Nanowire MOSFET variability: a 3D density gradient versus NEGF approach

As device dimensions shrink to the order of nanometres, quantum effects such as confinement and tunnelling start to play a significant role. Quantum confinement shifts the threshold voltage, and the leakage increases due to band-toband, source-to-drain and gate tunnelling. Such effects will have a strong impact on the performance of nanowire transistors actively researched at present as a possible replacement for bulk, UTB SOI and MG MOSFETs beyond the 22 nm technology generation. At the same time, it is expected that nanowire MOSFETs will suffer from strong variability problems since a single stray dopant or atomic scale interface roughness could have a dramatic effect on the device characteristics at small nanowire cross sections [1]. 3D quantum transport simulators based, among others, on the NEGF formalism, are the best vehicles for studying quantum effects in nanowire transistors [2,3]. However they are computationally expensive and when simulation of variability on a statistical scale has to be carried out, more economical quantum correction techniques have to be used [4]. Thus, such techniques have to be thoroughly compared and calibrated with respect to full scale quantum transport simulators to ensure physically meaningful and reliable results. At the same time, one of the most successful techniques to simulate electronic devices has been the Drift-Diffusion (DD) approach which essentially is the first momentum from the Boltzmann equation or a particle conservation equation. This is a classical local representation of the electron transport. The transition from the drift-diffusion approach to a quantum mechanical approach is a big step because of the complex and expensive computational techniques involve in the latter. Besides, the inclusion of scattering such as phonon scattering (dissipative) and impurity scattering (nondissipative) in a full 3D quantum transport context is a huge computational challenge. An alternative way is to incorporate quantum corrections in the very flexible, and less computationally expensive, DD simulator. One straightforward, and widely accepted, quantum correction technique is the density gradient (DG) approach. This approach mimics the quantum repulsion of the electrons from the surface of a confining potential by introducing a effective quantum potential, which is dependent on the gradient of the electron density. Here we carefully compare the density gradient approach with a mode-space non-equilibrium Green’s function (NEGF) simulator [3] before using it to simulate the random dopant induced variability in nanowire MOSFETs illustrated in Fig. 1. In the NEGF approach, Neumann boundary conditions (NBC) are used in the source and drain, as the potential adjusts the electron injection to preserve charge neutrality. We have also implemented such NBC in the density gradient simulator, rather than the usual Dirichlet boundary conditions (DBC) for the ohmic source/drain contacts used in DD simulations. The NBC work better in conjunction with the density gradient in the sense that they do not restrict the potential and electron concentration in the source and drain from following the distribution imposed by quantum mechanics. This is demonstrated in Fig. 2 which shows the potential at the source/drain contacts being pulled to the fixed potential by the DBC, introducing an artificial electric field at the contacts. With NBC the potential in the contacts is flat in both NEGF and DG cases reproducing a more realistic potential gradient there. The simulated nanowire transistor has a 10nm channel length, 4nm 4nm channel cross-section with a wrap-around gate, oxide thickness tox=1nm, source/drain doping of 10 20 cm and an undoped channel. In order to ensure comparable handling of the electrostatics by the two simulators, the classical drift-diffusion module of each simulator is first used to compare the corresponding ID-VG characteristics. The threshold voltages are matched, and the corresponding ID-VG curves are shown in Fig. 3. The use of density gradient typically requires calibration against more rigorous quantum mechanical simulations, in this case provided by our NEGF simulator. The effective mass in the density gradient equation is treated as a fitting parameter, and is adjusted to match the electron distributions obtained from the NEGF simulator, as shown in Fig. 4. We have implemented an anisotropic effective mass in the density gradient simulation, with one value of effective mass in the confinement (y-z) directions and a different one in the transport (x) direction. For the distributions shown in Fig. 4 we have used my * = mz * = 0.27and m x * = 5 . The value of m x * is large, and reducing this produces closer agreement in electron distributions at low gate voltage, however this high value is required to prevent density gradient from excessively lowering the potential barrier, which results in serious degradation of the subthreshold slope that is not in agreement with the NEGF simulations. The effective mass in the oxide,