 Characteristics of MOSFET using MATHEMATICA
Lecture : Information Devices
15 March 2005 Nagoya University Prof. K.Nakazato
Initialize : set constants
Clear@"Global`*"D;
Off@General::spellD; Off@General::spell1D;<< "Graphics`Graphics`"
H* constants *L
q = 1.60218 ´ 10 ^ -19
H* elementary charge @CD *L;
k = 1.38066 ´ 10 ^ -23
H* Boltzman constant @JKD *L;
T = 300
H* Absolute temperature @KD *L;
ni = 1.45 ´ 10 ^ 10
H* intrinsic carrier concentration @cm-3D *L;
e0 = 8.85418 ´ 10 ^ -14 H* permitivity in vaccum @FcmD *L;
es = 11.9 e0
H* silicon permitivity *L;
eox = 3.9 e0
H* oxide permitivity *L;
Nc = 2.8 ´ 10 ^ 19
H* effective density of state in conduction band @cm-3D *L;
Nv = 1.04 ´ 10 ^ 19
H* effective density of state in valance band @cm-3D *L;
H* input parameters *L
p0 = 2 ´ 10 ^ 17
H* substrate acceptor concentration @cm^-3D*L;
tox = 10 ^ -6
H* gate oxide thickness @cmD*L;
H* substrate property *L
n0 = ni ^ 2  p0;
H* donor concentration @cm^-3D*L;
LD = Sqrt@Hk T esL  Hp0 q ^ 2LD;
H* Debye length *L
Ec = k T  q Log@Nc  n0D;
H* conduction band edge *L
Ev = -k T  q Log@Nv  p0D;
H* valance band edge *L
Ei = HEc + EvL  2 + k T  H2 qL Log@Nv  NcD; H* intrinsic energy *L
H* gate oxide prperty *LCox = eox  tox;

H* gate capacitance per unit area *LFermi potential

Plot[{-k T/q ArcSinh[10^N/(2ni)],k T/q ArcSinh[10^N/(2ni)],Ei-Ec,Ei-Ev},{N,15,21},PlotRange->{-0.6,00.40.2

161718192021-0.2-0.4

Semiconductor Charge Qs as a function of surface potential p
F[x_, y_] := Sign[x] Sqrt[Exp[-x]+x-1 + y(Exp[x] - x - 1)];
Ex[p_,n0_]:= Sqrt[2] k T/q/LD F[q p/(k T), n0/p0];
Qs[p_,n0_]:=-es Ex[p,n0];
Print["P0=",N[p0]," [cm-3]\tEc=",Ec," [eV]\tEi=",Ei," [eV]\tEv=",Ev," [eV]"];
LogPlot[Abs[Qs[p,n0]], {p, -0.5, 1.5}];

2MOSusingMATHEMATICA.nb

P0=2. ´ 1017 @cm-3D

Ec=0.977754 @eVD

Ei=0.425001 @eVD

Ev=-0.102148 @eVD

0.01

0.0001

1. ´ 10-6

1. ´ 10-8-0.5

00.5

11.5

Energy profile : ps is surface potential
ps = 1 H* @VD *L;
xmax = 2 ´ 10 ^ -5; H* @cmD *L
Print@"LD=", LDD;
eval = NDSolve@8p '@xD + Ex@p@xD, n0D  0, p@0D  ps<, p, 8x, 0, xmax<D;Plot@8Evaluate@Ec - p@xD . evalD, Evaluate@Ev - p@xD . evalD<,8x, 0, xmax<, PlotRange ® 8-1.5, 1.5<, PlotLabel ® "energy"D;
Plot@Evaluate@p0 Exp@-q p@xD  k  TD . evalD, 8x, 0, xmax<,
PlotLabel -> "hole concentration"D;
W@p_D := -NIntegrate@1  Ex@x, n0D, 8x, p, k T  q Log@2D<D;
Print@"Depletion width=", W@psDD;
Plot@Evaluate@n0 Exp@q p@xD  k  TD . evalD,8x, 0, xmax<, PlotLabel -> "electron concentration"D;MOSusingMATHEMATICA.nb3

LD=9.21988 ´ 10-7energy1.5
10.5-60.00001

51́00.000015

0.00002-0.5-1-1.5
hole concentration
172´1017

1.5´1017

1´1016

5´10-60.00001 0.000015 0.00002

5´10Depletion width=7.69378 ´ 10-6
electron concentration
15

7´1015

6´10155´10154´1015

3´10152´1015

1´10-65´100.00001

0.000015

0.00002

Surface potential as a function of gate voltage
Vg[p_,n0_]:=-Qs[p,n0]/Cox + p;
pstart=-0.46; pend=1.1; pstep=0.01;
ListPlot[Table[{Vg[p,n0],p},{p,pstart,pend,pstep}],PlotJoined®True,PlotLabel®"Surface potental"];(*depletion width *)Pl1=ListPlot[Table[{Vg[p,n0],Sqrt[2es p/(q p0)]},{p,0,pend,pstep}],PlotJoined®True,DisplayFunction®I
Off[NIntegrate::singd]; Off[NIntegrate::inum]; Off[Graphics::gptn];
Pl2=ListPlot[Table[{Vg[p,n0],W[p]},{p,0,pend,pstep}],PlotJoined®True,DisplayFunction®Identity];
Pl3=ListPlot[Table[{Vg[p,n0],2Sqrt[es Ei/q/p0]},{p,0,pend,pstep}],PlotJoined®True,DisplayFunction®Id
Show[{Pl1,Pl2,Pl3},DisplayFunction®$DisplayFunction,PlotLabel®"Depletion width"];4MOSusingMATHEMATICA.nb

Surface potental
10.75
0.5
0.25-15-10-5

51015-0.25

Depletion width-68´10-66´10-6

4´10-6

2´10

0.5

1

1.5

2

2.5

3

3.5

MOS capacitanceFc[x_,y_]:=If[Abs[x]<10^-8,Sqrt[2(1+y)],(-Exp[-x]+1+y(Exp[x]-1))/F[x,y]];Cs[p_,n0_]:=es/LD Fc[(q p)/(k T), n0/p0]/Sqrt[2];pstart=-0.25; pend=1.2; pstep=0.01;
Plcl=ListPlot[Table[{Vg[p,n0],1/(Cox/Cs[p,n0]+1)},{p,pstart,pend,pstep}],PlotJoined®True,PlotRange®{

pstart=-0.25; pend=1.2; pstep=0.01;
Plch=ListPlot[Table[{Vg[p,n0],1/(Cox/Cs[p,0]+1)},{p,pstart,pend,pstep}],PlotJoined®True,PlotRange®{{

pstart=-0.15; pend=4;
Plcd=ListPlot[Table[{Vg[p,0],1/(Cox/Cs[p,0]+1)},{p,pstart,pend,pstep}],PlotJoined®True,PlotRange®{{Show[{Plcl,Plch,Plcd},PlotLabel->""];low frequency
10.8
0.6
0.40.2-3-2-1

12345MOSusingMATHEMATICA.nb

high frequency
10.80.60.40.2-3-2-1

1234545

deep depletion
10.8
0.6
0.4
0.2-3-2-1

12312

3

10.8
0.6
0.4
0.2-3-2Thredshold voltage

-1

4556

MOSusingMATHEMATICA.nb

PhiF@p_D := Module@8n, LD, Ec, Ev<,n = ni ^ 2  p;
LD = Sqrt@Hk T esL  Hp q ^ 2LD;
Ec = k T  q Log@Nc  nD;
Ev = -k T  q Log@Nv  pD;
HEc + EvL  2 + k T  2 Log@Nv  NcD
DVT@p_D := PhiF@pD + Sqrt@2 q p es 2 PhiF@pDD  Cox;
Plot@VT@10 ^ pD, 8p, 14, 18<, PlotRange ® 80, 3<D;
32.5
2
1.5
1
0.5

15

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