Bipolar quantum corrections in resolving individual dopants in atomistic device simulation

Transcription

1 Superlattices and Microstructures 34 (2003) Bipolar quantum corrections in resolving individual dopants in atomistic device simulation Gareth Roy, Andrew R. Brown, Asen Asenov,ScottRoy Device Modelling Group, Department of Electronics and Electrical Engineering, University of Glasgow, Glasgow G12 8LT, UK Available online 12 May 2004 Abstract In atomistic device simulation the resolving of discrete charges onto a fine-grained simulation mesh can lead to problems. The sharply resolved Coloumb potential can cause simulation artefacts to appear in classical simulation environments using Boltzmann or Fermi Dirac statistics. Various methods have been proposed in an effort to reduce or eliminate such artefacts as the localisation of mobile carriers by sharply resolved Coulomb wells, however they have met with limited success. In this paper we present an alternative approach for handling discrete charges in drift diffusion atomistic simulations by properly introducing the related quantum mechanical effects using the density gradient formalism for both electrons and holes. This eliminates the trapping of mobile charge in heavily doped regions of the device and the related artefacts in the simulated device characteristics Published by Elsevier Ltd Keywords: MOSFET; Density gradient; Atomistic; Charge assignment 1. Introduction The resolution of individual charges within an atomistic simulation [1 3] using a fine mesh creates problems [4]. Due to the use of Boltzmann or Fermi Dirac statistics in classical drift-diffusion simulations the electron or hole concentration follows the electrostatic potential gained from the solution of the Poisson equation. As a result a significant amount of mobile charge can become trapped (localised) by the sharply resolved Coulomb potential wells created by discrete dopant charges when assigned to a fine mesh. Corresponding author. Tel.: ; fax: address: a.asenov@elec.gla.ac.uk (A. Asenov) /$ - see front matter 2004 Published by Elsevier Ltd doi: /j.spmi

2 328 G. Royetal. / Superlattices and Microstructures 34 (2003) Fig. 1. 1D Poisson Schrödinger solution to the single particle problem showing energy levels and electron concentration. Such trapping is physically impossible since quantum confinement keeps the electron ground state high in the well. Fig. 1, whichdepicts a 1D Poisson Schrödinger solution, shows the Coulomb potential well and the bound energy states, illustrating this point. It is clear that the sharply peaked classical electron concentration is smeared due to quantum confinement effects. Another effect related to the local relationship between the potential and the electron concentration in classical simulations is the strong sensitivity of the quantity of trapped charge to the mesh size. If a finer mesh is used then the singularity in the Coulomb potential well is more sharply resolved which results in an increased amount of localised charge. The problem of charge trapping in classical atomistic simulations of a MOSFET is illustrated in Fig. 2.When thesource/drainregionsare atomistically doped,asillustrated in Fig. 2(a), electrons are trapped in the sharply resolved potential wells which can reduce the effective concentration of mobile carriers and thus artificially increase the source/drain access resistances in the simulation. When only the substrate is atomistically doped, as shown in Fig. 2(b), there are no electrons trapped in the heavily doped source/drain areas and the current flows in the valleys around the sharp potential peaks in the channel. Howeverthe holes in the undepleted part of the atomistically doped substrate can become trapped in the inverted potential wells generated by the acceptors. This leads to a modification of the depletion region of the device and directly affects the threshold voltage. Attempts to correct these problems in atomistic simulations have been made by charge smearing [5] orbysplitting the Coulomb potential into short- and long-range components based on screening considerations [6]. In the case of charge smearing a reduction in the amount of trapping, and in the mesh size sensitivity, can be achieved. This approach is however purely empirical and can result in a loss of resolution in respect to atomistic scale effects. The splitting of the Coulomb potential into short and long range components also suffers from drawbacks including the possible double counting of mobile charge screening. This method also relies upon a somewhat arbitrary cut-off point in order to separate the long- and short-range components.

3 G. Royetal. / Superlattices and Microstructures 34 (2003) Fig. 2. 2D potential distribution at the SiSiO 2 interface of a MOSFET: (a) continuous channel and atomistic source/drain doping; (b) atomistic channel and continuous source/drain doping. 2. Density gradient approach In this paper we present a quantum mechanically consistent approach to resolving individual discrete dopants in an atomistic drift diffusion simulation using the density gradient (DG) approximation [7, 8]. Typically in the DG simulation of n-channel MOSFETs it is sufficient to solve, self-consistently, Poisson s equation and the electron current continuity equation using the modified electron equation of state (1). 2b n 2 n n = φ n ψ + k BT q ( ) n ln n i where b n = 2 /12qm n. This should resolve automatically the problems associated with the trapping of electrons in the atomistically doped source/drain regions. In order to avoid the trapping of holes in the undepleted atomistically doped substrate we add the modified hole equation of state (2) tothe above system but without solving the hole current continuity equation. 2b p 2 p p = ψ φ p + k BT q (1) ( ) p ln. (2) p i

4 330 G. Royetal. / Superlattices and Microstructures 34 (2003) Fig. 3. 3D potential profile of a nm atomistic resistor used in this investigation. This modified system of equations is solved self-consistently using a Gummel iteration method, solving the Poisson equation with the modified electron and hole state equations and feeding this result into the current continuity equations. 3. The quantum resistor To illustrate the overall increase in resistance associated with charge trapping, we have simulated a resistor of dimensions nm. The resistor is modelled as aslab of silicon with a doping concentration cm 3.Thisisrepresentative of the doping concentrations in the source/drain region of a MOSFET device [9]. The potential distribution for one of the simulations is shown in Fig. 3. Fig. 4 illustrates the effect of the DG correction to the electron distribution. The quantum confinement in the sharp potential wells smoothes the overall electron concentration, reducing the sharper peaks as well as raising the concentration in the surrounding area. This means that more mobile charge is available to carry the current and so the overall resistance of the corresponding region in the simulations is reduced. Fig. 5 summarises the results for both continuous doping and for two methods of charge assignment, nearest grid point (NGP) and cloud in cell (CIC) [10], for a mesh spacing of 1 nm. In the atomistic case samples of 200 resistors with macroscopically different distributions were simulated and the results averaged. In the NGP approach all of the charge associated with a single dopant atom is assigned to a single mesh point, which results in a large peak in electron concentration at that node. The CIC scheme spreads the charge associated with a single dopant atom over the surrounding eight mesh points. In the case of the classical simulation the resistance of the device increases dramatically in the

5 G. Royetal. / Superlattices and Microstructures 34 (2003) Fig. 4. Electron concentration within an atomistic resistor using both classical and DG approaches. Fig. 5. I V characteristics of a nm resistor comparing uniform continuous doping with atomistic averages when using NGP and CIC charge assignment methods with and without DG. atomistic cases and this increase is greater when the NGP approach is used which does not smear the charge and produces sharper potential wells. The overall effect of applying DG is areduction in the amount of charge trapped around a dopant atom. This reduction in trapped charge leads to a decrease in the effective resistance of the slab. This brings the average atomistic current closer to the continuously doped case. Fig. 6 shows the I V characteristics of the resistor with various sizes of mesh spacing. The problem of charge trapping is closely associated with the mesh size. When the charge is assigned using NGP or CIC techniques a smaller mesh size results in larger charge density at that point. This results in more sharply resolved potential wells meaning more localised charge which does not contribute to the conduction. The DG corrections

6 332 G. Royetal. / Superlattices and Microstructures 34 (2003) Fig. 6. I V characteristics of a nm resistor comparing uniform continuous doping with atomistic averages for classical and DG simulations when varying the mesh spacing. Fig. 7. I V characteristics of a 50 nm MOSFET device comparing uniform continuous doping with atomistic averages for classical and DG simulations. Both linear and semi-log plots are shown. are insensitive to the mesh spacing which removes the mesh dependence present in classical DD simulations. Fig. 8 shows that the resistance depends on the mesh spacing in the classical resistor simulation case and that the inclusion of DG quantum corrections completely removes this sensitivity. 4. MOSFET simulation The improvements due to the addition of DG can also be seen in MOSFET device simulations. Fig. 7 shows the I D V G characteristics of a 50 nm MOSFET illustrating

7 G. Royetal. / Superlattices and Microstructures 34 (2003) Fig. 8. Carrier concentration in a 30 nm MOSFET for (a) a classical simulation and (b) the full DG approach for both electrons and holes. the inclusion of the quantum corrections in the source and drain regions. It shows the atomistic average of 200 devices compared to the associated I D V G curve corresponding to uniform continuous doping in both the classical and DG regime. It can be seen on the linear plot that the atomistic average current above threshold is lower compared to the uniform continuous case, which is due to the increased series resistance associated with the charge trapped in the source/drain regions. This effect is substantially reduced in the DG case. The DG treatment in the source/drain regions results in larger threshold voltage lowering in the atomistic simulations. The inclusion of DG corrections for holes also has a significant effect on the operation of amosfet device. Fig. 8 illustrates the carrier concentration distribution in a nm MOSFET with and without DG corrections for both electrons and holes. The inclusion of DG correction smoothes the electron and hole distribution. Fig. 9 illustrates the impact of the DG corrections on the hole concentration along a line parallel to the interface in the undepleted part of the substrate. Again it can be seen that the addition of DG causes a smoothing effect upon the concentration of holes lowering the sharp peaks, and causing more free carriers to flow into the surrounding system. The increased concentration of the mobile holes in the substrate due to the implemented DG corrections for holes also modifies the shape of the depletion region under the gate. 5. Conclusions In this paper we have shown that the resolution of individual atomistic dopants onto a fine mesh can lead to substantial trapping of charge in heavily doped regions. This in turn

8 334 G. Royetal. / Superlattices and Microstructures 34 (2003) Fig. 9. Graph of the hole concentration within the bulk ofamosfet with and without density gradient for holes included. artificially reduces the mobile charge concentration and increases the effective resistance of thecorresponding regions which introduces errors into the simulation results. In MOSFET simulation, for example, this artificially increases the access source/drain resistance and modifies the depletion layer width. The use of the density gradient formalism offers a physical solution to this problem whichreducestheunphysicalchargetrappingdueto theinclusionofquantumconfinement into the simulation. We have presented simulation results illustrating this beneficial effect in both simple resistor and complete MOSFET simulations. Acknowledgements This work was facilitated by IBM through a Shared University Research grant. This work is also supported by EPSRC through grant GR/R47325/01 and a platform grant GR/R89738/1. References [1] A. Asenov, IEEE Trans. Electron Devices 45 (1998) [2] A. Asenov, A.R. Brown, J.H. Davies, S. Saini, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 18 (1999) [3] A. Asenov, G. Slavcheva, A.R. Brown, J.H. Davies, S. Saini, IEEE Trans. Electron Devices 48 (2001) [4] T. Ezaki, T. Ikezawa, A. Notsu, K. Tanaka, M. Hane, Proc. SISPAD (2002) 91. [5] A. Asenov et al., Proc. SISPAD (2002) 87. [6] N. Sano, K. Matsuzawa, M. Mukai, N. Nakayama, Electron Devices Meeting, 2000, IEDM Tech. Dig. (2000) 275. [7] M.G. Ancona, G.I. Iafrate, Phys. Rev. B 39 (1989) [8] A. Asenov, J.R. Watling, A.R. Brown, D.K. Ferry, J. Comput. Electron. 1 (2002) [9] S. Inaba et al., IEDM Tech. Dig. 641 (2001). [10] R.W. Hockney, J.W. Eastwood, Computer Simulation Using Particles, IoP Publishing, Bristol, 1999.