Modeling and Computation of Gate Tunneling Current through Ultra Thin Gate Oxides in Double Gate MOSFETs with Ultra Thin Body Silicon Channel Bhadrinarayana L V 17 th July 2008 Microelectronics Lab, Indian Institute of Science.
Introduction Why Double Gate? Excellent short-channel effect (SCE) immunity and close to an ideal subthreshold slope (60 mv/dec), resulting in a very high on-off ratio. Better SCE immunity allows the use of low-doped/undoped channels, which improves channel carrier transport properties (mobility), eliminates dopant fluctuation problems. Double Gate (DG) FETs have emerged as the most promising technology for sub-50nm transistor design. However, analysis and control of the gate tunneling leakage in DGFET is necessary to fully exploit their advantages.
Outline Study and Implementation of existing Gate Leakage Model for MOSFET and implementing them in DG MOSFETs. Study and Implementation of Yuan Taur Model for Charge Distribution and Voltage Profile for DG-FET Modeling of Gate leakage using Yuan Taur Model ( Classical ) Constructing a Quantum model for charge distribution and corresponding gate leakage. Comparison and analysis of results.
Gate Current in n-channel MOSFETs Fowler-Nordheim tunneling Tch - Wentzel Kramers Brillouin Tunnelling coefficient A function of Oxide electric field C- correction factor A function of Sheet charge Density A- Constant
Results for n-mos Analytical approach to integrate the different components of direct tunneling current through ultrathin gate oxides in n-channel metal oxide semiconductor field-effect transistors Kingsuk Maitra
Double Gate MOS The gate current can be modeled to double gate using a similar approach Finding the Charge Distribution Oxide Electric Field. Using Fowler-Nordheim tunneling to find Gate current using the same method used in a nmos.
Charge Distribution and Oxide Electric Field Charge distribution and Electric Field can be modeled broadly in two different ways Classical Using Yuan Taur Model for DG MOS which involves solving Poisson equation assuming states are continuous Quantum Solving simultaneous Schroedinger and Poisson Equations for a finite barrier quantum well
Classical model -Yuan Taur Model
Yuan Taur Model for DG MOS Charge distribution and Oxide Electric field is obtained by solving one dimensional Poisson Equation Integrating twice we get Beta is found out using the boundary conditions.
Yuan Taur Model for DG MOS Which on simplification yields Beta is a function of y, independent of x Drain current is found out using Pao Sah's integral Writing in terms of Beta we get
Yuan Taur Model for DG MOS Drain Current is found out using boundary conditions The boundary conditions to find out Betad and Betas are solved numerically using Newton's method. Charge is written in terms of Beta Using the above expressions, Drain current is found out.
Yuan Taur Model for DG MOS Drain current is modeled is checked with Yuan Taur's results A Continuous, Analytic Drain-Current Model for DG MOSFETs Yuan Taur Simulated results
Charge profile and Voltage profile Using Sheet charge density and boundary conditions, Charge profile is found out as a function of distance
Leakage Current Gate current increases significantly after 0.4-0.5 V Given by the following expression Vgs=
Dependence on Body thickness As expected the gate leakage current is found to reduce slightly on reduction of body thickness.
Charge and Oxide voltages
Vds Effect The Voltage profile across the length of the device is independent of the drain voltage when the MOS is in saturation mode and hence the Leakage current profile is also
Quantum Model Obtained by solving Schroedinger and Poisson equations. The equations are numerically solved using Euler's Difference method. The wavefunction is assumed to go to zero at the boundaries (Though actually wavefunction penetration into the oxide takes place).
Quantum Model Increase in charge density in after 10^25/m^3 is less pronounced. The voltage at which a carrier charge density of 10^25/m^3 is achieved is strongly dependent of body thickness.
Quantum Model- Gate leakage Gate leakage dependence on Gate Oxide thickness and Body thickness
Comparison of Gate leakage Currents Slightly higher gate current in Classical Model
Comparison Classical Vs Quantum Dependence on well width is well pronounced in the case of quantum model when compared to that of the classical one
Comparison Classical Vs Quantum
Comparison Classical Vs Quantum The difference is found to be significant at small body thickness due to quantization effects. For all values of body thickness the Classical result yields 5-8 times more leakage current than quantum model
Comparison with nmos(classical) For all voltages gate current is lesser in DG MOS than nmos At lower voltages, gate current in DG MOS is significantly lesser
Conclusion Gate current is found to be less in DG MOS than NMOS at low voltages Classical Model cannot be used for Channel thicknesses lesser than 5nm as quantization becomes predominant. In DGMOS with ultra-thin Gate thicknesses, Gate current is strongly dependent on Channel thickness. Aggressive scaling of Body thickness will result in several orders of reduction in gate current at lower gate voltages
Future Research The above methods with small modifications can be used to model gate leakage current VDFETs(Vertrical Drain FET), FinFETs and GAA(Gate all around) Transistors More accurate dependence on drain voltage can be obtained by solving 2D Schroedinger equation considering coupled effects of Gate Oxide and Source-Drain Fields and Short Channel effects. Leakage current can more accurately modeled by allowing wave function penetration and allowing exponential decay of wave function inside the gate oxide.