I hereby declare that this thesis was done by me and that it or any part of it has not been. filed elsewhere for the award of any degree or diploma. Golam Rabbani, Roll No P, Session: October 2005 has been accepted as satisfactory in partial fulfillment of the requirements for the degree of 'MASTER OF SCIENCE IN ELECTRICAL AND ELECTRONIC ENGINEERING' on 9 October. Dr. I would like to express my sincere thanks to my advisor Professor Quazi Deen Mohd Khosru for his generous help throughout my M.Sc. study in EEE, BUET.
This thesis benefited from discussions with some of my colleagues-in-EEE;- BTJET--and-I----- am in their debt. Kawsar Alam for his stimulating discussions and enthusiasm about 1hesis.and_Sishir.Bhowmic -for-his-help-in-wl'ite-up-and-discussion on the finite difference method. SymmetriceDouble gate_MOSEET structure_studied.in_this_ worle_ The source/drain doping is about 1x1020 cm.3, the channel is internal and the transistor is assumed to be wide (y dimension is treated as many Jarge. and .only._plane-J" is considered in the simulation) .. a) The interaction of a device with a tank can be represented by a self-energy matrix L. b) The self-energy of the device itself (LS) together with the self-energies of the source (L, ) and drain (L,).
Curves without symbols are for unprimed valleys, while those with symbols are for primed valleys.
Abstract
Intr6duction
Overview of the Th~sis
In Chapter 2, we describe the numerical --techniques- used-in-developing a-2D-simulator- fm'- nanoscale-douhIe,.gate.MOSEETs.-. First, we implement the aquanturiJ ballistic transport model using fixed uncoupled mode space (FUMS) approach in the non-equilibrium Green's..function --fNEGF) form¥ism. In Chapter 3, we_presem.simulation resulis'6(QnL~orLDjfferentproperties-Gf-the nanoscale DG MOSFET demonstrated(i'\hfd~~h relevant plots.
The sample code of the 2D mesh generator developed for the simulation and the flow chart of the simulation process are given in Appendix A.
2D Simulation of Nanoscale Double Gate MOSFET
Solution_of 2D.Schrodinger. Equation
- The_lUl£onpledmodupace-QJMS)-approach
- Solution oUD Schl"iidinger--Equation
- SQlution ofthe_S}'Stem-m-Equations
In this section we show how the problem can be transformed into a number of ID problems using the new mode space [39) (subband decomposition) approach. The device is wide in the y-direction and there is no limiting potential in this direction, -SO-the-wave -functions-in -this--
MOSFET structure we simulate), which is much smaller than that required in the real space representation, (Nz xN x) x(N zN x) (Nz is -40 for the device structures simulated in this work). The use of the FUMS approach greatly improves the efficiency of the simulation and makes it a practical model for comprehensive device simulation and design. Since the spectral functions only depend on the longitudinal energy, they can be moved out of the integration sign.
Once self-consistency is achieved, the terminal current can be expressed as a function of the transmission coefficient [10]. Note that TSD is indented by transverse energy Ekj and can therefore be moved out of the integration sign. We will not discuss the assembly procedure for 1D as it is similar and yet simpler than the 2D case discussed in the next section for the 2D FEM formulation of the Poisson equation.
Its algorithm involves performing a variable change in n, namely expressing n in terms of potential and quasi-Fermi energy, Fn• The quasi-Fermi potential energy is calculated based on the old potential as. Realistic boundary conditions to be determined after constructing the system of equations, as was done for the ID Schr{\dinger equation. So the grid points in both dimensions must be on straight lines (grid spacing may be uneven).
Each node in domain0 has a local number in the triangle and is a vertex of (and, of course, the global number). That is, it connects a numbecto-.its .associated.global.number._le-has In .rows,_each .for_a ~triangle.-In.a row,e,e,a column contains the global number of the node with the column number as. Now, before the system of equations is ready to be solved, we need to set the .the_actualboundary.of .the.device conditions.
The reason for using a zero-field boundary condition instead of the usual Dirichlet boundary condition at the source and drain contacts is given in [49].
Results andDiscussions
- Eigen-E-nergies-and-Eigenfunctions
- Eleetron-Density--Profile
- DrJlin-Cur~ent
- Validity ofFUMS approach
Due to the confinement, the carriers are quantized in the vertical direction and subbands are formed at different energies. J:"t-shows-the-sub-band-el"gies at different-veItic-segments-of-the-device in_the transport or x-direction (Vg=Vd=O.OV). Electron penetration into the oxide is necessary due to the nanoscale thickness levels involved and has been duly taken into account in the calculations, although it is trivial to run the simulation without considering the penetration.
As_expected at this unbiased _state-PIg=Yd=oO,OV-:J;-.the_eigen_energy levels-are-symmetric about the center of the transport direction and they are higher in the undoped silicon-body regiofr.- The-different--simulation -pair- ameters are-given-within _fig\ITe,-IL clear from the figure that the non-primed energy level (without symbol) is at a lower energy compared~ to the _corresponding_primed-levelL_The_reason_is-that--the llllprimetl valley has higher effective mass in the density of states. As we progress along the channel, the self-energy levels decrease until they are flat in the drain. The area under.any .function jsJ; .buUhe_unprimed wav.efunct~ons-have-high-rnaxima:i)i)ecause of the silicon-to-oxide energy barrier, the eigenfunctions are zero in the oxide regions except near tbe oxide-silicon interface.
As mentioned before, the inclusion of the penetration effect in the calculation is necessary because the_oxide_thickness_is_included, we can .so clearly see that the penetration effect I? for the fundamental valley it is larger due to the lower density of the mass state. Figure 3,6 sliows.the-samething.ex-eept-that in.this.case V:g=V:d;;;(DV.,-As can be seen from the data cursors of both figures, there is a slight drop in amplitude' ~f different wavefunctions as the gate and drain voltages are increased.are far unfilled valleys, while those with the symbol are for filled valleys.
Then Fig, 3,9 shows variation of 2D electron concentration with Vg at two different drain voltages; upper part (V d=O,OV) and lower part (O,5V), As Vg increases, the electron density in the channel increases, In the case of Vd=O.sV; due to lower channel to gate voltage, concentration is lower in the drain side compared to the source side. As Vd increases, concentration on the drain side of the channel decreases; _but_concentratioo..at .the..sOlITCe-remains-constant.,or higher Vg concentration in the channel is higher as can be seen from a comparison of the upper (Vg=O,OV) and lower (Vg=O, 5~).part of'Eigc 3dO,. Fig.3.n Variation-of-3D electron density with silicon thickness,t;;; (a),(b), and {c).seed the 3&- - density for 1,;; values 1.5, 3.0, and 5.0 nm, respectively; while (d) shows the ground state eigenfunction along a vertical cut in the center of the source region in the x-direction.
The conduction band is almost flat in the two oxide regions; (c) and (d) show z;-directed 1D !;:c,for conditionsQ,ns (a) and (b?; respectively;. As V,g increases with fixed Vd (O.5V), Ee is almost constant in the source and drain region, out it-drops-4own-in thechanneIregion (Hg, :h-13) In this section we show the validity of our simulation method by comparing the simulation results of this work, called femne. Finite element method nanoelectronic transistor simulator), with similar results of the famous nanomos [45] in nanohub.
The theory behind the two simulators is similar, but not precise; nanomos uses the UMS approach to solve the ID SchrOdinger equation in the vertical direction (z-direction), while femnet uses the new FUMS approach to reduce the time required to solve the ID SchrOdinger equation . Naturally, the two simulators differ greatly in their PDE discretization; nanomos uses the simple FDM approach and femnet uses the more suitable FEM approach, FEM allows easy control of complex geometry and we can focus on an area where any variable can change quickly by using a dense mesh there. Yet the simulators produce similar results! ts-as-sooWfl-in-Figs, -320-322.
Conclusion
Summary
Future Work
The NEGF approach is a very powerful mathematical tool for dealing with how the quantum state temporally evolves under various interactions within any small system (or device at the quantum level). AS-device .scaling_continues,~0Ye1.stnlcturesl4esigns must ~¥ultimately accept .the.role often played by semiconductor-based transistors. Banoo, Direct solution of the Boltzmann transport equation in a nanometer Si device, PhD thesis, Purdue.Uni\Zersity, W.est.Lafay.ette,.rn, -2llg0. Iang,.I.King,_LBokor.and-C._Hu, "Gate.Length_Scaling and.Ibreshielded Voltage Control of Double-Gate MOSFETs," IEDM Tech.
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