Conclusion
4.2 Future Work
Furtherextensions)n this study may include:
I) The NEGF approach can_be.used_to-'limula1e_gate _oxide \eakage-characteristics-of -ID -MGS structures. _Currently, the widely used method in modeling gate leakage is the WKB approxlmation-15r-53}. In-this-method- quantum-transmissiOlls-tllr-ough.the oxide layers-are- - evaluated approximately in a post-process operation. The electrostatic profile is obtained separately~ without-considering _the_effec.ts.on_the__charge_distrihutionoftheJunneling-GlHTent.
In addition, incident electrons are represented by plane-waves, so the effects of 2D carriers in the inversion-layer-on-the gate tunneling may-not.be properlyasscssed.- 'fhc-NEGF method,.
however, can exactly solve the transport problem, when coupled to the Poisson equation, providing a way.ofselfoconsistently _assessing1he..gate leakage aIld_electr-Ostatics_profile,.This method also allows us to examine the energy spectrum of the leakage current, which distinguishes the-eontributions-fr-em-the 2Q discrete-states-andID-{;ontimuHls-states.- _
2) Different .scattering _modeLare available.and..scatter.can_he.includedin.the-simulation -as.it will give the method a real life touch. One method to study the dissipative transport can be the use oflhe--Biittiker-probe-baseG-SGattering- m-ode1s-wherc-scattering centers-are- treated-as--- reservoirs that change the energy or momentum of the carriers and not the total number of -carriers.in _the_system...lt _has.he.en _used.in.the nanoMO.8_[A5J,.and.1'ecentl}',in-a-nanewire
simulation [54]. Each scattering center is modeled through a perturbation strength characterized by a position dependent self-energy, which can be mapped onto an equivalent m-obility.
3) Strained silicon MOSFETs are being studied in recent years. The methods presented in this work can be extended for the study of both uniaxially and biaxially strained DG MOSFETs.
52
In this case, the simple effective mass approximation might not be true because the electron and hole effective masses change with strain as does the band alignment.
The NEGF approach is a very powerful mathematical tool for addressing how a quantum- state evolutes temporally under a varieties of interactions within any tiny system (or quantum level device).AS-device .scaling_continues,~0Ye1.stnlcturesl4esigns must ~¥entually-take.over.the role clMfently being played by semiconductor-based transistors. Some recent works have brought carbon-tuDe and- moleeule-cluster-llased. device_structur~s-into-.foous-[-S5-S8]~These new-areas--.
provide us plenty of opportunities to apply the NEGF approach at theoretical research levels. In principle, _theJ'lEGF _formalism~ot_on1y .enables_usJo_understand-the-microscopic-phenomena, but also enable us to exploit the hidden potential. Moreover, it should be noted that this approach is applicable in all non-equilibrium systems beyond electron devices [9, 59].
53
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58
Appendix A
A.12D meshgenel'ation-using-MA'fLAB- .
%==========================================================================
function mesh~dgmGs2d
%====~====================================================================
%mesh_dgmos2d - 2d mesh generator function for rectangular DGMOSFET
%This function is used in FEM solution of 2D Poisson equation
%
%Related parameters:
%---
%Lsd: dJ;.ain_or...source-J.-engt:ch
%Lg_top : gate length
%t top: top oxide thickness
%t-bot~ bottom oxide thickness
%t si: silicon thickness
%dx: x-grid spacing
%dy: y-grid spacing
%ns: no. of smaller grids
%
%Nn: number of nodes
%Ne: number of (triangular) elements
%x : x-coordinate of nodes (nexl)
%y : y-coordinate of nodes (nexl)
%te: connectivity matrix (3xne)
%Nb: no. of boundary points
%td: array of global. number of Nb's
%---7---
%---
global Lsd Lg_top Lg_bot t_top t_bot t_si dx dy ns dys global Lsda Lg_topa Lg_bota t_topa t_bota t sia
global Nn Ne Nx Ny Nb x y te td~row globF~ eps_top eps~bot eps_si-alpx alpy
%---
%Set up Nn,Ne,x,y:
%---~---
dys~dy/2;
~sd~round(Lsd/dx)*dx;
Lg_top~round(Lg_top/dx)*dx;
Lg_bot~round(Lg_bot/dx)*dx;
Lx~round((2*Lsd+Lg top)/dx)*dx;
t_top~round(t_top/dys)kdys7 t_bot~round(t_bot/dys)*dys;
t_si~round(t_si/dys)*dys;
Ly~ro\>nd.((t_top+t_si+t_bot) Idyl *dy;
%
Lsda-round(Lsd/dx);
Lg topa=round(Lg top/dx);
Lg=:bota~round( Lg=:bor/dx)., Nx~round(Lx/dx)+l
t_topa=2*ns+round((t_top-2*ns*dys)/dy);
t_bota~2*ns+round((t-bot-2*ns*dys)/dy);
59
t_sia~2*ns+round(lt si-2*ns*dys)/dy);
Ny~6*ns+round((Ly-6*ns*dys)/dy)+1 Nn=Nx*Ny
Ne~2':(Nx-l)*(Ny-l);
x=zeros (-Nh, -1).;
y=zeros(Nn,l);
for i=l:Nx
xcolli)~(i-l)*dx;
end
x=repmat(xcol',Ny,l);
fOI:-.j_~l:Ny __
if j<=ns+l
yrow(j)~(j-l)*dys;
elseif j>~(t_topa+l-ns+l)&j<~Jt_topa+l+ns) yrow(j)~yrow(j-l)+dys;
e~seif j>~(t_topa+t_sia+l-ns+l)&j<~(t_topa+t sia+l+ns) yrow(j)~yrow(j-l)+dys;
elseif j>-(-Ny--ns+-l-}&j<-Ny-- yrow(j)~yrowlj-l)+dys;
else
yrowJj )~yrow (j-1) +.dy_;
end end
yrow=yrow I;
yd=repma.t_(yrow, 1, Nx);
y=reshape {yd' 1Nn-,1)--i-
%---
%Set up teo
%---~---
te=zeros(3,Ne)i vel~[1;2_;Nx+l] ; ve2~ [2;Nx+z'-;Nx+-lj;_
for jj~l: (Ny-I)
for ii~1:2: (2*(Nx-l)) ee~(jj-l)*(2*(Nx-l))+ii;
if ii==l
te (:Ieel =vel;
te C=- 1_ee+ 1) =ve2i _ else
tel:,ee)~te(:,ee-2)+1;
te(:,ee+l)~te(:,ee-l)+l;
end end
.1vel=vel+Nx;
ve2=~e2+Nx;
end
%---
%Set up Nb,td:
%---
Nb~2*(Lg_topa+l);
td~tnp~[Lsda+l:Lsda+Lg_topa+lJ ';
(Nn- (Lsda'l-Lg_topa-'Fl)+1);-
td _bot~ [-INn'-(Lsda+Lg_ topa+ 1) +1) : INn- (Lsda+ 1) +1) ] ';
td~[td_top;td_bot];
60