IEE5328 Nanodevice Transport Theory and Computational Tools (Advanced Device Physics with emphasis on hands-on calculations) Lecture 3A: A Self-Consistent Solver of Poisson-Schrodinger Equations in a MOS System Prof. Ming-Jer Chen Dept. Electronics Engineering National Chiao-Tung University March 18, 2013 IEE5328 Prof. MJ Chen NCTU

Double-gate MOSFET Simulator: MOS Electrostatics Student: Ting-Hsien Yeh 葉婷銜 Advisor: Dr. Ming-Jer Chen

Structure Schematic double-gate n-MOSFET and its MOS band diagram. In this work, we set up a simulator called DG-NEP to deal with a symmetrical double-gate n-MOSFET structure. tox Body(p-type) N+ Oxide D S Gate tbody Vg

Flowchart for DG-NEP simulator Without Penetration Effect Start Setting the environment and physics parameters. Calculate Ef at equilibrium,and set Ev=0. Use Poisson’s equation to solve potential(V0). Use V0 to solve Schrodinger equation to obtain wave function and subband occupancy. Use updated concentration to get new potential by using Poisson’s equation. If |Vn+1-Vn|<1.0 × 10-12 eV Calculate charge density,voltage…. Yes No

Schrödinger and Poisson Self-consistent of DG-NEP The three-dimensional carriers (both electrons and holes) density: Poisson Equation:

Physical Model in DG-NEP The two-dimensional electron density The oxide voltage The total inversion layer charge density The flat band voltage The average inversion layer thickness The gate voltage The transverse effective field: Nano Electronics Physics Lab @ NCTU

Subband Energy and Wave-function For Tsi=30nm: We can find that our DG-NEP simulation results without penetration effect match Schred's ones.

Subband Energy and Wave-function For Tsi=10nm:

Subband Energy and Wave-function For Tsi=1.5nm:

The Comparison of Potentials and Electron Density Distributions with Those of Shoji, et al. (a) Tsi=30nm: (b) Tsi=5nm In this paper , ml=0.98m0 , mt=0.19m0 , mox=0.5m0 , Nsub=1x1015cm-3 [10] M. Shoji and S. Horiguchi, “Electronic structures and phonon limited electron mobility of double-gate silicon-on-insulator Si inversion layers,” J. Appl. Phys., vol. 85, no. 5, pp. 2722–2731, Mar. 1999.

The Comparison of Subband Energies with Those of Shoji, et al. (a) Eeff=1 × 105 V/cm (b) Eeff=5 × 105 V/cm For thick tSi, two of each subbands have almost the same energy due to the upper and lower inversion layers sufficiently separated as a distinct bulk inversion layer. As tSi decreases, the barrier between two inversion regions becomes lower and making the subband energies split.

Comparison with Gamiz, et al. (a) Non-primed subbands (b) Primed subbands [11] F. Gamiz and M. V. Fischetti, “Monte Carlo simulation of double-gate silicon-on-insulator inversion layers: The role of volume inversion, ” J. Appl. Phys., vol. 89, no. 10, pp. 5478–5487, May 2001.

Comparison with Gamiz, et al. Energy separation for two different body thicknesses

The Comparison of C-V with Alam, et al. and Schred. (a) Different substrate thickness (b) Different oxide thickness [14] M. K. Alam, A. Alam, S. Ahmed, M. G. Rabbani and Q. D. M. Khosru, “Wavefunction penetration effect on C-V characteristic of double gate MOSFET, ” ISDRS 2007, December 12-14, 2007, College Park, MD, USA.

The Comparison of C-V with Alam, et al. and Schred. (a) Different substrate thickness (b) Different oxide thickness [14] M. K. Alam, A. Alam, S. Ahmed, M. G. Rabbani and Q. D. M. Khosru, “Wavefunction penetration effect on C-V characteristic of double gate MOSFET, ” ISDRS 2007, December 12-14, 2007, College Park, MD, USA.