FOWLER-NORDHEIM TUNNELING

Remove["Global`*"]

Highlights

Abstract

Introduction

Off[General::spell,General::spell1];

Needs["Graphics`Arrow`"];

tera  = 10^12  ; micro = 10^-6;
giga  = 10^9   ; nano  = 10^-9; 
mega  = 10^6   ; pico  = 10^-12;
kilo  = 10^3   ; femto = 10^-15; 
milli = 10.^-3 ; atto  = 10^-18;

NSub = {
c   -> 2.99792458 10^8  , e     -> 1.6021773 10^-19,
h   -> 6.6260755 10.^-34, hbar  -> 1.05457 10.^-34 ,
K   -> 1.380658 10^-23  , me    -> 9.109389 10^-31 ,
mu0 -> 4. Pi 10^-7      , eps0  -> 8.85419 10^-12   };

ESi   =0.179 tera; rhoSi   =2330 ; nuSi   =0.28;
ESiO2 =0.06  tera; rhoSiO2 =2200 ; nuSiO2 =0.3;
ESi3N4=0.15  tera; rhoSi3N4=3100 ; nuSi3N4=0.24;
EAu   =0.08  tera; rhoAu   =19300; nuAu   =0.42;
ENi   =0.207 tera; rhoNi   =8900 ; nuNi   =0.41;
EW    =0.345 tera; rhoW    =19300; nuW    =0.28;
EPt   =0.17  tera; rhoPt   =21400; nuPt   =0.39;
EIr   =0.52  tera; rhoIr   =22500; nuIr   =0.3 ;
EHOPG =0.2   tera; rhoHOPG =2250 ; nuHOPG =0.3 ;
EPol  =0.001 tera; rhoPol  =1000 ; nuPol  =0.3 ;

xx     =0.8;
EPtIr  =xx 0.8 EPt+(1-xx) EIr;
rhoPtIr=xx rhoPt  +(1-xx) rhoIr;
nuPtIr =0.3;

opt= {
PlotRange  ->All,
AxesOrigin ->{0,0},
Frame      ->True,
GridLines  ->Automatic,           
DefaultFont->{"Times-Roman",8},
PlotStyle  ->{
Dashing[{.0001}],
Dashing[{.0200}],
Dashing[{.0200}],
Dashing[{.0200}],
Dashing[{.0200}],
Dashing[{.0200}]}};
opt1= {
PlotRange  ->All,
AxesOrigin ->{0,0},
Frame      ->True,
GridLines  ->Automatic,           
DefaultFont->{"Times-Roman",8}};

Operating Conditions

caseA = { 
phiAu -> 4.2  e, phiW  -> 3.5  e,
phiAl -> 3.17 e, phiMg -> 2.3  e,
phiSi -> 4.2  e, phiCr -> 3.51 e,
phiPt -> 3.01 e, Ecb   -> 3.2  e}/.NSub;

case = Join[NSub,caseA];

Fowler-Nordheim Current Density

Clear[kappa];
kappa[phi_,meff_] := 2 Pi/h Sqrt[2 meff phi];

Clear[C0];
C0[phi_,meff_] := 2/e Integrate[kappa[x,meff],{x,0,phi}]

Print["C0 = ",C0[phi,meff]]

Clear[B0];
B0[phi_] := e^3 /(8 Pi h phi);

Clear[J];
J[v_,phi_,meff_,d_] =
B0[phi] (v/d)^2*
Exp[-C0[phi,meff] d/v]/.case;

Numerical Example

Clear[Jx,phi];
Jx[phi_] := J[8,phi,0.8 me,3 nano]/.case

phix =
FindRoot[Jx[phi]==10^5,
{phi,4 e/.NSub}][[1,2]];
Print["experimental phi = ",phix/e/.NSub," [eV]"]

caseB = {
phi ->phix,
meff->0.8 me,
d   ->3 nano};

case1 = Join[caseB,case];

B1 = B0[phi]/.case1;
C1 = C0[phi,meff]/.case1;

Clear[J];
J[v_] := B1 (v/d)^2 Exp[-C1 d/v]/.case1;

Fig11x2 =
Plot[J[v]/10^4,
{v,5,8},
FrameLabel ->{"v [V]","J [pA/(100 nm^2)]"},
Evaluate[opt]];

Oxide Field and Applied Field

delphi = (phi - phiSi)/e;
Print["delphi = ",delphi/.caseB/.case," [V]"]

Oscillation Factor

Clear[k1];
k1[v_] = (2 meff e v/d/hbar^2)^(1/3);

Print["k1[8] = ",N[k1[8]] nano//.case1," [1/nm]"]

Clear[Lcb];
Lcb[v_,En_] := d-(phi+En) d/(e v);

Print["Lcb[8,0] = ",N[Lcb[8,0]]/nano/.case1," [nm]"]

Clear[k];
k[v_] = Sqrt[2 meff (phi+(e v-Ecb))]/hbar;

Print["k[8] = ",N[k[8] nano]//.case1," [1/nm]"]

b0 = .08;
Clear[b];
b[v_,En_] :=
b0 N[((AiryAi[-k1[v] Lcb[v,En]])^2+
(k1[v]/k[v])^2 (AiryAiPrime[-k1[v]*
Lcb[v,En]])^2)^(-1)]//.case1;

Fig11x3 =
Plot[Evaluate[{
b[v,0],
b[v,.2 e]}],
{v,5,8},
FrameLabel ->{"v [V]","b[v,En]"},
Evaluate[opt]];

Clear[Josc];
Josc[v_,En_] := b[v,En] J[v];

Fig11x4 =
Plot[Evaluate[{
J[v]/10^4,
Josc[v,0]/10^4}],
{v,5,8},
FrameLabel ->{"v [V]","Josc [pA/(100 nm^2)]"},
Evaluate[opt]];

Clear[G];
G[v_] := -v/d Log[Josc[v,0]/(B1 (v/d)^2)] nano/.case1

F[v_]  := v/d nano/.case1

Fig11x5 =
ParametricPlot[{
F[v],
G[v]},
{v,5,8},
FrameLabel ->{"F [V/nm]","G [V/nm]"},
Evaluate[opt]];

Print["C1 = ",N[nano C1/.case1]," [nm/V]"]

Averaged Oscillations

Clear[C2];
C2[phi_,v_] := C1 d/(phi v)/.case1

Clear[bave];
bave[v_]:=
C2[phi,v] NIntegrate[b[v,En]*
Exp[-C2[phi,v] En],
{En,0,(e v-phiSi)/.case1}];

tbl1 = Table[{v,bave[v]},{v,5,10,.5}];
Fig11x6 =
ListPlot[tbl1,
PlotJoined ->True,
FrameLabel ->{"v [V]","bave[v]"},
Evaluate[opt1]];