performance variability in wrap-round gate silicon nano...

Performance variability in wrap-round gate silicon nano-transistors: a 3D self-consistent NEGF study of ballistic flows for atomistically-resolved source and drain.

Antonio Martinez, John R. Barker, Marc Bescond*,

Andrew R. Brown and Asen Asenov

Nanoelectronics Research Centre and Device Modelling Group, Department of Electronics and Electrical Engineering, University of Glasgow, Glasgow G12 8LT, UK, *Minatec INPG, F-38016, Grenoble, France

E-mail: [email protected]

Abstract. A recently developed 3D self-consistent Non-equilibrium Green Function technique is used to study technologically identical silicon wrap-round gate 3D nanowire devices each of which has a different atomistically-resolved spatial distribution of dopants in the source and drain but in the absence of inhomogeneities in the channel/oxide interface and for an undoped channel. The simulations broadly confirm the conclusions of an earlier theoretical study of granularity effects on drive current that was limited by being two-dimensional and used a strongly inhomogeneous but continuum model of the doping profile in source and drain. However, the 3D nature of the flows makes it easier for charge to re-distribute around the regions of strong attractive potential. The stronger inhomogeneity effects in current density and charge density and the large inter-device variability predicted by 2D models are therefore over-estimates.

1. Introduction As the International Technology Roadmap for Semiconductors unfolds, the standard bulk MOSFET architecture will inevitably be replaced by alternative architectures in order to suppress short channel effects. In particular, there are attractive prospects for silicon nanowire MOSFET transistors with wrap-round gates [1]. However, fluctuations in the discrete number and spatial micro-locations of the dopants in the source/drain, stray dopants in the channel, and imperfections in the Si/SiO2 (and Si/High-k) interface became very important in nanometer scale devices. The particular configuration of microscopic inhomogeneity, from device to device produces fluctuations in the current and threshold voltage in an ensemble of technologically similar devices. Nano-wire MOSFET devices will involve channel regions (4-12 nm length) of nominally undoped small volume bordered by very highly doped (>> 1018 cm-3) regions corresponding to source and drain. There are many problems and advantages of these highly doped source and drain regions. In a parallel semi-classical study [2], one of us has predicted that for the short channel lengths considered here, the presence of any unwanted impurities in the channel which act as highly deleterious scattering centres may be strongly offset by polarisation of the source and drain regions (image charge effects) that strongly screen out the channel impurities. This advantage may be lost however, if random dopant aggregations occur as random

International Symposium on Advanced Nanodevices and Nanotechnology IOP PublishingJournal of Physics: Conference Series 109 (2008) 012026 doi:10.1088/1742-6596/109/1/012026

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doping extensions to the source and drain as reported in experimental studies using STM methodologies for 28 nm MOSFETs [3]. In these experiments the individual dopants were not resolved. The theoretical influence of dopant aggregations on the device performance of double-gate MOSFETs based on a 2D NEGF model was reported subsequently by us in [4]. In recent work [5], we reported theoretical studies of the effects of stray discrete dopants in the channel of model 3D silicon nanowire MOSFETs using a continuous doping model for source and drain. In the present paper we present preliminary results of a theoretical study of granularity effects in source and drain in silicon wrap-round gate nanowire model 3D devices each of which has a different atomistically-resolved spatial distribution of dopants in source and drain but in the absence of inhomogeneities in the channel/oxide interface and for an undoped channel. In the present paper the work is confined to examination of the potential, carrier density and current density behaviour. A particular aim is to evaluate our earlier predictions [4] in the light of a significantly more realistic models. One advantage of the self-consistent 3D non-equilibrium Green Function (NEGF) simulation tool used here is that polarisation effects are automatically included albeit at the level of semi-classical field theory. A full-many body treatment of polarisation, screening and carrier scattering is desirable and in principle also possible with NEGF tools [5], for example the Random Phase Approximation, but it is has an unacceptably high computational burden at present.

2. Theoretical model Consider Figure 1 which shows the architecture of our model 3D silicon wrap-round gate nano-wire device. The scale corresponds to a channel of dimensions 2.2 nm � 2.2 nm � 4 nm, oriented in the direction which we consider to be the x-axis. We assume a 1 nm gate oxide equivalent thickness. The technological doping in the extended source and drain region is assumed to be 1�1020 cm-3 and the channel is undoped. To model the effects of dopant fluctuations (positively ionized donors) in source and drain, we only consider a small portion of the source (or drain) immediately adjacent to the channel to be atomistically doped (discrete doping). These extension regions are 4 nm in length. Beyond the extension regions, the source and drain are treated as continuum doped ( 4 nm lengths). The total simulation domain is therefore of dimensions: 2.2 nm � 2.2 nm � 20 nm. The coupling to the surrounding contact region is treated by boundary self-energies as in the standard NEGF formalism. The 3D Poisson equation governs the electrostatics of the device and it is is solved in the external box of Figure 1. The Green function equations are solved self-consistently in the internal box. In such small devices it is not appropriate to use bulk effective masses. Instead we compute the confined masses (tensor components) from a tight-binding calculation of the band structure which is shown in Figure 2. For the assumed cross-sectional dimensions and material orientation, the confined masses are found as: m *L =1.07 me, m *T = 0.3 me , which may be compared with the bulk values: m *L = 0.92 me, m *T = 0.19 me . The use of bulk masses would only provide a qualitative description of the quantum transport in the nano-transistor.

For a source-drain doping concentration of ND=1 �1020 cm-3 there are on average just two donors

present in the source/channel interface region (which has a 4 nm extension) and similarly two donors in the drain region. A typical randomly generated configuration is shown in Figure 3. It is supposed that these atomistic regions adjoin large macroscopic contact reservoirs (simulated as continuum doped 4 nm extensions). To study variability effects we here consider spatial variations in the distribution of the discrete donors but not the number fluctuations (that case was considered in the simpler model described in [4]) nor surface roughness (considered in [6]). We have used a full self-consistent 3D NEGF simulator [6] in the effective mass approximation in order to capture the strong local variations in the potential due to the discrete donors. The present calculations are in the ballistic limit of no energy or momentum relaxation due to electron phonon interactions (a good approximation in these very short channel devices).


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Figure 1: Schematic of the wrap=round nanowire transistor aligned in the direction showing coordinate system.

Figure 2: 2D Band structure from Tight Binding calculations for a 2.17 nm � 2.17 nm silicon nanowire.

3. Methodology We follow the methodology outlined in detail in [6]. The 3D real-space Green’s function method transforms the double-time dependence of the standard Green functions to energy space, and all the transverse modes are implicitly included automatically without approximation. The formalism controls the injected carrier energies, but the modal decomposition does not appear explicitly in the formulation nor do the transversal eigenfunctions of the system. In essence, the NEGF method solves the full Schrödinger equation throughout the volume irrespective of the varying cross section. Within limits set by the convergence criteria and mesh discretization, the NEGF method generates both the exact open-boundary Green function and, most importantly, gives explicit current conservation. We have examined two different simulation approaches in order to understand the microscopic physics better. In the first approximate approach we do not determine the Green’s functions self-consistently, but use a frozen potential landscape: this gives the strongest quantum mechanical effects such as development of nodes in the density, strong vortex flows and tunnelling. In the second approach we use fully converged iterative (typically, 25 iterations) solutions to the coupled Poisson – Green function equations to obtain the self-consistent potential landscape with minimum electrostatic energy.

4. Simulation results Figures 4-7 show data computed for the discrete donor distribution of Figure 3. Figure 4 shows part of the electrostatic potential in a 3D contour map of the electrostatic potential, corresponding to: gate voltage = 0.5 V; source-drain voltage = 0.2 V. The location of the discrete donors is clearly visible.

Figure 3. Typical location of discrete donors in source and drain regions in model device (the continuous doping region is excluded from diagram) corresponding to data in Figs 4-6. Dimensions shown: 2.2 nm � 2.2 nm �12 nm.

Figure 4: Two different slices showing the iso-contours of the electrostatic potential �(x, y,z) corresponding to dopant configuration in Fig 3; source-drain voltage = 0.2 V. Gate voltage = 0.5 V .


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Figure 5 shows the carrier density n(x, y,z) corresponding the potential �(x, y,z) of Figure 4. Figure 6 shows the axial component of the current density jx (x,y,z) corresponding to the potential of Figure 4. An example of the one dimensional variation of the electron density and potential profile down the channel is shown in Figure 7; the cut is slightly off the central axis. The current density results have also been computed for a fixed electrostatic potential (non self-consistent NEGF) and significant back-flow occurs in the vicinity of the donors in the source region.

Figure 5. Two different slices showing the iso-contours of the carrier density n(x, y,z) corresponding to the dopant configuration in Fig 3; source-drain voltage = 0.2 V; gate voltage = 0.5 V.

Figure 6: Two different slices showing the iso-contours of the current density jx (x,y,z) corresponding to dopant configuration in Fig 3; source-drain voltage = 0.2 V ; gate voltage =0.5 V . The blue region indicates backflow along the channel, the red region indicates strong forward current along the channel.

By contrast, the corresponding fully converged self-consistent results (Figures 8-11) show no such

effects and the flow is meandering with no back flow in the total current density. Evidently, self-consistent current flow is found to more easily bypass obstacles without generating vortex flows whether at energy-resolved transport or at the level of fully energy-(thermal) averaged transport.

Figure 7. Snapshot of the electrostatic potential (blue-dashed curve) and electron density (red full curve) along the channel slightly off the central axis.

Figure 8. Snapshot of the electrostatic potential (blue-dashed curve) and electron density (red full curve) along the channel slightly off the central axis. Fully converged self-consistent data.

Figure 7 shows clearly the potential wells of a donor in the source and a donor in the drain. The

non-self-consistent charge distribution shows backflow effects near the left-most donor; this is associated with the attractive potential well that swings the classical orbits around and has a similar effect on the high wavelength carriers in this region. Backflow effects in current might therefore expected. At longer wavelengths which are active near the rightmost potential well in Fig. 7 there is no


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such effect as the carriers diffract around the obstacle. The peak in density then correlates with the minimum in potential. However, the fully self-consistent results (Figure 8) for the same parameters show that the minimisation of electrostatic energy and the re-distribution of charge act to promote charge neutrality as the dominant effect and effectively kills off the strong quantum effects. Basically the atomically resolved features visible in the data of Figures 5-7 are smeared out by the electrostatics as exhibited in the data of Figures 8-11.

Figure 9.Two different slices showing the iso-contours of the self-consistent electrostatic potential corresponding to the dopant configuration of Figure 3; source-drain voltage= 0.2 V; gate voltage = 0.5 V

Figure 10. Two different slices showing the iso-contours of the carrier density n(x, y,z) corresponding to the dopant configuration in Figure 3; source-drain voltage = 0.2 V; gate voltage = 0.5 V. Fully converged self-consistent data.

Figure 11. Two different slices showing the iso-contours of the current density jx (x,y,z) corresponding to dopant configuration in Figure 3; source-drain voltage = 0.2 V ; gate voltage =0.5 V . Fully converged self-consistent data.

5. Discussion and Conclusions The earlier 2D study [4] suggested that the occurrence of strong backflow in the vicinity of positive potential wells arising from granularity in source and drain leads to a strong variability in device drive current between technologically identical devices. The results of the present study show that for three-dimensional flows it is easier for the charge to re-distribute both dynamically (transport) and electrostatically. Backflow effects and atomic-scale variations are therefore reduced from those predicted in 2D models. It is likely that 2D models over-estimate the variability of granularity effects


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on device performance. It is hoped to validate these qualitative predictions in a planned future report on a full device performance analysis for large ensemble of devices using the 3D NEGF approach. 6. References [1] Goldberger J, Hochbaum A I, Fan R and Yang P 2006 Nano Lett. 6 973 [2] Barker J R, Towie E and Watling J R 2007 these proceedings [3] Fukotome H, Momiyama Y, Yoshida E, Okuno M, Itakura T and Aoyama T 2005 IEDM Technical Digest 57 [4] Martinez A, Barker J R, Svizhenko A, Anantram M P and Asenov A 2007 IEEE Transactions on Nanotechnology 6 438 [5] Gartner P, Banyai L and Haug H 2001 Phys.Re. B 62 7116 [6] Martinez A, Bescond M, Barker J R, Svizhenko A, Anatram M P, Millar C and Asenov A 2007 IEEE Transactions on Electron Devices 54 2213


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performance variability in wrap-round gate silicon nano...

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