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Quantum Corrections in the Simulation of Decanano MOSFETs A. Asenov*, A. R. Brown and J. R. Watling Device Modelling Group Dept. of Electronics and Electrical Engineering University of Glasgow, G12 8LT, UK * [email protected] ABSTRACT Quantum mechanical confinement and tunnelling play an important role in present and future generation decanano (sub 100 nm) MOSFETs and have to be properly taken into account in the simulation and design. Here we present a simple approach of introducing quantum corrections in a 3D drift diffusion simulation framework using the density gradient (DG) algorithm. We discuss the calibration of the DG approach in respect of quantum confinement effects in comparison with more comprehensive but computationally expensive quantum simulation techniques. We also speculate about the capability of DG to describe source-todrain tunnelling in sub 10 nm (nano) MOSFETS. The application of the DG approach is illustrated with examples of 3D statistical simulations of intrinsic fluctuation effects in decanano and nano-scale double gate MOSFETs. 1 INTRODUCTION MOSFETs scaled down to 15nm gate lengths have been successfully demonstrated [1] and 9nm MOSFETs are expected in mass production in 2016 according the new edition of the roadmap. There is a consensus, however, that scaling around and below 10 nm will require a departure from the traditional MOSFET architecture. Among the most promising nanometre scale transistor candidates are double gate MOSFETs [2]. The combination of thin gate oxides and heavy doping in the conventional MOSFETs, and the thin silicon body of the double-gate structures, will result in substantial quantum mechanical (QM) threshold voltage shift and transconductance degradation [3]. Below 10 nm gate-lengths direct source-to-drain tunnelling will rapidly became one of the major limiting factors for scaling [4]. Computationally efficient methods to include QM effects are required for the purpose of practical Computer Aided Design of this generation of devices. First order quantum corrections based on density gradients (DG) have already been introduced in 2-D [5] and 3-D [6] driftdiffusion simulations. In this paper we discuss the implementation of DG quantum corrections in a 3-D drift diffusion simulation network designed to study intrinsic parameter fluctuations introduced by the discreteness of charge and atomicity of matter. We start with the calibration of the DG approach in respect of quantum confinement effects in comparison with more sophisticated quantum simulations [2]. We also investigate to what extent the DG approach can describe, at least semi-quantitatively, the expected source-to-drain tunnelling in nano-scale devices. Finally we illustrate the application of DG corrected drift-diffusion simulations in the analysis of various sources of intrinsic fluctuations in decanano double gate MOSFETs. 2 THE DENSITY GRADIENT APPROACH We are motivated by the need to include quantum corrections in the atomistic simulation of intrinsic fluctuation effects in decanano MOSFETs introduced by atomicity of charge and matter [6-8]. The investigation of intrinsic fluctuation effects involves statistical 3-D simulations of large samples of macroscopically identical but microscopically different devices and is very computationally expensive. Therefore the computational efficiency of the quantum correction approach is of great importance and the use of the DG approach becomes an attractive option. The DG approach may be derived from the one particle Wigner function [9]: ∂f (k, r, t ) + v ⋅ ∇ r f (k, r, t ) − ∂t h∇ r ∇ k ∂f (k, r, t ) 2 f (k, r, t ) = V (r )sin h 2 ∂t coll (1) Quantum effects are included through the inherently non-local driving potential in the third term on the left-hand side. Expanding to first order in h, so that only the first nonlocal quantum term is considered, has been shown to be sufficiently accurate to model non-equilibrium quantum transport and also for the inclusion of tunnelling phenomena in particle based Monte Carlo simulators [10, 11]. The additional, non-classical, quantum correction term may be viewed as a modification to the classical potential and acts like an additional quantum force term in the particle simulations, similar in spirit to the Bohm interpretation. (2) 0.1 (3) Thus the unipolar drift-diffusion system of equations with QM corrections, which in many cases is sufficient for MOSFET simulations, becomes: ( ∇ ⋅ (ε∇ψ ) = −q p − n + N D+ − N A− 2bn ) kT n ∇2 n ln = φn − ψ + q ni n ∇ ⋅ (nµn ∇φn ) = 0 (4) (5) (6) We have carefully calibrated the DG approach against the results of a full-band 1-D Poisson-Schrödinger (PS) solver [2]. Although Poisson-Schrödinger simulations are more accurate they are not yet practical for 3-D device simulations. Fig. 1 shows the quantum mechanical threshold voltage shift for DG as a function of substrate doping compared with the results presented by Jallepalli in [2]. Simulation results obtained using the recently developed effective potential (EP) quantum correction approach [12] are also presented for comparison in this and the following two figures. Utilising a single value of the electron effective mass of 0.18m0, obtained from matching the PS results at doping concentration 1018 cm-3 the DG simulations follow precisely the PS results. Fig. 2 shows typical carrier concentration profiles obtained from the 1-D simulations. Both the DG and the EP simulations show a peak in the concentration away from the Si/SiO2 interface, although the EP produces a sharper drop-off at the Si/SiO2 interface compared to PS and DG. 1018 1019 Fig. 1. Threshold voltage shift due to quantum effects versus substrate doping. Results for Density Gradient and Effective Potential are compared to those obtained from Poisson Schrödinger. Jallepalli (Poisson-Schrödinger) Density Gradient Effective Potential 17 10 16 10 15 10 0 1 2 3 4 7 6 5 8 Depth from interface [nm] Fig. 2. Electron carrier concentration as a function of distance from the interface, for substrate doping of 5×1017cm-3. All have the same net sheet density. 10 -5 10 -6 10 -7 10 -8 ID [A] CALIBRATION 0 17 10 Substrate Doping [cm-3] The system of equations (4) – (6) is solved self-consistently using standard techniques. 3 Effective Potential Density Gradient Jallepalli (Poisson-Schrodinger) 0.2 where bn = h /(12qm*n ) , and all other symbols have their usual meaning. To avoid the discretisation of fourth order derivatives in (1) in multidimensional numerical simulations a generalised electron quasi-Fermi potential φ n is introduced as follows: Fn = nµn ∇φn 0.3 Electron Concentration [cm-3] ∇2 n Fn = nµn ∇ψ − Dn ∇n + 2µn ∇bn n 0.4 VT (QM) – VT (Classical) [V] The density gradient approximation may be derived in a manner similar to that for deriving the drift diffusion approximation from the Boltzmann Transport Equation and results in a quantum potential correction term in the standard drift-diffusion flux [5]. 10 -9 10 -10 10 -11 10 -12 10 -13 Classical Effective Potential Density Gradient -14 10 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 VG [V] Fig. 3. ID-VG characteristic obtained from both classical and quantum simulators for a 30nm×30nm n-MOSFET, with VD=0.01V and a substrate doping of 5×1018cm-3. 4 SOURCE-TO-DRAIN TUNNELLING -5 10 -6 10 -7 10 Density Gradient Classical -8 10 I D It still remains unclear to what extent the approximations involved in deriving the DG approach remove its ability to model the direct source-to-drain tunnelling expected in nanometre channel length MOSFETs. In search of a qualitative answer to this question we simulate a set of double gate MOSFETs with generic structure illustrated in Fig 4. subthreshold slope degrades significantly as the channel length is decreased, while in the classical simulations the subthreshold slope remains nearly constant with channel length. The degradation in the subthreshold slope in the DG simulations is consistent with the more elaborate quantum mechanical simulations performed by others [13]. These observations provide an indication that source-to-drain tunnelling is included, to some extent, in the DG simulations. [A] Fig. 3, shows an ID-VG characteristic for a 30nm×30nm n-MOSFET obtained from our 3-D quantum simulator. The threshold voltage shift between the classical and the quantum simulations and the overall shape of the current voltage characteristics obtained using the DG and EP approaches are very similar. -9 10 6 nm 8 nm 10 nm -10 10 Top Gate 12 nm 15 nm 20 nm -11 Source Undoped Channel n+ Drain Oxide n+ 30 nm 10 -12 10 -0.4 -0.2 0 G Oxide Bottom Gate Fig. 4. Schematic representation of the double-gate MOSFET structure considered in this work. Fig. 5 show the electron concentration distribution normal to the gate obtained from the DG simulations of a 30×30×1.5nm double-gate MOSFET. As expected, for such a thin Si body quantum confinement effects produce a peak in the distribution in the middle of the channel. 20 VG = 0.6 V VG = 0.4 V 10 10 0.4 0.6 [V] Fig. 6. ID-VG characteristics for a double gate structure, with gate lengths ranging from 30nm down to 6nm, obtained from our classical and density gradient simulations. VD=0.01V and VG is applied to both top and bottom gate contacts. Further evidence can be gained by looking at the temperature dependence of the subthreshold slope illustrated in Fig. 7. In the classical drift-diffusion simulations the subthreshold slope, essentially thermionic in nature, depends linearly on temperature. However, any current due to tunnelling will have a much weaker temperature dependence [14]. -3 10 -3 Electron concentration [cm ] 19 0.2 V 18 10 -4 VG = 0.2 V 17 10 -5 10 10 -6 10 16 [A] 10 15 I VG = 0.0 V 14 10 -7 10 D 10 Density Gradient Classical -8 10 -9 10 13 10 0 -10 1 0.5 1.5 Depth [nm] 10 300 K 200 K -11 77 K 10 -12 Fig. 5. Quantum (density gradient) electron concentration profile through the centre of a 30×30×1.5nm double-gate MOSFET. The oxide thickness is 1.5nm. Fig. 6 illustrates the current voltage characteristics for a set of double gate transistors with channel lengths in the range from 30 to 6 nm. In the DG simulations the 10 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 V G 0.2 0.3 0.4 0.5 [V] Fig. 7. ID-VG characteristics for an 8nm channel length double gate structure from classical and density gradient simulations, for a range of temperatures. VD=0.01V and VG is applied to both top and bottom gate contacts. 5 RANDOM DISCRETE DOPANTS Theoretically the double-gate MOSFETs do not require channel doping to operate and therefore are considered to be inherently resistant to random dopant induced parameter fluctuations. Here we investigate to what extent the random dopants in the source and drain region introduce intrinsic parameter fluctuations in such devices. for the shorter channel length device. This is associated with the effective channel length variation along the channel introduced by the random discrete dopant distribution in the source and the drain regions. 6 CONCLUSIONS The density gradient approach provides computationally efficient means for incorporating quantum corrections in multi-dimensional device simulations. It agrees well with the available data from Poisson-Schrödinger simulations. In the simulation of sub 10 nm double gate MOSFETs the density gradient approach shows behaviour qualitatively consistent with source-to-drain tunnelling. REFERENCES Fig. 8. Electrostatic Potential in a 30×30×5nm double-gate atomistic MOSFET at threshold. Fig. 9. Electron equiconcentration surface in a 30×30×5nm double-gate atomistic MOSFET with location of dopants shown. Figs. 8 and 9 illustrate the impact of the unavoidable discrete dopants in the source/drain region on the potential and the electron distribution in a 30×30×5 nm double-gate MOSFET. The carrier concentration in the source and drain region is modulated by the potential fluctuations associated with individual discrete dopants. The maximum in the carrier concentration is in the middle of the channel. Channel Dimensions (L × W × T) Threshold Voltage Fluctuations σVT [mV] 10 × 30 × 1.5 nm 1.07 9.56 7.13 30 × 30 × 5 nm 0.66 3.28 1.93 Off-current On-current Fluctuations Fluctuations σID [%] σID [%] Table 1. Intrinsic parameter fluctuations in 10 nm and 30 nm double-gate MOSFETs due to atomistic doping in the source and drain. The threshold voltage is ~200mV. As shown in Table 1 the corresponding effective channel length fluctuations introduce small threshold voltage fluctuations which increase from 0.66 mV to 1.07mV as the device is scaled from 30 to 10 nm. The on-state current, however, does exhibit significant fluctuations, particularly [1] S. Thompson, M. Alavi, R. Argavani, A. Brand, R. Bigwood, J. Brandenburg, B. Crew, V. Dubin, M. Hussein, P. Jacob, C. Kenyon, E. 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