METAL-INSULATOR-METAL 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

case = {
phi  -> 4 e,
epsr -> 5.}/.NSub;

case1 = Join[case,NSub];

Nphi = phi/e/.case1;

Tunneling Current Density

General Solution

Clear[A,J0,Jg,NJg];
A[ds_]             := 4 Pi/h Sqrt[2 me] ds;
J0[ds_]            := e/(2 Pi h ds^2);
Jg[v_,ds_,phibar_] := J0[ds]*
(phibar      Exp[-A[ds] Sqrt[phibar]]-
(phibar+e v) Exp[-A[ds] Sqrt[(phibar+e v)]]);

NJg[v_,ds_,phibar_]:= N[Jg[v,ds,phibar]/.case1]

Small Voltage Approximation

Clear[Jgsmall,NJgsmall];
Jgsmall[v_,s_]  := Jg[v,s,phi- e v/2];
NJgsmall[v_,s_] := N[Jgsmall[v,s]/.case1];

Clear[Jsmall,NJsmall];
Jsmall[v_,s_]  := Sqrt[2 me phi] (e/h)^2 v/s*
Exp[-4 Pi s/h Sqrt[2 me phi]];
NJsmall[v_,s_] := N[Jsmall[v,s]/.case1]

stest = 1 nano;

Fig10x2 =
Plot[Evaluate[{
micro NJgsmall[v,stest],
micro NJsmall[v,stest]}],
{v,.01,1},
FrameLabel ->{"v [V]","J [MA/m^2]"},
Evaluate[opt]];

vtest = 1.;

Fig10x3 =
Plot[Evaluate[{
micro NJgsmall[vtest,s nano],
micro  NJsmall[vtest,s nano]}],
{s,.9,1.1},
FrameLabel ->{"s [nm]","J [MA/m^2]"},
Evaluate[opt]];

Large Voltage Approximation

Clear[Jg1arge,NJglarge];
Jglarge[v_,s_]  := Jg[v,s phi/(e v),phi/2];
NJglarge[v_,s_] := N[Jglarge[v,s]/.case1];

Clear[Jlarge,NJlarge];
Jlarge[v_,s_] := e^3 F^2/(8 Pi h phi)*
Exp[-8 Pi phi/(3 h e F)*
Sqrt[2 me phi]]/.F->v/s;
NJlarge[v_,s_] := N[Jlarge[v,s]/.case1]

stest = 1 nano;

Fig10x4 =
Plot[Evaluate[{
micro NJglarge[v,stest],
micro NJlarge[v,stest]}],
{v,4,6},
FrameLabel ->{"v [V]","J [MA/m^2]"},
Evaluate[opt]];

vtest = 5.;

Fig10x5 =
Plot[Evaluate[{
micro NJglarge[vtest,s nano],
micro NJlarge[vtest,s nano]}],
{s,.9,1.1},
FrameLabel ->{"s [nm]","J [MA/m^2]"},
Evaluate[opt]];

The Image Potential

g = 1.15 e Log[2]/(8 Pi eps0 epsr)/.case1;
Print["g = ",N[g]/nano," [V nm]"]

Clear[vi];
vi[x_] := -g s/(x(s-x));

Print["vi[s/2] = ",N[vi[s/2]]/.s->1 nano," [V]"]

Barrier with an Image Potential

Clear[phii];
phii[v_,s_,x_] := phi-e v x/s-e g1 s/(x(s-x));

Clear[Nphii];
Nphii[v_,s_,x_] := N[phii[v,s,x]/e/.case1/.g1->g]

Clear[Nphii1];
Nphii1[v_,s_,x_] := If[x<s,Nphii[v,s,x],0]

stest = 1 nano;

Fig10x6 =
Plot[Evaluate[{
Nphii1[0,stest,x nano],
Nphii1[2,stest,x nano],
Nphii1[4,stest,x nano],
Nphii1[6,stest,x nano]}],
{x,.01,.99},
PlotRange  ->{0,Nphi}, 
FrameLabel ->{"v [V]","phi [eV]"},
Evaluate[opt]];

vtest = 5.;

Fig10x7 =
Plot[Evaluate[{
Nphii1[vtest,.8  nano,x nano],
Nphii1[vtest,.9  nano,x nano],
Nphii1[vtest,1.  nano,x nano],
Nphii1[vtest,1.1 nano,x nano]}],
{x,.01,.99},
PlotRange  ->{0,Nphi},
FrameLabel ->{"x [nm]","phi [eV]"},
Evaluate[opt]];

The Barrier Width

Clear[xi,xi1,xi2,dxi];
sol = Solve[Nphii[v,s,x]==0,x];
xi1[v_,s_] = Re[x/.sol[[1]]]/.case1/.g1->g;
xi2[v_,s_] = Re[x/.sol[[2]]]/.case1/.g1->g;
xi3[v_,s_] = Re[x/.sol[[3]]]/.case1/.g1->g;
dxi[v_,s_] = xi3[v,s]-xi2[v,s];

stest = 1 nano;

Fig10x8 =
Plot[Evaluate[dxi[v,stest]/nano],
{v,.01,5},
AxesOrigin ->{0,stest/nano},
FrameLabel ->{"v [V]","dxi [nm]"},             
Evaluate[opt]];

vtest = 5.;

Fig10x9 =
Plot[Evaluate[dxi[vtest,x nano]/nano],
{x,.9,1.1}, 
FrameLabel ->{"s [nm]","dxi [nm]"},             
Evaluate[opt]];

Average Barrier Height

Clear[phibari];
phibari[v_,s_] := 1/dxi[v,s]*
Integrate[Nphii[v,s,x],{x,xi2[v,s],xi3[v,s]}];

stest = 1 nano;
tbl1 = Table[{v,Re[phibari[v,stest]]},
{v,.01,5.01,1}];

Fig10x10 =
ListPlot[tbl1,
AxesOrigin ->{0,0},
PlotRange  ->{0,Nphi},
PlotJoined ->True,
FrameLabel ->{"v [V]","phibari [eV]"},             
Evaluate[opt1]];

vtest = 5.;
tbl2 = Table[{s,Re[phibari[vtest,s nano]]},
{s,.9,1.1,.05}];

Fig10x11 =
ListPlot[tbl2,
AxesOrigin ->{0,0},
PlotJoined ->True,
FrameLabel ->{"s [nm]","phibari [eV]"},             
Evaluate[opt1]];

Comparison of the Barriers

stest = 1 nano;
vtest = 5.;

s0    = stest/nano;
xa    = xi2[vtest,stest]/nano;
xb    = xi3[vtest,stest]/nano;
v1    = vtest;
p1    = Re[phibari[vtest,stest]];
p     = Nphi;

Barrier without an Image Potential

barrierGraphicsA =
Show[
Graphics[{
Line[{{-1,0},{0,0},{0,p},	 
{s0,p-v1},{s0,-v1},
{s0+1,-v1}}],		       
Text[FontForm["metal",{"Times-Roman",8}],{-.5,-3}],
Text[FontForm["phi",{"Times-Roman",8}],{-.2,p+.2}],	  
Text[FontForm["-v",{"Times-Roman",8}],{xb-.1,-v1}],
Text[FontForm["phi-v",{"Times-Roman",8}],{s0+.3,p-v1}],
Text[FontForm["insulator",{"Times-Roman",8}],{s0/2,-3}],
Text[FontForm["metal",{"Times-Roman",8}],{s0+.5,-3}]}],
Frame->True,PlotRange->{{-1,2},{-7,5}},
DisplayFunction->Identity];

Barrier with an Image Potential

barrierGraphicsB =
Plot[Evaluate[Nphii[v1,stest,x nano]],
{x,.001,s0-.001},
Frame     ->True,
PlotRange->{-6,4},
AxesOrigin->{-1,-6},
DefaultFont->{"Times-Roman",8},		
PlotStyle ->Dashing[{.01}],
DisplayFunction->Identity];

Average Barrier Height

barrierGraphicsC =
Show[
Graphics[	  			  
{Line[{{0,0},{xa,0},{xa,p1},{xb,p1},
{xb,-v1},{xb+1,-v1}}],
Text[FontForm["phibar",
{"Times-Roman",8}],
{.35,p1-.6}]}],
DisplayFunction->Identity];

The Plot of the Three Barriers

Fig10x12 =
Show[{
barrierGraphicsA,
barrierGraphicsB,
barrierGraphicsC},
Frame->True,
DefaultFont->{"Times-Roman",8},
FrameLabel ->{"s [nm]","phi [V]"},
DisplayFunction->$DisplayFunction];

Comparison of Tunneling Currents

The General Solution with Image Potential

Clear[Jgi];
Jgi[v_,s_] := NJg[v,dxi[v,s],Re[e phibari[v,s]]];

stest = 3 nano;
tbl3 = Table[{v,N[Log[10,v/Jgi[v,stest]]]},
{v,.01,6.01,1}];

Small Voltage Approximation without Image Potential

tbl4 = Table[{v,Log[10,N[v/NJgsmall[v,stest]]]},
{v,.01,1.01,.05}];

Large Voltage Approximation without Image Potential

tbl5 = Table[{v,Log[10,v/NJlarge[v,stest]]},
{v,4,6,.05}];

Plot of the Three Tunneling Currents

resGraphicsA =
ListPlot[tbl3,
PlotJoined->True,
FrameLabel->{"v [V]","log(v/i) [Ohm m^2]"},
DisplayFunction->Identity,
Evaluate[opt1]];

resGraphicsB =
ListPlot[tbl4,
PlotJoined->True,
PlotStyle  ->Dashing[{.02}],
DisplayFunction->Identity,
Evaluate[opt1]];

resGraphicsC =
ListPlot[tbl5,
PlotJoined->True,
PlotStyle ->Dashing[{.02}],
DisplayFunction->Identity,
Evaluate[opt]];

Fig10x13 = 
Show[{
resGraphicsA,
resGraphicsB,
resGraphicsC},
DisplayFunction->$DisplayFunction];

Apparent Barrier Height

kappa = N[2 Pi/h Sqrt[2 me phi]/.case1];
Print["kappa = ",kappa 10^(-10)," [1/Angstrom]"]

a = 2/me (h/(4 Pi))^2 10^20/e/.case1;
Print["a = ",N[a]," [eV Angstrom^2]"]

Clear[dlnJgi];
dlnJgi[v_,s_] := (Jgi[v,s+ds]/Jgi[v,s]-1.)/ds;

ds = .01 nano;

A0    = N[4 Pi/h Sqrt[2 me]/.case1];
stest = 1 nano;

tbl6 = Table[{v,
(dlnJgi[v,stest]/A0)^2/e/.case1},
{v,.01,5.01,1}];

Fig10x14 =
ListPlot[tbl6,
PlotRange ->{0,Nphi},
PlotJoined->True,
FrameLabel->{"v [V]","phiapp [V]"},
DisplayFunction->Identity,
Evaluate[opt]];
Show[{Fig10x14,Fig10x10},
DisplayFunction->$DisplayFunction];

vtest = 1.;

tbl7 =
Table[{s,
(dlnJgi[vtest,s nano]/A0)^2/e/.case1},
{s,.9,1.1,.1}];

Fig10x15 =
ListPlot[tbl7,
PlotJoined->True,
FrameLabel->{"s [nm]","phiapp [V]"},
Evaluate[opt1]];