistics and therefore becomes impractical. Table 1 Structure and characteristics of the simulated MOSFET L (nm) W (nm) T SiO2 (nm) N A (cm 3 ) V T (V) S (mv/dec) Fig. 4. Percentage reduction in drive-current upon introducing a single trapped negative charge in the centre of the channel. Results for Drift- Diffusion (DD) and Monte Carlo (MC) are shown for a 30 nm channel length n-mosfet at V D = 50 mv.

5 C. Alexander et al. / Solid-State Electronics 49 (2005) The introduction of the trapped charge results in different reductions in the drain current in the DD and MC simulations. In the DD simulations at low gate voltages the region in which the inversion layer density is reduced by the trapped charge is large due to the small amount of screening from the relatively low electron concentration in the inversion layer. This large reduction in net carrier density in the channel is responsible for the large reduction in the current. An increase in the gate voltage, and a corresponding increase in the inversion charge density, leads to more effective screening. The region affected by the trapped charge becomes more localised and the reduction in current is smaller. Results from MC simulations show a similar trend as DD, however the reduction in current is consistently larger. In this case, not only the electrostatics but also increased scattering from the screened Coulombic potential of the trapped charge plays a role in reducing the current. The velocity profile along the line between source and drain directly through the trapped charge is shown in Fig. 5. The velocity is plotted for three different gate voltages: 1.0 V, 0.8 V and 0.6 V, with and without the charge present. The reduction in average velocity, with carriers being scattered back towards the source, is clearly discernable. This reduction by additional scattering extends almost to the source and drain regions, well outside the screening radius of the potential around the trapped charge. In the worst case (at 0.6 V) this range is approximately 5 nm. The asymmetry in the source to drain velocity profile is attributed to the applied bias and the scattering by the trapped charge. In the absence of the trapped charge electrons injected into the channel from the source would propagate uniformly towards the drain. However, propagation in the potential of the trapped charge does not allow a straight path into the region beyond the location of the charge. Instead, this charge exclusion region is maintained by electrons being scattering into it. This occurs through both phonon mechanisms, randomising the velocity, and propagation in the point charge potential from other incident directions, favouring velocities directed towards the drain. This is in contrast to the velocity profile on the source side where the continual deceleration of injected electrons is seen. By examining the percentage reduction in current over a range of gate voltages, we determine that at gate voltages close to the subthreshold the electrostatics play a more important role and the agreement between the results from the DD and MC simulations is closer. At higher gate voltages the difference between DD and MC becomes larger, as under these conditions, the additional Coulombic scattering captured in the MC simulations plays an increasingly important role. For example at V G =1V the reduction of the current flow from the MC simulations is more than four times larger than the reduction from DD simulations. 4. Impact on devices An important problem affected by Coulombic scattering from random discrete impurities in the channel of nano-cmos devices is the magnitude of intrinsic parameter fluctuations. Analysis of this issue requires the simulation of a statistical sample of randomly generated, atomistically doped devices. To address this we have studied the relative change in the current for three nm MOSFETs with the same structure and dimension as the device described in the previous section but with different, fully atomistic, dopant configurations in the channel. These devices were chosen from an ensemble of 200 microscopically different devices and represent, the highest, lowest and average threshold voltage within the ensemble as determined by the Drift-Diffusion simulations. As before, we have calculated the percentage change in drive current, now due to the atomistic doping, at low drain voltage (50 mv) using both DD and MC simulations. Ionised impurity scattering rates are now not included in the MC simulation of these devices, as this scattering is now treated Table 2 DI D /I D [(I D atomistic I D continuous )/I D continuous ] for the 3 atomistic MOSFETs shown in Fig. 6, comparing Drift-Diffusion and Monte Carlo simulation results Fig. 5. Velocity profiles though the centre of the device with and without the trapped charge present for three different gate voltages. DI D /I D (low V T ) (%) DI D /I D (average V T ) (%) I D /I D (high V T ) Drift-Diffusion (%) Monte Carlo (%) V D = 0.05 V, V G =1V.

6 738 C. Alexander et al. / Solid-State Electronics 49 (2005) directly through the screened Coulombic potential of the individual discrete dopants in the channel. From the results of the simulations summarised in Table 2, it is clear that the ab initio inclusion of impurity scattering in the MC simulations results in a significant increase in the current variation. In the MC simulations we observe not only a larger reduction in current in the high threshold device but also a larger increase in current in the low threshold device as compared to the DD simulations. This is associated with increased scattering from a greater number of acceptors, or alternatively reduced scattering from a smaller number of acceptors, when compared to the average scattering rate implied within Drift-Diffusion through the use of concentration dependant mobility values. The potential distribution and the average carrier velocity distribution in the channel of the three devices are shown in Fig. 6. Here the influence of the dopants in shaping the charge flow can clearly be seen, with electrons flowing in the potential valleys around the acceptors where possible. The low numbers, and configuration, of acceptors in the low threshold voltage device allow the electrons to propagate relatively unimpeded. Conversely, in the high threshold voltage device, the large number of acceptors impedes the current flow, forcing the electrons to flow through a constricted region. In addition to the current percolation pattern the velocity around individual impurities located within the middle of the channel is affected over a range greater than the screening length. In some cases this extends from the source to the drain. Fig. 6. Electrostatic potential and associated channel velocity profiles for 3 atomistic n-mosfets with: (top) low, (middle) average and (bottom) high threshold voltages. (blue = low velocity; red = high velocity). For interpretation of colours please refer to the web version of this article.

7 C. Alexander et al. / Solid-State Electronics 49 (2005) Conclusions We have studied the impact of a single charge in the channel of a MOSFET by comparing Drift-Diffusion and Monte Carlo simulations. All simulations were carried out at a low applied drain bias in order to allow the frozen field approximation to be adopted in the Monte Carlo simulations. We have been able to resolve the relative importance of variations in electrostatics and carrier transport, due to increased scattering, in determining the current fluctuations over a wide range of applied gate voltages. The relative importance of each mechanism in reducing the drive current is inferred: In the case of a single trapped negative charge at low gate voltages, where there is little screening, electrostatics are shown to dominate. Whereas at high gate voltages the additional Coulombic scattering plays an increased role, resulting in nearly a five times greater reduction in current. We have also investigated a set of three randomly generated, atomistic devices. The devices were chosen to have the lowest, highest and average threshold voltage from a set of 200 atomistic devices as simulated by the Drift-Diffusion approach. The Monte Carlo simulations showed increased current variation in all cases. References [1] International Technology Roadmap for Semiconductors. Available from: SEMATECH, [2] Asenov A. Random dopant induced threshold voltage lowering and fluctuations in sub-0.1 lm MOSFETÕs: A 3-D atomistic simulation study. IEEE Trans Electron Dev 1998;45: [3] Asenov A, Brown AR, Davies JH, Kaya S, Slavcheva G. Simulation of intrinsic parameter fluctuations in decananometer and nanometer-scale MOSFETsÕ. IEEE Trans Electron Dev 2003; 50: [4] Stolk PA, Widdershoven FP, Klaassen DBM. Modeling statistical dopant fluctuations in MOS transistors. IEEE Trans Electron Dev 1998;45: [5] Asenov A, Brown AR, Davies JH, Saini S. Hierarchical approach to ÔatomisticÕ 3D MOSFET simulation. IEEE Trans Comput Aided Des Integr Circuits Syst 1999;18: [6] Frank DJ, Taur Y, Ieong M, Wong HSP. Monte Carlo modeling of threshold variation due to dopant fluctuationsõ symposium on VLSI technology Dig. Techno. Papers p [7] Hockney RW, Eastwood JW. Computer simulation using particles. IOP Publishing Ltd.; [8] Jacoboni C, Reggiani L. The Monte Carlo method for the solution of charge transport in semiconductors with applications to covalent materials. Rev Mod Phys 1983;55: [9] Gross WJ, Vasileska D, Ferry DK. A novel approach for introducing the electron electron and electron-impurity interactions in particle-based simulations. IEEE Electron Dev Lett 1999; 20: [10] Wordelman CJ, Ravaioli U. Integration of a particle particle particle-mesh algorithm with the ensemble Monte Carlo method for the simulation of ultra-small semiconductor devices. IEEE Trans Electron Dev 2000;47: [11] Barraud S, Dollfus P, Galdin S, Hesto P. Short-range and longrange Coulomb interactions for 3D Monte Carlo device simulation with discrete impurity distribution. Solid State Electron 2002; 46: [12] Gross WJ, Vasileska D, Ferry DK. Three-dimensional simulations of ultrasmall metal oxide semiconductor field-effect transistors: The role of the discrete impurities on the device terminal characteristics. J Appl Phys 2002;91: [13] Arokianathan CR, Asenov A, Davies JH. Mesh-based particle simulation of sub-0.1 lm FETs. Semicond Sci Technol 1998;13: A [14] Arokianathan CR, Davies JH, Asenov A. Ab-initio Coulomb scattering in atomistic device simulation. VLSI Des 1998;8: [15] Dollfus P, Bournel A, Galdin S, Barraud S, Hesto P. Effect of discrete impurities on electron transport in ultrashort MOSFET using 3D MC simulation. IEEE Trans Electron Dev 2004;51: [16] Thurber WR, Mattis RL, Liu YM, Filiben JJ. Resistivity-dopant density relationship for phosphorus-doped silicon. J Electrochem Soc 1980;127: [17] Shi Z, Mieville J-P, Dutoit M. Random telegraph signals in deep submicron n-mosfetõs. IEEE Trans Electron Dev 1994;41: [18] Simoen E, Dierick B, Claeys CL, Declerck GJ. Explaining the amplitude of RTS noise in submicrometer MOSFETs. IEEE Trans Electron Dev 1992;39: [19] Watling JR, Yang L, Borici M, barker JR, Asenov A. Degeneracy and high doping effects in deep sub-micron relaxed and strained Si n-mosfets. J Comp Electron 2003;2: [20] Asenov A, Balasubramaniam R, Brown AR, Davies JH. RTS amplitudes in decananometer MOSFETs. IEEE Trans Electron Dev 2003;50: