4-1 Conclusion
4-2 Future Works
4-1 Conclusion
In this thesis, two major topics are studied. One is the quantum mechanical
effects in MOS capacitors; another is the C-V and I-V characteristics of ultrathin MOS
structures.
In chapter 1, the experimental, measuring, and simulation methods of this work
are elaborated. The motivation and objectives of this research are also proposed. Our
goal is to propose an general theory for explaining and simulating the C-V and I-V
characteristics of both MOS(n) and MOS(p) capacitors with ultrathin oxide layers,
especially the unusual I-V behavior of MOS(p) reverse-biased current.
In chapter 2, we apply an exponential surface potential approximation in solving
the quantization energy levels in accumulation layer and strong inversion layer of MIS
or MOS structure. The approximation used in this paper is more precise than previous
researches which used a linear surface potential approximation. The wave function
under the exponential surface potential approximation is Bessel function of the first
kind, which is also different from the results of previous researches, the wave function
of which is in the form of Airy function. Finally, this work use the uncertainty principle
to calculate the density of states and carrier distribution, the results has a much better
consistency in the 2D to 3D transition region than previous researches. However, how
to decide the precise magnitude of wave packets in both real-space and k-space is still
an open question for future research.
In chapter 3, we extracted the C-V and I-V data of the ultrathin MOS structures
with different oxide thicknesses from experiments. We combine the surface
quantization effect of MOS and build a modified Tsu-Esaki tunneling model for
calculating the hole and electron tunneling current. Also, in order to analyze the deep
depletion phenomenon in MOS reverse-biased region, we establish the relation between
the generation current, tunneling current, and quasi-Fermi level in the depletion region.
We consider that the deep depletion phenomenon indicates the oxide is so leaky that
the minority carrier cannot further accumulate in the “inversion” region. Thus, we
qualitatively picture the band diagrams of different oxide thicknesses, which can
successfully explain the unusual behavior that “the thicker the oxide layer, the larger
the reverse-biased saturation current in ultrathin MOS(p) structures”. In addition, we
qualitatively describe the difference between the band diagrams of central area and edge
area of Aluminum gate. This difference is due to the fringing field effect and can
successfully explain the phenomenon that “the reverse-biased saturation current of
MOS(p) is proportional to perimeter of the gate rather than proportional to gate area”.
We further deduct that the hole tunneling current is the major component of the
reverse-biased current of MOS(p), while the electron tunneling current is the major components
of MOS(n) leakage current and the forward-biased current of MOS(p). The simulation
in this chapter shows that our theoretical model and qualitative illustration can explain
the experimental C-V and I-V characteristics of ultrathin MOS structures very well. In
the end of chapter 3, we apply our model to explain the results of two experiments:
MOS in illuminated environment and MOS after a long period negative voltage stress.
The qualitative explanations of our model can illustrate the C-V and I-V characteristics
of the two experiments very well.
4-2 Future works
There are three suggested works for the future studies.
The first one is to improve the calculating efficiency of the simulations in
chapter 3. Currently, a complete and convergent result cost more than 4 days to obtain
(CPU: Intel® Core™ i5-2450M [email protected]~3.1Ghz, all the threads are occupied).
If the MATLAB codes can be rewritten in C-based parallel computing codes, the
efficiency may be improved a lot.
The second one is the two frequency C-V correction mentioned in chapter 1.
This method is hindered by two facts: 1. it cannot correct heavily leaking cases; 2. as
mentioned in chapter one, the capacitance cannot be correctly corrected for large gate
area cases. There are some researches try to fix these problems, but their models, which
contains the usage of inductance and phase shifting elements, are lacking of physical
meanings.
The third suggested future work is related to the second one. For the oxide
thinner than 1.7 nm, accumulation capacitance is severely decreased and even becomes
negative (see Fig. 4-1). Although the net leakage of majority carriers in accumulation
region would cause a negative differential capacitance in measurement, the huge
negative capacitance phenomenon in Fig. 4-1 cannot be explained by the net leakage of
charges in accumulation region. Even all the accumulated charges are leaking out, it
still can’t generate such huge negative capacitance. Similar measuring results is
founded in previous research of Schottky diode [45]. When the oxide becomes
extremely thin, the MOS structure is gradually becoming a MS structure. Therefore, we
consider the huge differential capacitance as a nature of Schottky diode rather than
MOS structures. We find that the negative capacitance follows the empirical formula:
Eq. (4-1) is inducted from numerous measurements. A possible explanation is that the
carriers near the junction are in plasmatic movements. From the theory of the
carrier-concentration-reduction (CCR) waveguides [46], the effective dielectric constant is
2
differential constant would become negative. The physical explanation of Eq. (4-2) is
that the carriers’ movement is 180 degrees counter phase to applied signal. If somehow
the large amount of carriers near the Schottky junction also have the similar behavior,
we are able to explain Eq. (4-1).
-3 -2 -1 0 1 2
Fig. 4-1. The C-V characteristics for extremely leaky MOS structures. (a) MOS(p) (b)
MOS(n).
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Appendix I
The following is the MATLAB simulation code of case 2 in chapter 2. By
changing the doping conditions, the code can be used in cases 1 & 3 as well.
clear all;
mat = load(file);
Sn = [2.338 4.087 5.52 6.787 7.944 9.021 10.039 11.007 11.935 12.828 13.691 14.527 15.34 16.132 16.905 17.66 18.4008 19.1261 19.8379 20.5371
if besselj(0.01*x,2*k0*A)*besselj(0.01*(x-1),2*k0*A) < 0 && i<Lnum+1 nu(i) = 0.01*(x +
(0- besselj(0.01*x,2*k0*A))/(besselj(0.01*x,2*k0*A)-besselj(0.01*(x-1),2*k0*A)));
y = 0.001 : 0.001 : 2*k0*A;
CiSQ(i) =
( trapz( ( (2*A./y).*real(besselj(nu(i),y)).^2 ))*0.001 )^(-1);
Ei(i) = V0-(((hbar*nu(i))/(2*A))^2)/(2*e*meQ);
(x-Ei(i)/ElecF).*(2*meQ*e*ElecF/(hbar^2))^(1/3)) )).^2 )*1e-10 )^(-1);
Xn(i) = Ei(i)/ElecF; %unit m
for u= 1:count-1+Lnum2 ((y-6)*1e-10-Ei(u)/ElecF)*(2*meQ*e*ElecF/(hbar^2))^(1/3)) ))^2 + n(y);
end ((10+y-106)*1e-9-Ei(u)/ElecF)*(2*meQ*e*ElecF/(hbar^2))^(1/3)) ))^2 + n(y);
end 0.025875*log(1e15/1e10)=0.2982 ; 0.56-0.292 = 0.268
if y < 107
mat(6) = (1/(ox+10*Si))*(ox*mat(5)+10*Si*mat(7));
mat(5) = 0.5*(mat(4)+mat(6)); %20A is the effective thickness
n20(a) = Nsum(u)*CiSQ(u)*(real(besselj(nu(u),2*k0*A*exp(-((10+y-106)*1e-9)/(2*A)))))^2 + n20(a);
end
for u = count:count-1+Lnum2
n20(a) = Nsum(u)*CiSQ(u)*(real(airy( ((10+y-106)*1e-9-Ei(u)/ElecF)*(2*meQ*e*ElecF/(hbar^2))^(1/3)) ))^2 + n20(a);
end
xf = (psi-0.56+0.292)/ktq; % = kT/q 0.0259*log(3e15/1e10) = 0.3266 0.025875*log(e15/1.5e10)=0.3158
n30(a) = Nc*fermi(xf,0.5); % - hole; no need to multiply*(2/pi^0.5)
end
Appendix II
The following is the simulation code to generate the results in Fig. 3-4 (b) & (d).
By replacing “conc=1e15 n.type” into “conc=3e15 p.type”, this code can also generate the results shown in Fig. 3-4 (a) & (c).
go atlas set dox = 0.0017
loop steps = 11
mesh space.mult=1.0 set dox = $dox+0.0001
#
x.mesh loc=-80 spac=0.1 x.mesh loc=-75 spac=0.1 x.mesh loc=-70 spac=0.1 x.mesh loc=-0.00 spac=5 x.mesh loc=70 spac=0.1 x.mesh loc=75 spac=0.1 x.mesh loc=80 spac=0.1
#
y.mesh loc=-0.252 spac=0.05 y.mesh loc=-$dox spac=0.0001 y.mesh loc=0 spac=0.0001 y.mesh loc=0.1 spac=0.05 y.mesh loc=0.42 spac=0.01 y.mesh loc=5 spac=1
region number=1 x.min=-80 x.max=80 y.min=-0.252 y.max=5 material=si3n4 region number=2 x.min=-80 x.max=80 y.min=0 y.max=5 material=Silicon region number=3 x.min=-80 x.max=80 y.min=-$dox y.max=0 material=sio2
region number=4 x.min=-75 x.max=75 y.min=-0.252 y.max=-$dox material=Aluminum
# #1=gate #2=substrate
electrode name=gate number=1 x.min=-75 x.max=75 y.min=-0.252 y.max=-$dox electrode name=substrate number=2 substrate
#
doping uniform conc=1e15 n.type x.left=-80 x.right=80 y.top=0 y.bottom=5 models conmob srh auger bgn fldmob print
#
models fermi print temperature=300
#
contact name=gate workfunction=4.19
method newton itlimit=25 trap atrap=0.5 maxtrap=4 autonr nrcriterion=0.1 \ tol.time=0.005 dt.min=1e-25
#save outfile=structure.str output con.band val.band
############ CV solve init solve vgate=-2.05
log outfile=nCV@"dox".log
solve vgate=-2.05 vstep=0.05 vfinal=2 name=gate qscv save outfile=nCV@"dox".str
extract name = "CV curve" curve (v."gate", i."gate"*150) outfile="nCV_doxum.dat"
log off
############ IV
models gate1 qtunn.el qtunn.ho solve init
solve vgate=2
save outfile=nVg2@"dox".str log outfile=nIV@"dox".log
solve vgate=2 vstep=-0.1 vfinal=-2 name=gate save outfile=nVg-2@"dox".str
extract name="IV curve" curve (v."gate", i."gate"*150) outfile="nIV_doxum.dat"
log off
l.end quit
Appendix III
The following are the 1D and 2D Poisson-Schrodinger solver MATLAB codes used in
chapter 3. By changing the doping condition, it can be used on MOS(n) as well.
%%%%%%%%%%% 1-D solver %%%%%%%%%%%
meQ = 0.916*m0; %2-D electron effective mass in z-direction mhQ = 0.325*me;
Ce = 1.5;%current correction term(Gaussian distribution of thickness) Ch = 1.5;
ox = 3.9*(20/(j+0.5)); %effective dielectric constant for effective oxide thickness of 20A
Xn = [0 0 0];
file(6) = file(6)+fix(a/10)-10*fix(a/100);
file(7) = file(7)+rem(a,10);
file(7) = file(7)+fix(a/10)-10*fix(a/100);
file(8) = file(8)+rem(a,10);
CiSQ(i) =
( trapz( ( (2*A./y).*real(besselj(nu(i),y)).^2 ))*0.001 )^(-1);
En(i) = V0-(((hbar*nu(i))/(2*A))^2)/(2*e*meQ);
xf(Ec<En(3)) = (mat(6)-En(3)-0.88+dEFnAvg)/ktq;
n(7:106) = Nsum(1)*CiSQ(1).*(real(besselj(nu(1),2*k0*A*exp(-(((7:106)-6)*1e-10)/(2*A))))).^2 +
mat(7:106) = 0.5*(mat(6:105)+mat(8:107)-K1);
mat(106) = ((10/11)*mat(105)+(1/11)*mat(107)) - (10/11)*K1(100);
psi = mat(107:Xr-1);
xf = (psi+dEFnAvg-0.88)/ktq; % = kT/q 0.025875*log(1e15/1.5e10) = 0.2874 0.025875*log(1e15/1e10)=0.2982
n(107:Xr-1) = Nc*fermi(xf,0.5)+Na; % already multiply by (2/pi^0.5) in Fermi function
K2 = (h^2)*e*(n(107:Xr-1))/epSi;
mat(107:Xr-1) = 0.5*(mat(106:Xr-2)+mat(108:Xr)-K2);
Eox = e*(sum(n(6:106)*1e-8) + sum(n(107:Xr-1)*1e-7))/(ox*8.85e-14);
mat(6) = 0.1*(8*mat(6) + 2*(mat(5)-1e-7*Eox));
w = (10+y-106)*h;
TPe = exp(constE*dox1*(2/(3*Vox3)).*((Es+Be-E1).^1.5-(Es+Be+Vox3-E1).^1.5) );
Knz = (1/hbar)*(2*meQ*e*En(1))^(0.5);
norm = ((1/hbar).*(2*meQ*e.*(E1-Es)).^(0.5)-Knz)./deltaKnz(1);
Rn = exp(-(norm.^2));
TPe = exp(constE*dox1*(2/(3*Vox3)).*((Es+Be-E2).^1.5-(Es+Be+Vox3-E2).^1.5) );
Knz = (1/hbar)*(2*meQ*e*En(2))^(0.5);
norm = ((1/hbar).*(2*meQ*e.*(E2-Es)).^(0.5)-Knz)./deltaKnz(2);
Rn = exp(-(norm.^2));
TPe = exp(constE*dox1*(2/(3*Vox3)).*((Es+Be-E3).^1.5-(Es+Be+Vox3-E3).^1.5) );
NE = e*ktq*log( (1+exp(-(E3-EFn)/ktq)) ./ (1+exp(-(E3-EFm)/ktq)) );
sec3 = trapz(e*TPe.*NE)*1e-4; %Rn=1
Jte = Ce*0.0001*4*e*pi*me*(sec1+sec2+sec3)/((2*pi*hbar)^3); %C 0.0001
m2->cm2
TPh = exp(constH*dox1*(2/(3*Vox2)).*((Bh+E4-Ev).^1.5-(Bh+Vox2+E4-Ev).^1.5) );
NE = e*ktq*log( (1+exp((E4-EFp)/ktq)) ./ (1+exp((E4-EFm)/ktq)) );
sec4 = trapz(e*TPh.*NE)*1e-4; %Rn=1
Jth = -Ch*0.0001*4*e*pi*mh*(sec4)/((2*pi*hbar)^3); 0.0001 m2->cm2 Jall = Jth+Jte;
para = [Vox2 w Jrg Jte Jth Jdiff dEFn Jall Nsum(1) Nsum(2); Nsum(3) En(1)
En(2) En(3) nu(1) nu(2) nu(3) CiSQ(1) CiSQ(2) CiSQ(3)];
%%%%%%%%%%% 2-D solver %%%%%%%%%%%
file(6) = file(6)+fix(a/10)-10*fix(a/100);
file(7) = file(7)+rem(a,10);
file3 = '2DP00A000.txt';
file3(4) = file3(4)+fix(j/10);
file3(5) = file3(5)+rem(j,10);
file3(7) = file3(7)+fix(a/100);
file3(8) = file3(8)+fix(a/10)-10*fix(a/100);
file3(9) = file3(9)+rem(a,10);
file(7) = file(7)+fix(a/10)-10*fix(a/100);
file(8) = file(8)+rem(a,10);
file3(9) = file3(9)+fix(a/10)-10*fix(a/100);
file3(10) = file3(10)+rem(a,10);
n(7:Xr-1,y) = Nc*fermi(xf,0.5) + Na*(mat(7:Xr-1,y)>0.0001)';
end
Length = fix(100-10*(0.01*(1:100)).^2);
for y = 2:split
n(7:6+Length(y),y) = (1-0.001*y)*n(fix(7:100/Length(y):106),1);
end
for pp = 1:5
n(6:56,split+pp) = n(6:56,split)*(1.1-0.2*pp);
end
mat(6,2:Yr-1) = (1/(10*ox+102*Si))*(10*ox*mat(5,2:Yr-1) + 100*Si*mat(7,2:Yr-1) + Si*mat(6, 1:Yr-2) + Si*mat(6, 3:Yr));
mat(5,2:Yr-1) = 0.25*(mat(4,2:Yr-1) + mat(6,2:Yr-1) + mat(5, 1:Yr-2) + mat(5, 3:Yr));
mat(4,split+1:Yr-1) = (1/(1+3*ox))*(mat(3,split+1:Yr-1) +
ox*(mat(5,split+1:Yr-1) + mat(4, split:Yr-2) + mat(4, split+2:Yr)));
% the air part
mat(3,split+1:Yr-1) = 0.25*(mat(2,split+1:Yr-1) + mat(4,split+1:Yr-1) + mat(3, split:Yr-2) + mat(3, split+2:Yr));
mat(2,split+1:Yr-1) = mat(3,split+1:Yr-1);
n(1:Xr,2:Yr) = n(1:Xr,2:Yr) - (mat(1:Xr,2:Yr)<1e-5).*n(1:Xr,2:Yr);
mat(1:Xr,2:Yr) = mat(1:Xr,2:Yr) - (mat(1:Xr,2:Yr)<1e-5).*mat(1:Xr,2:Yr);
TPh = exp(constH*dox1*(2/(3*Vox2)).*((Bh+E4-Ev).^1.5-(Bh+Vox2+E4-Ev).^1.5) );
NE = e*ktq*log( (1+exp((E4-EFp)/ktq)) ./ (1+exp((E4-EFm)/ktq)) );
sec4 = trapz(e*TPh.*NE)*1e-4; %Rn=1
Jth(pos) = -Ch*0.0001*4*e*pi*mh*(sec4)/((2*pi*hbar)^3); %0.0001 m2->cm2 end
cfile(8) = cfile(8)+fix(rd/10)-10*fix(rd/100);